Big Ideas Math Answers Grade 8 Chapter 2 Transformations

Big Ideas Math Answers Grade 8 Chapter 2

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Big Ideas Math Book 8th Grade Answer Key Chapter 2 Transformations

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Performance

Lesson: 1 Translations

Lesson: 2 Reflections

Lesson: 3 Rotations

Lesson: 4 Congruent Figures

Lesson: 5 Dilations

Lesson: 6 Similar Figures

Lesson: 7 Perimeters and Areas of Similar Figures

Chapter 2: Transformations 

Transformations STEAM Video/Performance

STEAM Video

Shadow Puppets

Some puppets are controlled using strings or wires. How else can a puppet be controlled?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 1.1

Watch the STEAM Video “Shadow Puppets.” Then answer the following questions.

Question 1.
Tory and Robert are using a light source to display puppets on a screen. Tory wants to show the pig jumping from the floor to the window. Should she use a translation, reflection, rotation, or dilation? Explain.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 1.2

Answer:
In the situation given, if we translate first, we move the pre-image closer to the center of dilation than if we translate second. That will result in a different image.

Question 2.
How can Tory show the pig getting smaller as it jumps out the window?

Performance Task

Master Puppeteer

After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given the coordinates of a kite being used bya puppeteer.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 1.3
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 1.4
You will be asked to identify transformations for given movements of the kite. When might a puppeteer want to use a reflection?

Transformations Getting Ready for Chapter 2

Getting Ready for Chapter 2

Chapter Exploration
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 2

Question 1.
Work with a partner. Form each triangle on a geoboard.

  • Which of the triangles are congruent to the triangle at the right?
  • Measure the sides of each triangle with a ruler. Record your results in a table.
  • Write a conclusion about the side lengths of triangles that are congruent.

Big Ideas Math Answers Grade 8 Chapter 2 Transformations 3
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 4

Answer: option d

When two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position

Vocabulary

The following vocabulary terms are defined in this chapter. Think about what the terms might mean and record your thoughts.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 5

Lesson 2.1 Translations

EXPLORATION 1
Work with a partner.
a. For each figure below, draw the figurein a coordinate plane. Then copy the figureonto a piece of transparent paper and slide the copy to a new location in the coordinate plane. Describe the location of the copy compared to the location of the original.

  • point
  • triangle
  • line segment
  • rectangle
  • line

Big Ideas Math Answers Grade 8 Chapter 2 Transformations 6
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 6.1
b. When you slide figures, what do you notice about sides, angles, and parallel lines?
c. Describe the location of each point below compared to the point A(x, y).
B(x + 1, y + 2)
C(x – 3, y + 4)
D(x – 2, y + 3)
E(x + 4, y – 1)
d. You copy a point with coordinates (x, y) and slide it horizontally a units and vertically b units. What are the coordinates of the copy?

2.1 Lesson

Try It

Tell whether the blue figure is a translation of the red figure.

Question 1.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 7

Answer:
Here in the given figure, we can see that the shape of both red and blue figures is the same but the size is different. The red figure slide to form a blue figure but it is not the same size. So blue figure is not the translation of red figure.

Question 2.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 8

Answer:
Here in the given figure, we can see that the shape of both red and blue figures is the same and also the size is the same. The red figure slide to form exactly blue figure. So blue figure is the translation of red figure.

Try It

Question 3.
WHAT IF?
The red triangle is translated 4 units left and 2 units up. What are the coordinates of the image?

Answer:
Big Ideas Math Grade 8 Chapter 2 Answer Key img_6
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: A(-2,1) B(2,5), C(1,2) and a = -4, b = 2
A'(-2+a, -2+b) = A'(-2-4, 1+2) = A'(-6,3)
B'(2+a, 5+b) = B'(2-4, 5+2) = B'(-2,7)
A'(1+a, 2+b) = C'(1-4, 2+2) = C'(-3,4)
Hence the coordinate of image are A'(-6,3), B'(-2,7), C'(-3,4).

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

IDENTIFYING A TRANSLATION
Tell whether the blue figure is a translation of the red figure.

Question 4.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 9

Answer:
Here in the given figure, we can see that the shape of both red and blue figures is the same but the size is different. The red figure slide to form a blue figure but it is not the same size. So blue figure is not the translation of red figure.

Question 5.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 10

Answer:
Here in the given figure, we can see that the shape of both red and blue figures is the same, and also the size is the same. The red figure slide to form the exactly blue figure. So blue figure is the translation of red figure.

Question 6.
The vertices of a triangle are A(2, 2), B (0, 2), and C (3, 0). Translate the triangle 1 unit left and 2 units up. What are the coordinates of the image?

Answer:
Bigideas Math Answers Grade 8 Chapter 2 img_7
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: A(2,2) B(0,2), C(3,0) and a = -1, b = 2
A'(2+a, 2+b) = A'(2-1, 2+2) = A'(1,4)
B'(0+a, 2+b) = B'(0-1, 2+2) = B'(-1,4)
A'(3+a, 0+b) = C'(3-1, 0+2) = C'(2,2)
Hence the coordinate of image are A'(1,4), B'(-1,4), C'(2,2).

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
A neighborhood planner uses a coordinate plane to design a new neighborhood. The coordinates A(1, -1), B(1, -2), and C (2, -1) represent House A, House B, and House C. The planner decides to place a playground centered at the origin, and moves the houses to make space. House A is now located at A'(3, -4). What are the new coordinates of House B and House C when each house is moved using the same translation? Justify your answer.

Answer:
Big Ideas Math Answers 8th Grade Chapter 2 img_8
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: A(1,-1) B(1,-2), C(2,1) and A'(3,-4)
A'(1+a, -1+b) = A'(3,-4) So, a = 2, b = -3
New coordinates of the houses are
B'(1+a, -2+b) = B'(1+2, -2-3) = B'(3,-5)
C'(2+a, -1+b) = C'(2+2, -1-3) = C'(4,-4)
Hence the coordinate of image are B'(3,-5),C'(4,-4)

Question 8.
The locations of a quarterback and a wide receiver on a football field are represented in a coordinate plane. The quarterback throws the football to the point (6, -2). Use a translation to describe a path the wide receiver can take to catch the pass.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 11

Answer:
Coordinate of Receiver: (1,3) and football point:(-6,-2)
Horizontal shift: a = x2-x1 = 6 – 1 = 5
Verrical shift: b = y2 – y1 = -2 – 3 = -5
Hence the path which receiver will take 5 unit right and 5 unit down.

Translations Homework & Practice 2.1

Review & Refresh

Solve the equation for y.

Question 1.
6x + y = 12

Answer:
Given
6x + y = 12
Subtract 6x from both sides
y = 12 – 6x
Now arranging the terms
y = -6x + 12
y = 6 (-x + 2)
Thus y = 6(-x + 2)

Question 2.
9 = x + 3y

Answer:
Given,
9 = x + 3y
3y = 9 – x
Dividing by 3 on both sides
y = (9 – x)/3
y =  \(\frac{9}{3}\) – \(\frac{x}{3}\)
Now arranging the terms
y = – \(\frac{x}{3}\) + 3
Thus y = – \(\frac{x}{3}\) + 3

Question 3.
\(\frac{1}{3}\)x + 2y = 8

Answer:
Given,
\(\frac{1}{3}\)x + 2y = 8
Subtracting x/3 from both sides
2y = 8 – \(\frac{x}{3}\)
Now arranging the terms
2y = – \(\frac{x}{3}\) + 8
y = – \(\frac{x}{6}\) + 4

Question 4.
You put $550 in an account that earns 4.4% simple interest per year. How much interest do you earn in 6 months?
A. $1.21
B. $12.10
C. $121.00
D. $145.20

Answer: $12.10

Explanation:
Given:
You put $550 in an account that earns 4.4% simple interest per year.
Principal amount: P = $550
Rate of Interest: r = 4.4%
Time: t = 6 months = 0.5 year
We know that formula for Simple Interest is SI = prt/100
SI = (550 × 4.4 × 0.5)/100
SI = 1210/100
SI = 12.10
Hence the simple interest is $12.10
Thus the correct answer is option B.

Concepts, Skills, & Problem Solving

DESCRIBING RELATIONSHIPS
For each figure, describe the location of the blue figurerelative to the location of the red figure. (See Exploration 1, p. 43.)

Question 5.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 12

Answer: The path which the receiver will take is 6 units right and 3 units down.

Explanation:
Coordinate of Point A: (-3,2) and Point A’: (3,-5)
Horizontal shift: a = x2 – x1 = 3 – (-3) = 3 + 3 = 6
Vertical shift: b = y2 – y1 = -5 – (-2) = -5 + 2 = -3
Hence, The path which the receiver will take is 6 units right and 3 units down.

Question 6.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 13

Answer: The path which receive will take is 5 units left and 2 units down.

Explanation:
Coordinate of point A: (3,-2) and point A’: (-2,-4)
Horizontal shift: a = x2 – x1 = -2 – (3) = -2 – 3 = -5
Vertical shift: b = y2 – y1 = -4 – (-2) = -4 + 2 = -2
Hence, The path which receive will take is 5 unit left and 2 unit down.

IDENTIFYING A TRANSLATION
Tell whether the blue figure is a translation of the red figure.

Question 7.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 14

Answer:
Blue figure is the translation of red figure.

Explanation:
Here in the given figure, we can see that the shape of both blue and red figures are the same, and also the size of both the figure are the same. Also, the orientation of the blue figure is the same as the red figure. This means that the red figure sides to form the blue figure. So, the blue figure slides to form the red figure.

Question 8.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 15

Answer:
Blue figure is not the translation of the red figure.

Explanation:
Here in the given figure, we can see that the shape of both blue and red figures are the same, and also the size of both the figure are the same. But the orientation of the blue figure is different from the red figure. This means that the blue figure is not the translation of the red figure.

Question 9.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 16

Answer:
Blue figure is not the translation of the red figure.

Explanation:
Here in the given figure, we can see that the shape of both blue and red figures are the same, and also the size of both the figure are the same. But the orientation of the blue figure is different from the red figure. This means that the blue figure is not the translation of the red figure. The blue figure is the mirror image of the red figure.

Question 10.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 17

Answer:
Blue figure is the translation of red-figure.

Here in the given figure, we can see that the shape of both blue and red-figure are the same, and also the size of both the figure are the same. Also, the orientation of the blue figure is the same as the red figure. This means that the red figure slides to form the blue figure. so the figure slides to form the red figure.

Question 11.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 18

Answer:
Blue figure is the translation of red-figure.

Explanation:
Here in the given figure, we can see that the shape of both blue and red figures are the same, and also the size of both the figures are the same. Also, the orientation of the both figure is the same as the red figure. This means that the red figure slides to form the blue figure. So blue figure slides to form the red figure.

Question 12.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 19

Answer: Blue figure is not the translation of the red-figure.

Explanation:
Here in the given figure we can see that the shape of both blue and red figure are same and the size of both the figure are not same. The red figure are smaller as compared to the blue figure. This means that blue figure is not the translation of red figure.

TRANSLATE A FIGURE
The vertices of a triangle are L(0, 1), M(1, -2), and N(-2, 1). Draw the figure and its image after the translation.

Question 13.
1 unit left and 6 units up

Answer:
Big Ideas Math Answers Grade 8 Chapter 2 Transformations img_1

explanation:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to the x-coordinate and ‘b’ is added to the y-coordinates of the vertices.
A(x,y) – A'(x+a, y+b)
The value ‘a’ and ‘b’ will be positive if the shift is Right and Vertical up and the value of ‘a’ and ‘b’ will be negative if the shift is left and vertical down.
Given:L(0,1),M(1,-2),N(-2,1)anda=-1,b=6
L'(0+a,1+b)=l'(0-1,1+6)=L'(-1,7)
M'(1+a,-2+b)=M'(1-1,1+6)=M'(0,4)
N'(-2+a,1+b)=N'(-3-1,1+6)=N'(-4,7)
Hence,the coordinate of image are L'(-1,7),M'(0,4),N'(-4,7)

Question 14.
5 units right

Answer:
We know that to translate a figure ‘a’ units horizontal and ‘b’ units vertically in the coordinate plane, ‘a’ is added to X-coordinate and ‘b’ is added to Y-coordinate of the vertices.
a(x,y),=A'(x+a,y+b)
the value ‘a’ and ‘b’ will be positive if the shift is right and vertical up and the value of ‘a’ and ‘b’ will be negative if shift is left and vertical down.
Given:L(0,1),M(1,-2),N(-2,1)and a=5,b=0
L(0+a,1+b)=L'(0+5,1+0)=L'(5,1)
M'(1+a,-2=b)=M'(1+5,-2+0)=M'(6,-2)
N'(-2+a,1+b)=N'(-2+5,1+0)=N'(3,1)
Hence the coordinate of image are L'(5,1),M'(6,-2),N'(3,1).

Question 15.
(x + 2, y + 3)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given: L(0,1), M(1,-2), N(-2,1) and (x+2,y+3)
So the value of: a = 2 and b = 3
L'(o + a, 1 + b) = L'(0 + 2,1 + 3) = L'(2,4)
M'(1+a, -2 + b) = M'(1 + 2, -2 + 3) = M'(3, 1)
N'(-2 + a, 1 + b) = N'(-2 + 2, 1 + 3) = N'(0, 4)
Hence the coordinate of the image is L'(2,4), M'(3,1), N'(0,4)

Question 16.
(x – 3, y – 4)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given: L(0,1), M(1,-2), N(-2,1) and (x-3, y-4)
So the value of a = -3 and b = -4
L'(0 + a, 1 + b) = L'(0-3,1-4) = L'(-3,-3)
M'(1 + a, -2 + b) = M'(1 – 3, -2-4) = M'(-2,-6)
N'(-2 + a, 1 + b) = N'(-2 – 3, 1 – 4) = N'(-5, -3)
Hence the coordinate of the image are L'(-3,-3), M'(-2,-6), N'(-5, -3)

Question 17.
YOU BE THE TEACHER
Your friend translates point A 2 units down and 1 unit right. Is your friend correct? Explain your reasoning.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 20

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given points,
A(3, 1) and a = 1, b = -2
A'(3+a, 1+b) = A'(3+1, 1-2) = A'(4, -1)
So, the point A’ translated by my friend is wrong. He has reversed the x and y coordinate for translation.
Hence the correct translate point is A'(4,-1)

Question 18.
TRANSLATING A FIGURE
Translate the triangle 4 units right and 3 units down. What are the coordinates of the image?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 21

Answer:
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations img_3
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given points from graph: J(-1,3), K(-1,1), L(-4,1)
and a = 4, b = -3
J'(-1 + a, 3 + b) = J'(-1+4,3-3) = J'(3,0)
K'(-1 + a, 1 + b) = K'(-1 + 4, 1 – 3) = K'(3,-2)
L'(-4 + a, 1 + b) = L'(-4 + 4, 1 – 3) = L'(0,-2)
Hence the coordinate of image is J'(3,0), K'(3,-2), L'(0,-2)

Question 19.
TRANSLATING A FIGURE
Translate the figure 2 units left and 4 units down. What are the coordinates of the image?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 22

Answer:
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations img_4
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given: A(-1,4), B(2,3), C(3,0), D(-1,-1)
and a = -2, b = -4
A'(-1+a, 4+b) = A'(-1-2, 4-4) = A'(-3,0)
B'(2+a, 3+b) = B'(2-2, 3-4) = B'(0,-1)
C'(3+a, 0+b) = C'(3-2, 0-4) = C'(1,-4)
D'(-1+a, -1+b) = D'(-1-2, -1-4) = D'(-3,-5)
The coordinate of image are A'(-3,0), B'(0,-1), C'(1,-4), D'(-3,-5)

DESCRIBING A TRANSLATION
Describe the translation of the point to its image.

Question 20.
(3, 2) → (1,0)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Coordinate of the point A:(3,-2) and the image point A’:(1,0)
Horizontal shift: a = x2 – x1 = 1 – 3 = -2
Vertical shift: b = y2 – y1 = 0 – (-2) = 0 + 2 = 2
Hence the translation path will be 2 units left and 2 units up.

Question 21.
(-8, -4) → (-3, 5)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Coordinate of the point A:(-8,-4) and the image point A’:(-3,5)
Horizontal shift: a = x2 – x1 = -3 – (-8) = 5
Vertical shift: b = y2 – y1 = 5 – (-4) = 9
Hence the translation path will be 5 units left and 9 units up.

Question 22.
REASONING
You can click and drag an icon on a computer’s desktop. Is this an example of a translation? Explain.

Answer:
Yes, the dragging of an icon on a computer’s desktop is an example of translation.
Because when dragging an icon on desktop the icon directly slides and is stored in its new position. While dragging the icon there is no change in shape and size of the icon, thus fulfilling the criteria of translation.

Question 23.
MODELING REAL LIFE
The proposed location for a new oil platform is represented in a coordinate plane by a rectangle with vertices A(1, 3), B(1, 4), C(4, 4), and D(4, -3). An inspector recommends moving the oil platform 4 units right and 2 units down. Find the coordinates of the image. Then draw the original figureand the image in the coordinate plane.

Answer:
BIM 8th Grade Answers Chapter 2 Transformations img_5
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given: A(1,-3), B(1,4), C(4,4), D(4,-3) and a = 4, b = -2
A'(1+a, -3+b) = A'(1+4, -3-2) = A'(5,-5)
B'(1+a, 4+b) = B'(1+4, 4-2) = B'(5,2)
C'(4+a, 4+b) = C'(4+4, 4-2) = C'(8,2)
D'(4+a, -3+b) = D'(4+4, -3-2) = D'(8,-5)
Hence the coordinate of image are A'(5,-5), B'(5,2), C'(8,2), D'(8,-5)

Question 24.
PROBLEM SOLVING
A school of fish translates from point F to point D.
a. Describe the translation of the school of fish.
b. Can the fishing boat make the same translation? Explain.
c. Describe a translation the fishing boat could make to get to point D.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 22.1

Answer:
a. Coordinate of the point F: (-3,2) and the point D: (2,3)
Horizontal shift: a = x2 – x1 = 2 – (-3) = 2 + 5 = 7
Vertical shift: b = y2 – y1 = 3 – 2 = 1
Hence the path of translation is 5 unit Right and 1 unit Up.
b. No, the fishing boat (point B) cannot make the same translation as by fish (point F). Because in path between from point B to point D the is an island which will interrupt the translation of fishing boat.
c. Coordinate of the point B:(-2,-1) and the point D: (2,3)
Horizontal shift: a = x2 – x1 = 2 – (-2) = 2 + 2 = 4
Verical shift: b = y2 – y1 = 3 -(-1) = 3 + 1 = 4
Hence the path of translation is 4 unit Right and 4 unit Up.

Question 25.
REASONING
The vertices of a triangle are A(0, -3), B(2, -1), and C(3, -3). You translate the triangle 5 units right and 2 units down. Then you translate the image 3 units left and 8 units down. Is the original triangle identical to the final image? Explain your reasoning.

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
the value of a and b will be positive if the shift is Right and vertical Up and the value of a and b will be negative if the shift is left and vertical down.
Given: A(0,-3), B(2,-1), C(3,-3) and a1 = 5, b1 = -2
A'(0+a, -3+b) = A'(0+5, -3-2) = A'(5,-5)
B'(2+a, -1+b) = B'(2+5, -1-2) = B'(7,-3)
C'(3+a, -3+b) = C'(3+5,-3-2) = C'(8,-5)
Hence the coordinate of the first image are A'(5,-5), B'(7,-3), C'(8,-5)
Given: A'(5,-5), B'(7,-3), C'(8,-5) and a2 = -3, b2 = -8
A”(5+a, -5+b) = A”(5-3, -5-8) = A”(2,-13)
B”(7+a, -3+b) = B”(7-3, -3-8) = B”(4,-11)
C”(8+a, -5+b) = C”(8-3,-5-8) = C”(5,-13)
Hence the coordinate of the first image are A”(2,-13), B”(4,-11), C”(5,-13)
a = a1+a2 = 5 – 3 = 2, and b = b1 + b2 = -2 – 8 = -10
A'(0+a, -3+b) = A'(0+2, -3-10) = A'(2,-13)
B'(2+a, -1+b) = B'(2+2, -1-10) = B'(4,-11)
C'(3+a, -3+b) = C'(3+2,-3-10) = C'(5,-13)
Hence the original triangle is identical to the final image. This is because we can use both the translation by finding the resultant translation.
For final translation we can use:(x+2, y-10)

Question 26.
DIG DEEPER!
In chess, a knight can move only in an L-shaped pattern:

  • two vertical squares, then one horizontal square;
  • two horizontal squares, then one vertical square;
  • one vertical square, then two horizontal squares; or
  • one horizontal square, then two vertical squares.

Write a series of translations to move the knight from g8 to g5.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 23

Answer:
The series of translation to move knight from g8 to g5
1. Move 1 units Right to h8 and then 2 units Down to h6
2. Move 2 units Left to f6 and then 1 unit Up to f7
3. Move 2 units Down to f5 and then 1 unit Right to g5

Lesson 2.2 Reflections

Reflecting Figures

Work with a partner.
a. For each figure below, draw the figure in the coordinate plane. Then copy the axes and the figure onto a piece of transparent paper. Flip the transparent paper and align the origin and the axes with the coordinate plane. For each pair of figures, describe the line of symmetry.

  • point
  • triangle
  • line segment
  • rectangle
  • line

Big Ideas Math Answers Grade 8 Chapter 2 Transformations 24
b. When you reflect figures, what do you notice about sides, angles, and parallel lines?
c. Describe the relationship between each point below and the point A(4, 7) in terms of reflections.
d. A point with coordinates (x, y) is reflected in the x-axis. What are the coordinates of the image?
e. Repeat part(d) when the point is reflected in the y-axis

2.2 Lesson

Try It

Tell whether the blue figure is a reflection of the red figure.

Question 1.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 25

Answer: Blue figure is not the reflection of the red figure

Explanation:
By seeing the above figure we can say that the blue figure is not the mirror image of the red figure. Thus Blue figure is not the reflection of the red figure.

Question 2.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 26

Answer: Blue figure is the reflection of the red figure

Explanation:
By seeing the above figure we can say that the blue figure is the mirror image of the red figure. If the red figure is flipped it would form the shape of the blue figure. Thus Blue figure is the reflection of the red figure

Try It

Question 3.
The vertices of a rectangle are A(-4, -3), B(-4, -1), C(-1, -1), and D(-1, -3). Draw the figure and its reflection in (a) the x-axis and (b) the y-axis.

Answer:
Given,
The vertices of a rectangle are A(-4, -3), B(-4, -1), C(-1, -1), and D(-1, -3).
Reflection about the x-axis:
A(x,y) = A'(x,-y)
A(-4, -3) = A'(-4,3)
B(-4, -1) = B'(-4,1)
C(-1, -1) = C'(-1,1)
D(-1, -3) = D'(-1,3)
Reflection through x-axis:
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations img_9(i)
Reflection through y-axis:
A(x,y) = A'(-x,y)
A(-4, -3) = A'(4,-3)
B(-4, -1) = B'(4,-1)
C(-1, -1) = C'(1,-1)
D(-1, -3) = D'(1,-3)
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations img_9(ii)

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
REFLECTING A FIGURE
The vertices of a triangle are J(-3, -5), K(-2, 2), and L(1, -4). Draw the figure and its reflection in
(a) the x-axis and
(b) the y-axis.

Answer:
When a point is reflected about the x-axis then the y coordinate becomes the opposite.
A(x,y) = A'(x,-y)
The vertices of a triangle are J(-3, -5), K(-2, 2), and L(1, -4).
Reflection about the x-axis:
J(-3, -5) = J'(-3,5)
K(-2, 2) = K'(-2,-2)
L(1, -4) = L'(1,4)
Big Ideas Math Answers Grade 8 Chapter 2 Transformations img_10(i)
when a point is reflected about the y-axis then the x coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Reflection about the y-axis:
J(-3, -5) = J'(3,-5)
K(-2, 2) = K'(2,2)
L(1, -4) = L'(-1,-4)
BIM Grade 8 Answers Chapter 2 Transformations img_10(ii)

Question 5.
WHICH ONE DOESN’T BELONG?
Which transformation does not belong with the other three? Explain your reasoning.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 27

Answer: 3rd figure is different from other figures. Because all the other three pictures are reflections of each other except the third one. The third picture is pointed in the same direction but all the other three figures are in opposite direction.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
You design a logo using the figure shown at the left. You want both the x-axis and the y-axis to be lines of reflection. Describe how to use reflections to complete the design. Then draw the logo in the coordinate plane.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 28

Answer:
When a point is reflected about the x-axis then the y coordinate becomes the opposite.
A(x,y) = A'(x,-y)
A(-4,2), B(-2,2), C(0,0), D(-2,0)
Reflection about the x-axis:
A(-4,2) = A'(-4,-2)
B(-2,2) = B'(-2,-2)
C(0,0) = C'(0,0)
D(-2,0) = D'(-2,0)
when a point is reflected about the y-axis then the x coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Reflection about the y-axis:
A(-4,2) = A”(4,2)
B(-2,2) = B”(2,2)
C(0,0) = C”(0,0)
D(-2,0) = D”(2,0)
Now to complete the Logo again we have to take a reflection of the image figure about the y-axis. In this way, the logo will be symmetric about both axis.
A”(4,2) = A”‘(4,-2)
B”(2,2) = B”‘(2,-2)
C”(0,0) = C”‘(0,0)
D”(2,0) = D”‘(-2,0)
Bigideas Math Answer Key Grade 8 Chapter 2 img_11

Question 7.
DIG DEEPER!
You hit the golf ball along the path shown, so that its final location is a reflection in the y-axis of its starting location.
a. Does the golf ball land in the hole? Explain.
b. Your friend tries the shot from the same starting location. He bounces the ball of the wall at the point (-0.5, 7) so that its path is a reflection. Does the golf ball land in the hole?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 29

Answer:
a. Coordinates of the location of golf ball = (2,4)
Coordinates of location of hole = (-3,4)
Location of ball after reflection through y-axis = (2,4) = (-2,4)
But the location of the hole is (-3,4)
So the ball will not go into a hole and it will miss the hole by 1 unit.
Hence the ball will not go into the hole.
b. Yes, when the ball bounces at the point (-0.5,7) then it will land in the hole.

Reflections Homework & Practice 2.2

Review & Refresh

The vertices of a quadrilateral are P(-1, -1), Q(0, 4), R(3, 1), and S(1, -2). Draw the figure and its image after the translation.

Question 1.
7 units down

Answer:
Big Ideas Math 8th Grade Answer Key for Chapter 2 img_12
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinates of the vertices.
A(x,y) = A'(x+a,y+b)
the value ‘a’ and ‘b’ will be positive if shift is right and vertical up and the value of ‘a’ and ‘b’ will be negative if shift is left and vertical down.
Given,
P(-1,-1)
Q(0,4)
R(3,1)
S(1,-2) and a = 0, b = -7
P'(-1+a,-1+b) = P'(-1+0,-1-7) = P'(-1,-8)
Q'(0+a,4+b) = Q'(0+4,4-7) = Q'(4,-3)
R'(3+a, 1+b) = R'(3+0,1-7) = R'(3,-6)
S'(1+a,-2+b) = S'(1+0,-2-7) = S'(1,-9)
Thus the coordinate of the image is P'(-1,-8), Q'(4,-3), R'(3,-6), and S'(1,-9)

Question 2.
3 units left and 2 units up

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinates of the vertices.
A(x,y) = A'(x+a,y+b)
the value ‘a’ and ‘b’ will be positive if shift is right and vertical up and the value of ‘a’ and ‘b’ will be negative if shift is left and vertical down.
Given,
P(-1,-1)
Q(0,4)
R(3,1)
S(1,-2) and a = -3, b = 2
P'(-1+a,-1+b) = P'(-1-3,-1+2) = P'(-4,1)
Q'(0+a,4+b) = Q'(0-3,4+2) = Q'(-3,6)
R'(3+a, 1+b) = R'(3-2,1+2) = R'(0,3)
S'(1+a,-2+b) = S'(1-3,-2+2) = S'(-2,0)
Thus the coordinate of the image are P'(-4,1), Q'(-3,6), R'(0,3) and S'(-2,0)
Big Ideas Math Answers Grade 8 Ch 2 Transformations img_11

Question 3.
(x + 4, y – 1)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinates of the vertices.
A(x,y) = A'(x+a,y+b)
the value ‘a’ and ‘b’ will be positive if shift is right and vertical up and the value of ‘a’ and ‘b’ will be negative if shift is left and vertical down.
Given,
P(-1,-1)
Q(0,4)
R(3,1)
S(1,-2) and a = 4, b = -1
P'(-1+a,-1+b) = P'(-1+4,-1-1) = P'(3,-2)
Q'(0+a,4+b) = Q'(0+4,4-1) = Q'(4,3)
R'(3+a, 1+b) = R'(3+4,1-1) = R'(7,0)
S'(1+a,-2+b) = S'(1+4,-2-1) = S'(5,-3)
Thus the coordinate of the image are P'(3,-2), Q'(4,3), R'(7,0) and S'(5,-3)
Big ideas math answers grade 8 chapter 2 transformations img_12

Question 4.
(x – 5, y – 6)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinates of the vertices.
A(x,y) = A'(x+a,y+b)
the value ‘a’ and ‘b’ will be positive if shift is right and vertical up and the value of ‘a’ and ‘b’ will be negative if shift is left and vertical down.
Given,
P(-1,-1)
Q(0,4)
R(3,1)
S(1,-2) and a = -5, b = -6
P'(-1+a,-1+b) = P'(-1-5,-1-6) = P'(-6,-7)
Q'(0+a,4+b) = Q'(0-5,4-6) = Q'(-5,-2)
R'(3+a, 1+b) = R'(3-5,1-6) = R'(-2,-5)
S'(1+a,-2+b) = S'(1-5,-2-6) = S'(-4,-8)
Thus the coordinate of the image are P'(-6,-7), Q'(-5,-2), R'(-2,-5) and S'(-4,-8)
BIM 8th Grade Answer Key Chapter 2 Transformations img_13

Tell whether the angles are complementary, supplementary or neither.

Question 5.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 30

Answer:
108° + 82° = 190°
Thus the angle is neither supplementary nor complementary.

Question 6.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 31

Answer: Complementary

Explanation:
43° + 47° = 90°
Two angles are called complementary when their measures add to 90 degrees.

Question 7.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 32

Answer:
38° + 62° = 100°
Hence the given angle is neither supplementary nor complementary.

Question 8.
36 is 75% of what number?
A. 27
B. 48
C. 54
D. 63

Answer: B. 48

Explanation:
Let x be the unknown value.
75% of x = 36
75% × x = 36
75/100 × x = 36
3/4x × x = 36
3x = 36 × 4
3x = 144
x = 144/3
x = 48
Thus the correct answer is option B.

Concepts, Skills, &Problem Solving
DESCRIBING RELATIONSHIPS
Describe the relationship between the given point and the point A(5, 3) in terms of reflections. (See Exploration 1, p. 49.)

Answer:
We know that when a point is reflected about x-axis then y-coordinate becomes the opposite.
P(x,y) = P'(x,-y)
We know that when a point is reflected about y-axis then x-coordinate becomes opposite.
P(x,y) = P'(-x,y)
Given: A(5,3), B(5,-3)
Hence the point A is reflected about the x-axis to get point B.

IDENTIFYING A REFLECTION
Tell whether the blue figure is a reflection of the red figure.

Question 12.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 33

Answer: No

Explanation:
The blue figure is not the mirror image of the red figure. If the red figure were flipped then the right of the blue and red figure should be facing each other. So, the blue figure is not a reflection of red figure.

Question 13.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 34

Answer: Yes

Explanation:
The blue figure is the mirror image of the red figure. If the red figure were flipped it will result in the blue figure. So, the blue figure is a reflection of red figure.

Question 14.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 35

Answer: Yes

Explanation:
The blue figure is the mirror image of the red figure. If the red figure were flipped it will result in the blue figure. So, the blue figure is a reflection of red figure.

Question 15.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 36

Answer: No

Explanation:
The blue figure is not the mirror image of the red figure. If the red figure were flipped then the right of the blue and red figure should be facing each other. So, the blue figure is not a reflection of red figure.

Question 16.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 37

Answer: Yes

Explanation:
The blue figure is the mirror image of the red figure. If the red figure were flipped it will result in the blue figure. So, the blue figure is a reflection of red figure.

Question 17.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 38

Answer: No

Explanation:
The blue figure is not the mirror image of the red figure. If the red figure were flipped then the right of the blue and red figure should be facing each other. So, the blue figure is not a reflection of red figure.

REFLECTING FIGURES
Draw the figure and its reflection in the x-axis. Identify the coordinates of the image.

Question 18.
A(3, 2), B(4, 4), C(1, 3)

Answer:
We know that when a point is reflected about the x-axis then y-coordinate becomes the opposite.
A(x,y) = A'(x,-y)
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
A(3, 2), B(4, 4), C(1, 3)
Reflection about the x-axis:
A(3, 2) = A'(3,-2)
B(4, 4) = B'(4,-4)
C(1, 3) = C'(1,-3)
Thus the coordinate of the image are A'(3,-2), B'(4,-4), C'(1,-3)
Bigideas Math Answers Grade 8 Chapter 2 img_14

Question 19.
M(-2, 1), N(0, 3), P(2, 2)

Answer:
We know that when a point is reflected about the x-axis then y-coordinate becomes the opposite.
A(x,y) = A'(x,-y)
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
M(-2, 1), N(0, 3), P(2, 2)
Reflection about the x-axis:
M(-2, 1) = M'(-2,-1)
N(0, 3) = N'(0,-3)
P(2, 2) = P'(2,-2)
Thus the coordinate of the image are M'(-2,-1), N'(0,-3), P'(2,-2)
Big Ideas Math Grade 8 Ch 2 Answer Key img_15

Question 20.
H(2, -2), J(4, -1), K(6, -3), L(5, -4)

Answer:
We know that when a point is reflected about the x-axis then y-coordinate becomes the opposite.
A(x,y) = A'(x,-y)
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
H(2, -2), J(4, -1), K(6, -3), L(5, -4)
Reflection about the x-axis:
H(2, -2) = H'(-2,-1)
J(4, -1) = J'(4,1)
K(6, -3) = K'(6,3)
L(5, -4) = L'(5,4)
Thus the coordinate of the image are H'(-2,-1), J'(4,1), K'(6,3) and L'(5,4)
Big Ideas Math Answers Grade 8 Chapter 2 Transformations img_16

Question 21.
D(-2, -5), E(0, -1), F(2, -1), G(0, -5)

Answer:
We know that when a point is reflected about the x-axis then y-coordinate becomes the opposite.
A(x,y) = A'(x,-y)
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
D(-2, -5), E(0, -1), F(2, -1), G(0, -5)
Reflection about the x-axis:
D(-2, -5) = D'(-2,5)
E(0, -1) = E'(0,1)
F(2, -1) = F'(2,1)
G(0, -5) = G'(0,5)
Thus the coordinate of the image are D'(-2,5), E'(0,1), F'(2,1), G'(0,5)
Big ideas Math Answers Grade 8 Chapter 2 Transformations img_17

REFLECTING FIGURES
Draw the figure and its reflection in the y-axis. Identify the coordinates of the image.

Question 22.
Q(-4, 2), R(-2, 4), S(-1, 1)

Answer:
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
Q(-4, 2), R(-2, 4), S(-1, 1)
Reflection about the x-axis:
Q(-4, 2) = Q'(4,2)
R(-2, 4) = R'(2,4)
S(-1, 1)= S'(1,1)
Thus the coordinate of the image is Q'(4,2), R'(2,4), S'(1,1)
Big Ideas Math Grade 8 Chapter 2 solution Key img_18

Question 23.
T(4, -2), U(4, 2), V(6, -2)

Answer:
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
T(4, -2), U(4, 2), V(6, -2)
Reflection about the y-axis
T(4,-2) = T'(-4,-2)
Y(4,2) = U'(-4,2)
V(6,-2) = V'(-6,-2)
Thus the coordinates of the figure are T'(-4,-2), U'(-4,2), V'(-6,-2)
Big Ideas Math Grade 8 Chapter 2 transformations answer key img_19

Question 24.
W(2, -1), X(5, -2), Y(5, -5), Z(2, -4)

Answer:
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
W(2, -1), X(5, -2), Y(5, -5), Z(2, -4)
Reflection about the y-axis:
W(2,-1) = W'(-2,-1)
X(5,-2) = X'(-5,-2)
Y(5,-5) = Y'(-5,-5)
Z(2,-4) = Z'(-2,-4)
Thus the coordinates of the figure are W'(-2,-1), X'(-5,-2), Y'(-5,-5), Z'(-2,-4)
Big Ideas Math Grade 8 2nd Chapter Answer Key for Transformations img_20

Question 25.
J(2, 2), K(7, 4), L(9, -2), M(3, -1)

Answer:
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given,
J(2, 2), K(7, 4), L(9, -2), M(3, -1)
Reflection about the y-axis
J(2, 2) = J(-2,2)
K(7, 4) = K'(-7,4)
L(9, -2) = L'(-9,-2)
M(3, -1) = M'(-3,-1)
Thus the coordinates of the figure are J(-2,2), K'(-7,4), L'(-9,-2), M'(-3,-1)
BIM Grade 8 Solution Key Chapter 2 Transformations img_21

Question 26.
REASONING
Which letters look the same when reflected in the line?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 39

Answer:
The letters which will look the same after being reflected through horizontal line are
B, C, D, E, H, I, K, O, X

STRUCTURE
The coordinates of a point and its image after a reflection are given. Identify the line of reflection.

Question 27.
(2, -2) → (2, 2)

Answer:
When a point is reflected about the x-axis then the y coordinate becomes the opposite.
A(x,y) = A'(x,-y)
when a point is reflected about the y-axis then the x coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given A(2, -2) → A'(2, 2)
Here we can see that x-coordinate of both A & A’ is the same but the y-coordinate of A’ is just the opposite of A. This means that A’ is the reflection of A about the x-axis.
Hence the point A is reflected about the x-axis to get point A’.

Question 28.
(-4, 1) → (4, 1)

Answer:
When a point is reflected about the x-axis then the y coordinate becomes the opposite.
A(x,y) = A'(x,-y)
when a point is reflected about the y-axis then the x coordinate becomes the opposite.
A(x,y) = A'(-x,y)
Given A(-4, 1) → A'(4, 1)
Here we can see that y-coordinate of both A & A’ is the same but the x-coordinate of A’ is just the opposite of A. This means that A’ is the reflection of A about y-axis.
Hence the point A is reflected about the y-axis to get point A’.

Question 29.
(-2, -5) → (4, -5)

Answer:
Given,
A(-2, -5) → A'(4, -5)
We observe that y-coordinate of both A and A’ is same but the x-coordinate of A’ is not opposite of A. This means that A’ is the reflection of A about a line x = a.
a = (x2+x1)/2 = (4-2)/2 = 2/2 = 1
Hence the point (-2,-5) is reflected about the line x = 1 to get point (4,-5)

Question 30.
(-3, -4) → (-3, 0)

Answer:
Given,
B(-3, -4) → B'(-3, 0)
We observe that x-coordinate of both B and B’ is the same but the y-coordinate of B’ is not the opposite of B. This means that A’ is the reflection of A about a line x = a.
b = (y2+y1)/2 = (0-4)/2 = -4/2 = -2
Hence the point (-3,-4) is reflected about the line y = -2 to get point (-3,0)

TRANSFORMING FIGURES
Find the coordinates of the figure after the transformations.

Question 31.
Translate the triangle 1 unit right and 5 units down. Then reflect the image in the y-axis.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 40

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if the shift is Right and Vertical Up and the value of a and b will be negative if the shift is left and vertical Down.
Given:
R(-4,1)
S(-4,4)
T(-2,1)
a = 1 and b = -5
R(-4,1) = R'(-4+a, 1+b) = R'(-4+1, 1-5) = R'(-3, -4)
S(-4,4) = S'(-4+a, 4+b) = S'(-4+1, 4-5) = S'(-3, -1)
T(-2,1) = T'(-2+a, 1+b) = T'(-2+1, 1-5) = T'(-1, -4)
Thus the coordinates of the image are R'(-3, -4), S'(-3, -1), T'(-1, -4)

Question 32.
Reflect the trapezoid in the x-axis. Then translate the image 2 units left and 3 units up.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 41

Answer:
When a point is reflected about the x-axis then the y coordinate becomes the opposite.
A(x,y) = A'(x,-y)
Given, W(-2,-2), X(-2,1), Y(2,1), and Z(4,-2)
Now reflection about the x-axis:
W(-2,-2) = W'(-2,2)
X(-2,1) = X'(-2,-1)
Y(2,1) = Y'(2,-1)
Z(4,-2) = Z'(4,2)
Thus the coordinates of the image: W'(-2,2), X'(-2,-1), Y'(2,-1), Z'(4,2)
Now translating the above image point:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if the shift is Right and Vertical Up and the value of a and b will be negative if the shift is left and vertical Down.
Given:
W(-2,-2), X(-2,1), Y(2,1), and Z(4,-2)
a = -2 and b = 3
W(-2,2) = W”(-2+a, 2+b) = W”(-2-2, 2+3) = W”(-4, 5)
X(-2,-1) = X”(-2+a, -1+b) = X”(-2-2, -1+3) = X”(-4, 2)
Y(2,-1) = Y”(2+a, -1+b) = Y”(2-2, -1+3) = Y”(0, 2)
Z(4,2) = Z”(4+a, 2+b) = Z”(4-2, 2+3) = Z”(2, 5)
Thus the coordinates of the image: W”(-4, 5), X”(-4, 2), Y”(0, 2), Z”(2, 5)

Question 33.
REASONING
In Exercises 31 and 32, is the original figure identical to the final image? Explain.

Answer: Yes, in exercises 31 and 32 the original figure is identical to the final image. Because the type of transformation used is reflection and translation. The shape and size of the image figure do not change when there is reflection or translation. The only position of the image changes in both cases when compared to the position of the original figure.

Question 34.
CRITICAL THINKING
Hold a mirror to the left side of the photo of the vehicle.
a. What word do you see in the mirror?

Answer: The word which we will see in the mirror will be AMBULANCE. Because the word is written in mirror image form on the vehicle.

b. Why do you think it is written that way on the front of the vehicle?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 42

Answer: Ambulance

Explanation:
It is written in that way because the ambulance will be behind any vehicle then the word “AMBULANCE” will correctly appear in the Rear-view mirror of the front vehicle.

Question 35.
DIG DEEPER!
Reflect the triangle in the line y = x. How are the x- and y-coordinates of the image related to the x- and y-coordinates of the original triangle?
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 43

Answer:
When a point is reflected about the line y = x then both x and y-coordinate become opposite.
A(x, y) = A'(-x, -y)
Given,
D(-1,-3)
E(-1,1)
F(-3,1)
Reflection about the line y = x
D(-1,-3) = D'(1,-3)
E(-1,1) = E'(1,-1)
F(-3,1) = F'(3,-1)
Hence the coordinates of the image: D'(1,-3), E'(1,-1), F'(3,-1)
Big Ideas Math Grade 8 Chapter 2 Solution Key img_21

Lesson 2.3 Rotations

EXPLORATION 1
Work with a partner.
a. For each figurebelow, draw the figure in the coordinate plane. Then copy the axes and the figure onto a piece of transparent paper. Turn the transparent paper and align the origin and the axes with the coordinate plane. For each pair of figures, describe the angle of rotation.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 44

  • point
  • triangle
  • line segment
  • rectangle

Big Ideas Math Answers Grade 8 Chapter 2 Transformations 45
b. When you rotate figures, what do you notice about sides, angles, and parallel lines?
c. Describe the relationship between each point below and the point A(3, 6) in terms of rotations.
d. What are the coordinates of a point P(x, y) after a rotation 90° counterclockwise about the origin? 180°? 270°?

2.3 Lesson

Try It
Tell whether the blue figure is a rotation of the red figure about the origin. If so, give the angle and direction of rotation.

Question 1.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 46

Answer: Yes blue figure is the rotation of red figure about the origin.

Explanation:
When we rotate the red figure 180 degrees clockwise or anti-clockwise about the origin we will get the same figure as the blue figure.

Question 2.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 47

Answer: Blue figure is not the rotation of the red figure.

Explanation:
When the red figure is rotated about the origin in any direction the distance of the center point of both the red figure and the blue figure will be the same from the origin point (0,0). The distance between the center of the object and the center of rotation always remains the same.
Thus Blue figure is not the rotation of the red figure.

Try It

The vertices of a figure are given. Rotate the figure as described. Find the coordinates of the image.

Question 3.
J(-4, -4), K(-4, 2), L(-1, 0), M(-2, -3); 180° about the origin

Answer:
When a point is rotated 180 degrees about the origin then both x and y-coordinates become opposite.
A(x, y) = A'(-x, -y)
Given, J(-4, -4), K(-4, 2), L(-1, 0), M(-2, -3)
Rotation about the origin
J(-4, -4) = J'(4,4)
K(-4, 2) = K'(4,-2)
L(-1, 0) = L'(1,0)
M(-2, -3) = M'(2,3)
Hence the coordinate of the image are J'(4,4), K'(4,-2), L'(1,0), M'(2,3)

Question 4.
P(-3, 2), Q(6, 1), R(-1, -5); 90° counterclockwise about the origin

Answer:
When a point is rotated 90 degrees about the origin then both x and y-coordinates become opposite.
A(x, y) = A'(-y, x)
Given,
P(-3, 2), Q(6, 1), R(-1, -5)
Rotation about the origin
P(-3, 2) = P'(-2,-3)
Q(6, 1) = Q'(-1,6)
R(-1, -5) = R'(5,-1)
Hence the coordinate of the image is P'(-2,-3), Q'(-1,6), R'(5,-1)

Question 5.
A(5, 3), B(4, -1), C(1, -1); 90° clockwise about the origin

Answer:
When a point is rotated 270 degrees counterclockwise about the origin then both x and y-coordinates gets interchanged and the x-coordinate becomes the opposite.
A(x, y) = A'(y, -x)
Given,
A(5, 3), B(4, -1), C(1, -1)
Rotation about the origin
A(5, 3) = A'(3,-5)
B(4, -1) = B'(-1,-4)
C(1, -1) = C'(-1,-1)
Hence the coordinate of the image are A'(3,-5), B'(-1,-4), C'(-1,-1)

Try It

Question 6.
The vertices of a triangle are P(-1, 2), Q(-1, 0), and R(2, 0). Rotate the triangle 180° about the origin, and then reflect it in the x-axis. What are the coordinates of the image?

Answer:
When a point is rotated 180 degrees about the origin then both x and y-coordinates become opposite.
A(x, y) = A'(-x, -y)
Given,
P(-1, 2), Q(-1, 0), and R(2, 0)
Rotation about the origin
P(-1, 2) = P'(1,-2)
Q(-1, 0) = Q'(1,0)
R(2, 0) = R'(-2,0)
Hence the coordinate of the image is P'(1,-2), Q'(1,0), R'(-2,0)
Now reflecting above image point about x-axis:
When a point is reflected about the x-axis then the y-coordinate becomes opposite.
A(x, y) = A'(x, -y)
Given,
P'(1,-2), Q'(1,0), R'(-2,0)
Rotation about the origin
P'(1,-2) = P”(1,2)
Q'(1,0) = Q”(1,0)
R'(-2,0) = R”(-2,0)
Hence the coordinate of the image are P'(1,2), Q'(1,0), R'(-2,0)

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
IDENTIFYING A ROTATION
Tell whether the blue figure is a rotation of the red figure about point P. If so, give the angle and direction of rotation.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 48

Answer:
Yes, the blue figure is the rotation of the red figure about the origin.

Explanation:
Because when we will rotate the red figure 90 degrees anti-clockwise about the origin we will get the same figure as the blue figure.
By this, we can say that the blue figure is the result of the rotation of red figure by 90 degrees in the clock or anti-clockwise direction.

Question 8.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 49
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 50

Answer:
The statement which different from all other 3 statement is:
What are the coordinates of the image after a 270 degrees clockwise rotation about the origin?
Now coordinate of both the image are:
The rotation of an object 90 degrees clockwise is equal to the rotation of 270 degrees counterclockwise.
we know that when a point is rotated 270 degrees counterclockwise about origin then both coordinate gets interchanges and x-coordinate becomes opposite.
A(x, y) = A'(y, -x)
Given,
A(2, 4)
B(4, 4)
C(4, 1)
Rotating 90 degrees clockwise about the origin
A(2,4) = A'(4,-2)
B(4, 4) = B'(4,-4)
C(4, 1) = C'(1,-4)
Hence the coordinate of the image are: A'(4,-2), B'(4,-4), C'(1,-4)
Image of statement which different from all 3 statement
The rotation of an object 270 degrees clockwise is equal to the rotation of 90 degrees counterclockwise.
we know that when a point is rotated 90 degrees counterclockwise about origin then both coordinate gets interchanges and x-coordinate becomes opposite
P(x, y) = P'(-y, x)
Given,
A(2, 4)
B(4, 4)
C(4, 1)
Rotating 90 degrees clockwise about the origin
A(2, 4) = A'(-4, 2)
B(4, 4) = B'(-4, 4)
C(4, 1) = C'(-1, 4)
Hence the coordinate of the image are: A'(4,-2), B'(-4,4), C'(-1,4)
Third statement “what are the coordinates of the image after a 270 degrees clockwise rotation about origin?” is different.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
You move the red game piece to the indicated location using a rotation about the origin, followed by a translation. What are the coordinates of the vertices of the game piece after the rotation? Justify your answer.
Big Ideas Math Answers Grade 8 Chapter 2 Transformations 51

Answer:
To move the red game piece in the indicated location the game piee should be rotated 90 degrees in clockwise direction. The rotation of red game piece 90 degrees clockwise is same as the rotation of 270 degrees counterclockwise.
we know that when a point is rotated 270 degrees counterclockwise about origin then both coordinate gets interchanges and x-coordinate becomes opposite.
A(x,y) = A'(y, -x)
Conner point of red game piece:
A(0,-1), B(0,0), C(1,0), D(1,1), E(-2,1), F(-2,0), G(-1,0), H(-1,-1)
Rotating 90 degrees clockwise about the origin:
A(0,-1) = A'(-1,0)
B(0,0) = B'(0,0)
C(1,0) = C'(0,-1)
D(1,1) = D'(1,-1)
E(-2,1) = E'(1,2)
F(-2,0) = F'(0,2)
G(-1,0) = G'(0,1)
H(-1,-1) = H'(-1,1)
Hence the coordinate of corner of red game piece are A'(-1,0), B'(0,0), C'(0,-1), D'(1,-1), E'(1,2), F'(0,2), G'(0,1), H'(-1,1)

Question 10.
DIG DEEPER!
Skytypingis a technique that airplanes use to write messages in the sky. The coordinate plane shows a message typed in the sky over a city, where the positive y-axis represents north. What does the message say? How can you transform the message so that it is read from north to south?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 52

Answer: HELLO

Explanation:
The message above on the coordinate plane can be transformed from north to south by rotating the image 90 degrees anticlockwise.

Rotations Homework & Practice 2.3

Review & Refresh

Tell whether the blue figure is a reflection of the red figure.

Question 1.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 53

Answer: Yes, the blue figure is the reflection of red figure.

Explanation:
Because the blue figure is the exact mirror image of the red figure. If the red figure will be flipped it will result in the blue figure. So, the blue figure is the reflection of red figure.

Question 2.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 54

Answer: No, the blue figure is not the reflection of red figure.

Explanation:
Because the blue figure is not the mirror image of the red figure. If the red figure will be flipped it will not result in the blue figure. So blue figure is not reflection of red figure.

Find the circumference of the object. Use 3.14 or \(\frac{22}{7}\) for π.

Question 3.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 55

Answer:
Given diameter of disk D = 28 cm
Circumference of the circular disk is π × D
C = π × 28
C = 22/7 × 28
C = 22 × 4
C = 88 cm
Thus the circumference is 88 cm.

Question 4.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 56

Answer:
Given the diameter of disk D = 11.4 in
Circumference of the circular disk is π × D
C = π × 11.4
C = 22/7 × 11.4
C = 3.14 × 11.4
C = 35.796 in
Thus the circumference is 35.796 in

Question 5.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 57

Answer:
Given diameter of disk r = 0.5 ft
Circumference of the circular disk is 2π × r
C = 2π × 0.5
C = 6.28 × 0.5
C = 3.14 ft
Thus the circumference is 3.14 ft

Concepts, Skills, &Problem Solving

DESCRIBING RELATIONSHIPS
Describe the relationship between the given point and the point (2, 7) in terms of rotations. (See Exploration 1, p. 55.)

Question 6.
B(7, -2)

Answer:
Given,
A(7, 2) = B(7, -2)
Here we can see that after rotation x and y coordinate are interchanged and the y-coordinate is opposite. And we know that when a point is rotated 270 degrees counterclockwise about origin then both coordinates get interchanged and the x-coordinate becomes opposite.
P(x, y) = P'(y, -x)
Hence the above rotation is 270 degrees counterclockwise about the origin.

Question 7.
C(-7, 2)

Answer:
Given,
A(7, 2) = C(-7, 2)
Here we can see that after rotation x and y coordinate are interchanged and the y-coordinate is opposite. And we know that when a point is rotated 270 degrees counterclockwise about origin then both coordinate gets interchanged and x-coordinate becomes opposite.
P(x, y) = P'(-y, x)
Hence the above rotation is 90 degrees counterclockwise about the origin.

Question 8.
D(-2, -7)

Answer:
Given,
A(2, 7) = C(-2, -7)
Here we can see that after rotation x and y coordinate are interchanged and the y-coordinate is opposite. And we know that when a point is rotated 270 degrees counterclockwise about origin then both coordinate gets interchanged and x-coordinate becomes opposite.
P(x, y) = P'(-x, -y)
Hence the above rotation is 180 degrees counterclockwise about the origin.

IDENTIFYING A ROTATION
Tell whether the blue figure is a rotation of the red figure about the origin. If so, give the angle and direction of rotation.

Question 9.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 58

Answer: No the blue figure is not the rotation of the red figure.

Explanation:
Because if the blue triangle were the result of the rotation of the red triangle then the hypotenuse of the blue triangle should have been parallel to the x-axis. so, it is not the case of rotation.

Question 10.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 59

Answer: Yes, the blue figure is the result of the rotation of the red figure.

Explanation:
Because if the red figure is rotated 90 degrees in a counterclockwise direction it will result in a blue figure.

Question 11.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 60

Answer: Yes, the blue figure is the result of the rotation of the red figure.

Explanation:
If the red figure is rotated 180 degrees in counterclockwise or clockwise direction it will result in blue figure.

Question 12.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 61

Answer: Yes, the blue figure is the result of the rotation of the red figure.

Explanation:
If the red figure is rotated 90 degrees in a clockwise direction it will result in blue figure.

ROTATING A FIGURE
The vertices of a figure are given. Rotate the figure as described. Find the coordinates of the image.

Question 13.
A(2, -2), B(4, -1), C(4, -3), D(2, -4)
90° counterclockwise about the origin

Answer:
We know that when a point is rotated 90 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(-y, x)
Given,
A(2, -2), B(4, -1), C(4, -3), D(2, -4)
Rotating 90 degrees counterclockwise about the origin
A(2, -2) = A'(2,2)
B(4, -1) = B'(1,4)
C(4, -3) = C'(3,4)
D(2, -4) = D'(4,2)
Hence the coordinates of the image are A'(2,2), B'(1,4), C'(3,4), D'(4,2)

Question 14.
F(1, 2), G(3, 5), H(3, 2) 180° about the origin

Answer:
We know that when a point is rotated 180 degrees counterclockwise or clockwise direction about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(-x, -y)
Given,
F(1, 2), G(3, 5), H(3, 2)
Rotating 180 degrees about the origin
F(1, 2) = F'(-1,-2)
G(3, 5) = G'(-3,-5)
H(3, 2) = H'(-3,-2)
Hence the coordinates of the image are F'(-1,-2), G'(-3,-5), H'(-3,-2)

Question 15.
J(-4, 1), K(-2, 1), L(-4, -3)
90° clockwise about the origin

Answer:
The rotation of an object 90 degrees clockwise is equal to the rotation of 270 degrees counterclockwise.
We know that when a point is rotated 270 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(y, -x)
Given,
J(-4, 1), K(-2, 1), L(-4, -3)
Rotating 90 degrees clockwise about the origin
J(-4, 1) = J'(1,4)
K(-2, 1) = K'(1,2)
L(-4, -3) = L'(-3,4)
Hence the coordinates of the image are J'(1,4), K'(1,2), L'(-3,4)

Question 16.
P(-3, 4), Q(-1, 4), R(-2, 1), S(-4, 1)
270° clockwise about the origin

Answer:
The rotation of an object 270 degrees clockwise is equal to the rotation of 90 degrees counterclockwise.
We know that when a point is rotated 90 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(-y, x)
Given,
P(-3, 4), Q(-1, 4), R(-2, 1), S(-4, 1)
Rotating 90 degrees clockwise about the origin
P(-3, 4) = P'(-4,-3)
Q(-1, 4) = Q'(-4,-1)
R(-2, 1) = R'(-1,-2)
S(-4, 1) = S'(-1,-4)
Hence the coordinates of the image are P'(-4,-3), Q'(-4,-1), R'(-1,-2), S'(-1,-4)

Question 17.
W(-6, -2), X(-2, -2), Y(-2, -6), Z(-5, -6)
270° counterclockwise about the origin

Answer:
We know that when a point is rotated 270 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(y, -x)
Given,
W(-6, -2), X(-2, -2), Y(-2, -6), Z(-5, -6)
Rotating 90 degrees clockwise about the origin
W(-6, -2) = W'(-2,6)
X(-2, -2) = X'(-2,2)
Y(-2, -6) = Y'(-6,2)
Z(-5, -6) = Z'(-6,5)
Hence the coordinates of the image are W'(-2,6), X'(-2,2), Y'(-6,2), Z'(-6,5)

Question 18.
A(1, -1), B(5, -6), C(1, -6)
90° counterclockwise about the origin

Answer:
We know that when a point is rotated 90 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(-y, x)
Given,
A(1, -1), B(5, -6), C(1, -6)
Rotating 90 degrees clockwise about the origin
A(1, -1) = A'(1,1)
B(5, -6) = B'(6,5)
C(1, -6) = C'(6,1)
Hence the coordinates of the image are A'(1,1), B'(6,5), C'(6,1)

Question 19.
YOU BE THE TEACHER
The vertices of a triangle are A(4, 4), B(1, -2), and C(-3, 0). Your friend finds the coordinates of the image after a rotation 90° clockwise about the origin. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 62

Answer:
We know that when a point is rotated 270 degrees counterclockwise about the origin then both coordinates gets interchanged and y-coordinate becomes opposite.
P(x, y) = P'(y, -x)
Given,
A(4, 4), B(1, -2), and C(-3, 0).
Rotating 90 degrees clockwise about the origin
A(4, 4) = A'(4,-4)
B(1, -2) = B'(-2,-1)
C(-3,0) = C'(0,3)
Hence the coordinates of the image are A'(4,-4), B'(-2,-1), C'(0,3)
By this I can say that my friend is not correct.

Question 20.
PROBLEM SOLVING
A game show contestant spins the prize wheel shown. The arrow remains in a fixed position while the wheel rotates. The wheel stops spinning, resulting in an image that is a rotation 270° clockwise about the center of the wheel. What is the result?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 62.1

Answer: Free spin

Explanation:
The arrow is located at 90 degrees in the counterclockwise direction of free spin. So when the wheel is rotated 270 degrees in a clockwise direction the arrow will be on the free spin column.

PATTERN
A figure has rotational symmetry if a rotation of 180° or less produces an image that fits exactly on the original figure. Determine whether the figure has rotational symmetry. Explain your reasoning.

Question 21.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 63

Answer: Yes the given figure has rotational symmetry.

Explanation:
The given figure in the problem is rotated 120 degrees in any direction clockwise or counterclockwise then it will produce the same identical image. Since 120 degrees is less than 180 degrees so it will have rotational symmetry.

Question 22.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 64

Answer: No the given figure does not have rotational symmetry.

Explanation:
The given figure in the problem will produce the same identical image only when it is rotated 360 degrees. Since 360 degrees is greater than 180 degrees so it will not have rotational symmetry.

Question 23.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 65

Answer: Yes the given figure has rotational symmetry.

Explanation:
The given figure in the problem will produce the same identical image only when it is rotated 180 degrees. Since the maximum angle for rotational symmetry is 180 degrees so it will have rotational symmetry.

USING MORE THAN ONE TRANSFORMATION
The vertices of a figure are given. Find the coordinates of the image after the transformations given.

Question 24.
R(-7, -5), S(-1, -2), T(-1, -5)
Rotate 90° counterclockwise about the origin. Then translate 3 units left and 8 units up.

Answer:
We know that when a point is rotated 90 degrees counterclockwise about origin then both coordinates gets interchanges and y-coordinate becomes opposite.
P(x,y) = P'(-y,x)
Given, R(-7, -5), S(-1, -2), T(-1, -5)
Rotating 90 degrees counterclockwise about the origin
R(-7,-5) = R'(5,-7)
S(-1, -2) = S'(2,-1)
T(-1, -5) = T'(5,-1)
The coordinate of the image are R'(5,-7), S'(2,-1), T'(5,-1)
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given, R'(5,-7), S'(2,-1), T'(5,-1) and a = -3, b = 8
R'(5+a, -7+b) = R”(5-3, -7+8) = R”(2,1)
S'(2+a, -1+b) = R”(2-3, -1+8) = S”(-1,7)
R'(5+a, -1+b) = R”(5-3, -1+8) = T”(2,7)
The coordinate of the image are R”(2,1), S”(-1,7), T”(2,7)

Question 25.
J(-4, 4), K(-3, 4), L(-1, 1), M(-4, 1) Reflect in the x-axis, and then rotate 180° about the origin.

Answer:
We know that when a point is reflected about x-axis then y-coordinate becomes opposite.
A(x, y) = A'(x, -y)
Given J(-4, 4), K(-3, 4), L(-1, 1), M(-4, 1)
Reflection about the x-axis:
J(-4, 4) = J'(-4,-4)
K(-3, 4) = K'(-3,-4)
L(-1, 1) = L'(-1,-1)
M(-4, 1) = M'(-4,-1)
The coordinate of the image are J'(-4,-4), K'(-3,-4), L'(-1,-1), M'(-4,-1)
Now rotating the above image 180 degrees about the origin.
We know that when a point is reflected about x-axis then y-coordinate becomes opposite.
A(x, y) = A'(-x, -y)
J'(-4,-4), K'(-3,-4), L'(-1,-1), M'(-4,-1)
Rotating 180 degrees about the origin:
J'(-4,-4) = J”(4,4)
K'(-3,-4) = K”(3,4)
L'(-1,-1) = L”(1,1)
M'(-4,-1) = M”(4,1)
The coordinate of the image are J”(4,4), K”(3,4), L”(1,1), M”(4,1)

CRITICAL THINKING
Describe two different sequences of transformations in which the blue figure is the image of the red figure.

Question 26.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 66

Answer:
Two different ways of translating a red figure in to blue figure:
1. First rotate the red figure 90 degrees in the counterclockwise direction and then translate that image 5 units towards the left to get the blue figure.
2. First rotate the red figure 90 degrees in a clockwise direction and then translate that image 1 unit towards the Right and 5 units Up to get the blue figure.

Question 27.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 67

Answer:
Two different ways of translating a red figure in to blue figure:
1. First rotate the red figure 90 degrees in the counterclockwise direction and then translate that image 1 unit towards the left and 1 Down to get the blue figure.
2. First rotate the image in the x-axis and then translate that image 4 units towards the left and 2 units Up to get the blue figure.

Question 28.
REASONING
A trapezoid has vertices A(-6, -2), B(-3, -2), C(-1, -4), and D(-6, -4).
a. Rotate the trapezoid 180° about the origin. What are the coordinates of the image?

Answer:
A(x, y) = A'(-x, -y)
Given,
A(-6, -2), B(-3, -2), C(-1, -4), and D(-6, -4).
Rotating 180 degrees about the origin:
A(-6, -2) = A'(6,2)
B(-3, -2) = B'(3,2)
C(-1, -4) = C'(1,4)
D(-6, -4) = D'(6,4)
The coordinates of the image of trapezoid vertices are A'(6,2), B'(3,2), C'(1,4), D'(6,4)

b. Describe a way to obtain the same image without using rotations.

Answer:
In the above question, we can see that the coordinates of all the vertices of the trapezoid are negative and all the coordinates of the image vertices are positive. So there is another way to get the vertices of the image.
First, reflect the trapezoid in the x-axis and then in the y-axis or first reflect the trapezoid in the y-axis and then in the x-axis.

ROTATING A FIGURE
The vertices of a figure are given. Rotate the figure as described. Find the coordinates of the image.

Question 29.
D(2, 1), E(2, -2), F(-1, 4)
90° counterclockwise about vertex D

Answer:
P(x, y) = P'(-(y – b) + a, (x -a) + b)
Given,
D(2, 1), E(2, -2), F(-1, 4)
(a, b) = (2, 1)
Rotation about the point D(2,1)
D(2, 1) = D'(2,1)
E(2, -2) = E'(-(-2-1) + 2, (2 – 2) + 1) = E'(5, 1)
F(-1, 4) = F'(-(4 – 1) + 2, (-1 – 2) + 1) = F'(-1, -2)
Hence the coordinate of the image: D'(2,1), E'(5, 1), F'(-1, -2)

Question 30.
L(-4, -3), M(-1, -1), N(2, -2)
180° about vertex M

Answer:
When a point is rotated 180 degrees counterclockwise about a given point (a, b) then its both x and y coordinate becomes opposite and ‘b’ and ‘a’ are subtracted from x and y coordinate respectively.
P(x, y) = P'(-(x – a) + b, -(y – b) + a)
Given,
L(-4, -3), M(-1, -1), N(2, -2)
Rotation about the point M(-1, -1):
L(-4, -3) = L'(-(-4 + 1) – 1, -(-3 + 1) – 1) = L'(2, 1)
M(-1, -1) = M'(-1, -1)
N(2, -2) = N'(-(2 + 1) – 1, -(-2 + 1) – 1) = N'(-4, 0)
Hence the coordinate of the image are L'(2, 1), M'(-1, -1), N'(-4, 0)

Question 31.
W(-5, 0), X(-1, 4), Y(3, -1), Z(0, -4)
270° counterclockwise about vertex W

Answer:
When a point is rotated 270 degrees counterclockwise about a given point (a, b) then its both x and y coordinate becomes opposite and ‘b’ and ‘a’ are subtracted from x and y coordinate respectively.
P(x, y) = P'(-(x – a) + b, -(y – b) + a)
Given,
W(-5, 0), X(-1, 4), Y(3, -1), Z(0, -4)
Rotation about the point W(-5, 0):
W(-5, 0) = W'(-5, 0)
X(-1, 4) = X'((4 – 0) – 5, -(-1 + 5) + 0) = X'(-1, -4)
Y(3, -1) = Y'((-1 – 0) – 5, -(3 + 5) + 0) = Y'(-6, -8)
Z(0, -4) = Z'((-4 – 0) – 5, -(0 + 5) + 0) = Z'(-9, -5)
Hence the coordinate of the image are W'(-5, 0), X'(-1, -4), Y'(-6, -8), Z'(-9, -5)

Question 32.
D(-3, -4), E(-5, 2), F(1, -1), G(3, -7)
270° clockwise about vertex E.

Answer:
When a point is rotated 90 degrees counterclockwise about a given point (a, b) then its both x and y coordinate becomes opposite and ‘b’ and ‘a’ are subtracted from x and y coordinate respectively.
P(x, y) = P'(-(x – a) + b, -(y – b) + a)
Given,
D(-3, -4), E(-5, 2), F(1, -1), G(3, -7)
Rotation about the point E(-5, 2):
D(-3, -4) = D'(-(-4 – 2) – 5, -(-3 + 5) + 2) = D'(1, 4)
E(-5, 2) = E'(-5, 2)
F(1, -1) = F'(-(-1 – 2) – 5, (1 + 5) + 2) = F'(-2, 8)
G(3, -7) = G'(-(-7 – 2) – 5, (3 + 5) + 2) = G'(4, 10)
Hence the coordinate of the image are D'(1, 4), E'(-5, 2), F'(-2, 8), G'(4, 10)

Question 33.
LOGIC
You want to find the treasure located on the map at Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 68. You are located at Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 69. The following transformations will lead you to the treasure, but they are not in the correct order. Find the correct order. Use each transformation exactly once.

  • Rotate 180° about the origin.
  • Reflect in the y-axis.
  • Rotate 90° counterclockwise about the origin.
  • Translate 1 unit right and 1 unit up.

Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 68.1

Answer:
The correct order of transformation to get the treasure are:

  • Rotate 180° about the origin.
  • Rotate 90° counterclockwise about the origin.
  • Reflect in the y-axis.
  • Translate 1 unit right and 1 unit up.

Question 34.
DIG DEEPER!
You rotate a triangle 90° counterclockwise about the origin. Then you translate its image 1 unit left and 2 units down. The vertices of the final image are (-5, 0), (-2, 2), and (-2, -1). What are the vertices of the original triangle?

Answer:
When a point is rotated 90 degrees counterclockwise about a given point (a, b) then its both x and y coordinate becomes opposite.
Let the three vertices of the triangle be: (x1, y1), (x2, y2), (x3, y3)
P(x, y) = P'(-y, x)
Rotating 90 degrees counterclockwise about the origin:
A(x1, y1) = A'(-y1, x1)
B(x2, y2) = B'(-y2, x2)
C(x3, y3) = C'(-y3, x3)
Now translating the image of the vertex for the final image
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x, y) = A'(x + a, y + b)
Given,
A'(-y1, x1), B'(-y2, x2), C'(-y3, x3) and a = -1, b = -2
A'(-y1 + a, x1 + b) = A”(-y1 – 1, x1 – 2)
B'(-y2 + a, x2 + b) = B”(-y2 – 1, x2 – 2)
C'(-y3 + a, x3 + b) = C”(-y3 – 1, x3 – 2)
The given coordinate of vertex point of final image are: (-5, 0), (-2, 2) and (-2, -1)
Now comparing the coordinate of the final image
(-y1 – 1, x1 – 2) = (-5, 0) so y1 = 4 and x1 = 2
(-y2 – 1, x2 – 2) = (-2, 2) so y2 = 1 and x2 = 4
(-y3 – 1, x3 – 2) = (-2, 1) so y3 = 1 and x3 = 1
Hence the vertices of original triangle are (2, 4), (4, 1) and (1, 1)

Lesson 2.4 Congruent Figures

EXPLORATION 1

Work with a partner.
a. For each pair of figures whose vertices are given below, draw the figures in a coordinate plane. Then copy one of the figures onto a piece of transparent paper. Use transformations to try to obtain one of the figures from the other figure.

  • A(-5, 1), B(-5, -4), C(-2, -4) and D(1, 4), E(1, -1), F(-2, -1)
  • G(1, 2), H(2, -6), J(5, 0) and L(-1, -2), M(-2, 6), N(-5, 0)
  • P(0, 0), Q(2, 2), R(4, -2) and X(0, 0), Y(3, 3), Z(6, -3)
  • A(0, 4), B(3, 8), C(6, 4), D(3, 0) and
    F(-4, -3), G(-8, 0), H(-4, 3), J(0, 0)
  • P(-2, 1), Q(-1, -2), R(1, -2), S(1, 1) and
    W(7, 1), X(5, -2), Y(3, -2), Z(3, 1)

Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 69.1
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 69.2
b. Which pairs of figures in part(a) are identical? Explain your reasoning.
c. FigureA and FigureB are identical. Do you think there must be a sequence of transformations that obtains Figure A from Figure B? Explain your reasoning.

2.4 Lesson

Try It

Question 1.
A triangle has vertices X(0, 4), Y(4, 4), and Z(4, 2). Is △XYZ congruent to any of the triangles in Example 1? Explain.

Answer:
Big Ideas Math Grade 8 Chapter 2 transformation key img_22
After plotting the triangle XYZ on the coordinate plane we can say that the triangle XYZ is congruent to triangle PQR among all the given triangle in the figure. In fact, if triangle PQR is rotated 90 degrees clockwise of 270 degrees counterclockwise it will result in the triangle XYZ.

Try It

Question 2.
Describing a different Sequence of rigid motions between the figures.

Answer:
Different sequence of rigid motion to get the blue figure from the red figure are:
1. First rotate the red figure 90 degrees clockwise and the origin.
2. Then translate the image 4 units Right and 1 unit Up.

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
IDENTIFYING CONGRUENT FIGURES
Use the coordinate plane shown.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 70
a. Identify any congruent figures.

Answer: a. After seeing the figure we can say that triangle ABCD is congruent to triangle JKLM.

b. A rectangle has vertices W(4, 1), X(4, 2), Y(1, 2), and Z(-1, -1). Is Rectangle WXYZ congruent to any of the rectangles in the coordinate plane? Explain.

Answer:
Big ideas math Grade 8 ch 2 solution key img_23
Rectangle WXYZ is not congruent to any of the rectangles in the given figure because rectangle WXYZ is square of 3 units sides and the other rectangle in the figure does not have all the sides of 3 units.

RIGID MOTIONS
The red figure is congruent to the blue figure. Describe a sequence of rigid motions between the figures.

Question 4.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 71

Answer:
The sequence of the rigid motions from the red-figure to the blue figure:
1. First we will rotate the red figure 180 degrees clockwise or anticlockwise about the origin because the given red figure in the 4th quadrant and the blue figure is in the 2nd quadrant.
2. Then we will translate the image 1 unit left because one vertex (-1,-4) of red figure is on the negative side of the x-axis.

Question 5.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 72

Answer:
The sequence of the rigid motions from red figure to blue figure:
1. First we will rotate the red figure 90 degrees clockwise about the origin because the given red figure in the 1st quadrant.
2. Then we will translate the image 3 units right and 1 unit down.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
In the coordinate plane at the left, each grid line represents 50 feet. Each figure represents a pasture.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 73
a. Are the figures congruent? Use rigid motions to justify your answer.

Answer: No the blue figure and red figure are not congruent.

Explanation:
By reflecting the red figure about the y-axis and translating the image 4 units Up we will not the same blue figure. So both figure are not congruent to each other.

b. How many feet of fencing do you need to enclose each pasture?

Answer:
Given the length of each grid line = 50 feet
Total feet of fencing = 50 × total number of grid line along the boundary
For red figure fencing: 50 × 12 = 600 feet
For blue figure fencing: 50 × 12 = 600 feet

Question 7.
A home decorator uses a computer to design a floor tile. How can the decorator transform the tile as shown?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 74

Answer:
First, rotate the given tiles about 90 degrees in the clockwise direction and then take the mirror image about the vertical axis.

Congruent Figures Homework & Practice 2.4

Review & Refresh

The vertices of a figure are given. Rotate the figure as described. Find the coordinates of the image.

Question 1.
A(1, 3), B(2, 5), C(3, 5), D(2, 3)
90° counterclockwise about the origin

Answer:
When a point is rotated 90 degrees about the origin then both x and y-coordinates become opposite.
A(x, y) = A'(-y, x)
Given,
A(1, 3), B(2, 5), C(3, 5), D(2, 3)
Rotating 90 degrees counterclockwise about the origin:
A(1, 3) = A'(-3, 1)
B(2, 5) = B'(-5, 2)
C(3, 5) = C'(-5, 3)
D(2, 3) = D'(-3, 2)
Hence the coordinate of the image are A'(-3, 1), B'(-5, 2), C'(-5, 3), D'(-3, 2)

Question 2.
F(-2, 1), G(-1, 3), H(3, 1)
180° about the origin

Answer:
When a point is rotated 180 degrees about the origin then both x and y-coordinates become opposite.
A(x, y) = A'(-x, -y)
Given,
F(-2, 1), G(-1, 3), H(3, 1)
Rotating 90 degrees counterclockwise about the origin:
F(-2, 1) = F'(2,-1)
G(-1, 3) = G'(1,-3)
H(3, 1) = H'(-3,-1)
Hence the coordinate of the image are F'(2,-1), G'(1,-3), H'(-3,-1)

Factor the expression using the greatest common factor.

Question 3.
4n – 32

Answer:
4n – 32
Take 4 as a common factor.
4(n – 8)
Thus the greatest common factor is 4(n – 8)

Question 4.
3w + 66

Answer:
3w + 66
Take 3 as a common factor.
3(w + 22)
Thus the greatest common factor is 3(w + 22)

Question 5.
2y – 18

Answer:
2y – 18
Take 2 as a common factor.
2(y – 9)
Thus the greatest common factor is 2(y – 9).

Concepts, Skills, & Problem Solving
TRANSFORMING FIGURES
The vertices of a pair of figures are given. Determine whether the figures are identical. (See Exploration 1, p. 63.)

Question 6.
G(0, 0), H(3, 2), J(1, -2) and L(-1, 0), M(2, 2), N(0, -3)

Answer:
Big Ideas Math 8th Grade Solution Key Chapter 2 img_24
After plotting the triangles GHJ and LMN we can say that the triangle LMN are bigger compared to the other triangle. Thus both the triangles are not identical.

Question 7.
A(-2, -1), B(-2, 2), C(-1, 1), D(-1, -2) and F(-2, 0), G(-1, 1), H(2, 1), J(1, 0)

Answer:
Answer Key for BIM Grade 8 Chapter 2 tranformations img_25
By seeing both the quadrilaterals ABCD and FGHJ we can say that they are identical.

IDENTIFYING CONGRUENT FIGURES
Identify any congruent figures in the coordinate plane.

Question 8.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 75

Answer:
On observing the diagram in the given figure we can see that the shape and size of pentagon ABCDE and pentagon FKJHG are the same. The length of each side of both the pentagon is the same. Thus they are congruent.

Question 9.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 76

Answer:
By seeing the above figure we can say that the shape and size of parallelogram EFGH and parallelogram BCDA are the same. The length of each side of both the parallelogram are same. Parallelogram BCDA can be obtained by rotating parallelogram EFGH 90 degrees clockwise and translating its image. Hence the parallelogram, EFGH is congruent to BCDA.

DESCRIBING A SEQUENCE OF RIGID MOTIONS
The red figure is congruent to the blue figure. Describe a sequence of rigid motions between the figures.

Question 10.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 77

Answer:
The sequence of rigid motions between the red and blue figures are:
1. First we will rotate the red figure 90 degrees clockwise about the origin because the given red figure is in the 2nd quadrant and the blue figure is in the 1st quadrant.
2. Then we will translate the image 1 unit left and 1 unit Down because one vertex of the red figure is at (-1, 1)

Question 11.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 78

Answer:
The sequence of rigid motions between the red and blue figures are:
1. First we will rotate the red figure 180 degrees clockwise or anticlockwise about the origin because the given red figure is in 4th quadrant and the blue figure is in the 2nd quadrant.
2. Then we will translate the image 1 unit Right and 1 unit Down because one vertex of red figure is at (2, -2)

Question 12.
YOU BE THE TEACHER
Your friend describes a sequence of rigid motions between the figures. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 79

Answer:
When a point is reflected about x-axis then the y-coordinate becomes opposite.
A(x, y) = A'(x, -y)
Coordinates of red figure are A(1, -1), B(3, -1), C(4, -3), D(2, -3)
Reflection about the x-axis:
A(1, -1) = A'(1, 1)
B(3, -1) = B'(3, 1)
C(4, -3) = C'(4, 3)
D(2, -3) = D'(2, 3)
Now translating the above image point 5 unit left.
Coordinate of the vertex of blue figure are: A”(-4, 1), B”(3,1), C”(-1, 3), D”(-3,3)
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if the shift is Right and Vertical Up and the value of a and b will be negative if the shift is left and vertical Down.
Given:
A(1, 1), B(3, 1), C(4, 3), D(2, 3) and a = -5, b = 0
A”(1+a, 1+b) = A”(1-5, 1+0) = A”(-4, 1)
B”(3+a, 1+b) = B”(3-5, 1+0) = B”(-2,1)
C”(4+a,3+b) = C”(4-5, 3+0) = C”(-1, 3)
D”(2+a,3+b) = D”(2-5, 3+0) = D”(-3,3)
Hence the coordinate of image are A”(-4, 1), B”(-2,1), C”(-1, 3), D”(-3,3)
Since the coordinate of the vertex of the blue is the same in both ways.
We can say that the blue figure is obtained by the rigid motion of the red figure.

NAMING CORRESPONDING PARTS
The figures are congruent. Name the corresponding angles and the corresponding sides.

Question 13.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 80

Answer:
Corresponding sides of the congruent figure are
AD = EH
AB = EF
BC = FG
CD = GH
Corresponding angles of the congruent figure are
∠A = ∠E
∠B = ∠F
∠C = ∠G
∠D = ∠H

Question 14.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 81

Answer:
Corresponding sides of the congruent figure are
PQ = WV
QR = VZ
RS = ZY
ST = YX
TP = XW
Corresponding angles of the congruent figure are
∠P = ∠W
∠Q = ∠V
∠R = ∠Z
∠S = ∠Y
∠T = ∠X

Question 15.
MODELING REAL LIFE
You use a computer program to transform an emoji. How can you transform the emoji as shown?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 82

Answer:
First, take the reflection of that emoji about vertical line and then rotate that image 90 degrees clockwise to get that given emoji.

Question 16.
CRITICAL THINKING
Two figures are congruent. Are the areas of the two figures the same? the perimeters? Explain your reasoning.

Answer:

  • The size of both figures should be the same.
  • The shape of both the figures should be the same.
  • All the corresponding angles should be the same.
  • Both the area and perimeter of two congruent figures are the same.

Question 17.
DIG DEEPER!
The houses are identical.
a. What is the length of side LM?

Answer:
Length of LM = length of CD
length of CD = 32 feet
So, the length of LM is 32 feet

b.Which angle of JKLMN corresponds to ∠D?
Answer:
∠D = ∠M
Thus ∠M corresponds to ∠D

c. Side AB is congruent to side AE. What is the length of side AB? What is the perimeter of ABCDE?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 83.

Answer:
AE = JN
The length of JN is 20 ft
So, the length of AE = 20 ft
Perimeter of ABCDE = AB + BC + CD + DE + EA
Perimeter of ABCDE = 20 + 12 + 32 + 12 + 20 = 96 feet
Thus the Perimeter of ABCDE  is 96 feet

Question 18.
REASONING
Two constellations are represented by the figures in the coordinate plane shown. Are the figures congruent? Justify your answer.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 84

Answer:
The above figure can be tranformed into below figure by rotating the figure 180 degrees clockwise or counterclockwise about the origin and translating the image 8 units Right and 8 units Up to get the above figure.

Lesson 2.5 Dilations

Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 85

EXPLORATION 1

Work with a partner. Use geometry software.
a. Draw a polygon in the coordinate plane. Then dilate the polygon with respect to the origin. Describe the scale factor of the image.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 86
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 86.1
b. Compare the image and the original polygon in part(a). What do you notice about the sides? the angles?
c. Describe the relationship between each point below and the point A(x, y) in terms of dilations.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 87
d. What are the coordinates of a point P(x, y) after a dilation with respect to the origin by a scale factor of k?

2.5 Lesson

Try It

Tell whether the blue figure is a dilation of the red figure.

Question 1.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 88

Answer:
No the blue figure is not the dilation of the red figure.

Explanation:
Blue and red figure has same size and same shape but the blue figure is reflection about vertical axis. So, the lines corresponding vertices of both figure does not meet at a point. This means that the blue figure is not the dilation of the red figure.

Question 2.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 89

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of red figure are congruent to the angles of red figure. Also their will be lines connecting the corresponding verrtices meeting at a point. This means that blue figure is dilation of the red figure.

Try It

Question 3.
WHAT IF?
Triangle ABC is dilated by a scale factor of 2. What are the coordinates of the image?

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: A (1, 3), B (2, 3), C (2, 1) and scale factor = 2
Dilating the figure by scale factor of 2
A (1, 3) = A'(1 . 2, 3 . 2) = A'(2, 6)
B (2, 3) = B'(2 . 2, 3 . 2) = B'(4, 6)
C (2, 1) = C'(2 . 2, 1 . 2) =  C'(4, 2)
Hence the coordinates of the image are A'(2, 6), B'(4, 6),  C'(4, 2)

Try It

Question 4.
WHAT IF?
Rectangle WXYZ is dilated by a scale factor of \(\frac{1}{4}\). What are the coordinates of the image?

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the rectangle: W(-4, -6), X(-4, 8), Y(4, 8), Z(4, -6)
scale factor = 0.25
W(-4, -6) = W'(-4 × 0.25, -6 × 0.25) = W'(-1, -1.5)
X(-4, 8) = X'(-4 × 0.25, 8 × 0.25) = X'(-1, 2)
Y(4, 8) = Y'(4 × 0.25, 8 × 0.25) = Y'(1, 2)
Z(4, -6) = Z'(4 × 0.25, -6 × 0.25) = Z'(1, -1.5)
Hence the coordinates of the image are W'(-1, -1.5), X'(-1, 2), Y'(1, 2), Z'(1, -1.5)

Try It

Question 5.
WHAT IF?
Trapezoid ABCDis dilated using a scale factor of 3, and then rotated 180° about the origin. What are the coordinates of the image?

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
a is the scaling factor
Given points of trapezoid: A(-2, -1), B(-1,1), C(0,1), D(0,-1) scale factor = 3
Dilating the figure by scale factor of 3
A(-2, 1) = A'(-2 . 3, -1 . 3) = A'(-6, -3)
B(-1, 1) = B'(-1 . 3, 1 . 3) = B'(-3, 3)
C(0, 1) = C'(0 . 3, 1 . 3) = C'(0, 3)
D(0, -1) = D'(0 . 3, -1 . 3) = D'(0, -3)
Thus the coodrinate of the image are A'(-6, -3), B'(-3, 3), C'(0, 3), D'(0, -3)
when a point is rotated 180 degrees about the origin then both x and y coordinate becomes opposite.
P(x, y) = P'(-x, -y)
Image points: A'(-6, -3), B'(-3, 3), C'(0, 3), D'(0, -3)
Rotating 180 degrees about the origin:
A'(-6, -3) = A”(6, 3)
B'(-3, 3) = B”(3, -3)
C'(0, 3) = C”(0, -3)
D'(0, -3) = D”(0, 3)
Thus the coodrinate of the image are A”(6, 3), B”(3, -3), C”(0, -3), D”(0, 3)

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

IDENTIFYING A DILATION
Tell whether the blue figure is a dilation of the red figure.

Question 6.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 90

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of red figure are congruent to the angles of red figure. Also their will be lines connecting the corresponding vertices meeting at a point. This means that blue figure is dilation of the red figure.

Question 7.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 91

Answer: No the blue figure is not the dilation of the red figure.

Explanation:
Blue and red figure has same size and same shape but the blue figure is reflection about vertical axis. So, the lines corresponding vertices of both figure does not meet at a point. This means that the blue figure is not the dilation of the red figure.

Question 8.
DILATING A FIGURE
The vertices of a rectangle are J(4, 8), K(12, 8), L(12, 4), and M(4, 4). Draw the image after a dilation with a scale factor of \(\frac{1}{4}\). Identify the type of dilation.

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
a is scale factor
Given, vertices of a rectangle are J(4, 8), K(12, 8), L(12, 4), and M(4, 4), scale factor = 0.25
J(4, 8) = J'(4 × 0.25, 8 × 0.25) = J'(1, 2)
K(12, 8) = K'(12 × 0.25, 8 × 0.25) = K'(3, 2)
L(12, 4) = L'(12 × 0.25, 4 × 0.25) = L'(3, 1)
M(4, 4) = M'(4 × 0.25, 4 × 0.25) = M'(1, 1)
Hence the coordinates of the image are J'(1, 2), K'(3, 2), L'(3, 1), M'(1, 1)
Big Ideas Math Answers Grade 8 Chapter 2 Transformations img_26

Question 9.
VOCABULARY
How is a dilation different from other transformations?

Answer:
The difference between dilation and other transformations are

  • In the case of dilate the size of the figure after the dilation either decrease or increase but the shape of the figure before and after dilation remains same. Also after the dilation the corresponding angles will be congruent.
  • In case of other transformations such as Rotation, translation, reflection the shape and size of figure before and after transformation remains the same.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
A photograph is dilated to fit in a frame, so that its area after the dilation is 9 times greater than the area of the original photograph. What is the scale factor of the dilation? Explain.

Answer: The scale factor of length and breadth will be 3.

Explanation:
Given,
The area after the dilation is 9 times greater than the area of the original photograph.
Area = length × breadth
p = 3 × 3
Hence the scale factor of length and breadth will be 3.

Question 11.
DIG DEEPER!
The location of a water treatment plant is mapped using a coordinate plane, where each unit represents 1 foot. The plant has vertices (0, 0), (0, 180), (240, 180), and (240, 0). You dilate the figure with a scale factor of \(\frac{1}{3}\). What are the coordinates of the image? What do you need to change so that the image accurately represents the location of the plant? Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 92

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
a is the scaling factor
Location of water treatment plant: A(0, 0), B(0, 180), C(240, 180), D(240, 0)
scale factor = \(\frac{1}{3}\)
Dilating the figure by scale factor of \(\frac{1}{3}\)
A(0, 0) = A'(0 × \(\frac{1}{3}\), 0 × \(\frac{1}{3}\)) = A'(0, 0)
B(0, 180) = B'(0 × \(\frac{1}{3}\), 180 × \(\frac{1}{3}\)) = B'(0, 60)
C(240, 180) = C'(240 × \(\frac{1}{3}\), 180 × \(\frac{1}{3}\)) = C'(80, 60)
D(240, 0) = D'(240 × \(\frac{1}{3}\), 0 × \(\frac{1}{3}\)) = D'(80, 0)
Hence the coordinates of the image are A'(0, 0), B'(0, 60), C'(80, 60), D'(80, 0)

Dilations Homework & Practice 2.5

Review & Refresh

The red figure is congruent to the blue figure. Describe a sequence of rigid motions between the figures.

Question 1.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 93

Answer:
Sequence of rigid motion between the red and blue figure are
1. First rotate the blue figure 90 degrees in counterclockwise direction about the orgin because blue figure in 1st quadrant and red figure is in 3rd quadrant.
2. Then translate the image 1 unit left and 4 units down.

Question 2.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 94

Answer:
Sequence of rigid motion between the red and blue figure are
1. First reflect the blue fiure about x-axis. The image after the reflection will lies in the 3rd quadrant with same orientation.
2. Then translate the image 5 units Right.

Tell whether the ratios form a proportion.

Question 3.
3 : 5 and 15 : 20

Answer:
When two ratios are equal then it is called as proportion.
Given,
Given 3 : 5 and 15 : 20
3/5 and 15/20
3/5 and 3/4
Since the above two ratio are not equal hence they are not proportion

Question 4.
2 to 3 and 12 to 18

Answer:
When two ratios are equal then it is called as proportion.
Given,
2 to 3 and 12 to 18
2/3 and 12/18
2/3 and 2/3
Since the above two ratio are equal hence they are proportion.

Question 5.
7 : 28 and 12 : 48

Answer:
When two ratios are equal then it is called as proportion.
Given,
7 : 28 and 12 : 48
7/28 and 12/48
1/4 and 1/4
Since the above two ratio are equal hence they are proportion.

Concepts, Skills, &Problem Solving

DESCRIBING RELATIONSHIPS
Describe the relationship between the given point and the point A(8, 12) in terms of dilations. (See Exploration 1, p. 69.)

Question 6.
B(16, 24)

Answer:
Given a point and its image: A(8,12), B(16, 24)
Here we can see that both x-coordinate and y-coordinate of image point have increased to double.
This means that in this case, the image figure has become larger by the scale factor of 2 with respect to the origin.
Hence the dilation scale factor is 2.

Question 7.
C(2, 3)

Answer:
Given a point and its image: A(8,12), C(2, 3)
Here we can see that both x-coordinate and y-coordinate of image point has decreased to one-fourth.
This means that in this case, the image figure has become smaller by the scale factor of 0.25 with respect to the origin.
Thus the dilation scale factor is 1/4.

Question 8.
D(6, 9)

Answer:
Given a point and its image: A(8,12), D(6, 9)
Here we can see that both x-coordinate and y-coordinate of image point has decreased to three-fourth.
This means that in this case the image figure has become smaller by the scale factor of 0.75 with respect to the origin.
Thus the dilation scale factor is 3/4.

IDENTIFYING A DILATION
Tell whether the blue figure is a dilation of the red figure.

Question 9.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 95

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of the red figure are congruent to the angles of the red figure. Also there will be lines connecting the corresponding vertices meeting at a point. This means that blue figure is a dilation of the red figure.

Question 10.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 96

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of the red figure are congruent to the angles of the red figure. Also there will be lines connecting the corresponding vertices meeting at a point. This means that blue figure is a dilation of the red figure.

Question 11.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 97

Answer: No the blue figure is not the dilation of the red figure.

Explanation:
The Blue and red figure has same size and same shape but the blue figure is a reflection of vertical axis. So, the lines corresponding vertices of both figure does not meet at a point. This means that the blue figure is not the dilation of the red figure.

Question 12.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 98

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of red figure are congruent to the angles of red figure. Also their will be lines connecting the corresponding vertices meeting at a point. This means that blue figure is dilation of the red figure.

Question 13.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 99

Answer: Yes, the blue figure is a dilation of the red figure.

Explanation:
We observe that all the angles of red figure are congruent to the angles of red figure. Also their will be lines connecting the corresponding vertices meeting at a point. This means that blue figure is dilation of the red figure.

Question 14.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 100

Answer: No the blue figure is not the dilation of the red figure.

Explanation:
Blue and red figure has same size and same shape but the blue figure is reflection about vertical axis. So, the lines corresponding vertices of both figure does not meet at a point. This means that the blue figure is not the dilation of the red figure.

DILATING A FIGURE
The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation.

Question 15.
A(1, 1), B(1, 4), C(3, 1); k = 4

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: A(1, 1), B(1, 4), C(3, 1) and scale factor = 4
Dilating the figure by scale factor by 4
A(1, 1) = A'(1 × 4, 1 × 4) = A'(4, 4)
B(1, 4) = B'(1 × 4, 4 × 4) = B'(4, 16)
C(3, 1) = C'(3 × 4, 1 × 4) = C'(12, 4)
Hence the coordinate of the image are A'(4, 4), B'(4, 16), C'(12, 4)
Big Ideas Math 8th Grade Solution Key Chapter 2 Transformations img_27

Question 16.
D(0, 2), E(6, 2), F(6, 4); k = 0.5

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: D(0, 2), E(6, 2), F(6, 4) and scale factor = 0.5
Dilating the figure by scale factor by 0.5
D(0, 2) = D'(0 × 0.25, 2 × 0.25) = D'(0, 1)
E(6, 2) = E'(6 × 0.25, 2 × 0.25) = E'(3, 1)
F(6, 4) = F'(6 × 0.25, 4 × 0.25) = F'(3, 2)
Hence the coordinate of the image are D'(0, 1), E'(3, 1), F'(3, 2)
BIM 8th grade solution key for chapter 2 transformations img_28

Question 17.
G(-2, -2), H(-2, 6), J(2, 6); k = 0.25

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: G(-2, -2), H(-2, 6), J(2, 6) and scale factor = 0.25
G(-2, -2) = G'(-2 × 0.25, -2 × 0.25) = G'(-0.5, -0.5)
H(-2, 6) = H'(-2 × 0.25,6 × 0.25) = H'(-0.5, 1.5)
J(2, 6) = G'(2 × 0.25, 6 × 0.25) = J'(0.5, 1.5)
Hence the coordinate of the image are G'(-0.5, -0.5), H'(-0.5, 1.5), J'(0.5, 1.5)
Big Ideas Math Book Answers Grade 8 Chapter 2 Transformations img_29

Question 18.
M(2, 3), N(5, 3), P(5, 1); k = 3

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: M(2, 3), N(5, 3), P(5, 1) and scale factor = 3
M(2, 3) = M'(2 × 3, 3 × 3) = M'(6, 9)
N(5, 3) = N'(5 × 3, 3 × 3) = N'(15, 9)
P(5, 1) = P'(5 × 3, 1 × 3) = P'(15, 3)
Hence the coordinate of the image are M'(6, 9), N'(15, 9), P'(15, 3)
Big Ideas Math Grade 8 Chapter 2 Solution Key img_30

Question 19.
Q(-3, 0), R(-3, 6), T(4, 6), U(4, 0); k = \(\frac{1}{3}\)

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given,
Q(-3, 0), R(-3, 6), T(4, 6), U(4, 0) and scale factor = \(\frac{1}{3}\)
Q(-3, 0) = Q'(-3 × \(\frac{1}{3}\), 0 × \(\frac{1}{3}\)) = Q'(-1, 0)
R(-3, 6) = R'(-3 × \(\frac{1}{3}\), 6 × \(\frac{1}{3}\)) = R'(-1, 2)
T(4, 6) = T'(4 × \(\frac{1}{3}\), 6 × \(\frac{1}{3}\)) = T'(4/3, 2)
U(4, 0) = U'(4 × \(\frac{1}{3}\), 0 × \(\frac{1}{3}\)) = U'(4/3, 0)
Hence the coordinate of the image are Q'(-1, 0), R'(-1, 2), T'(4/3, 2), U'(4/3, 0)
Big Ideas Math Grade 8 ch 2 transformations answer key img_31

Question 20.
V(-2, -2), W(-2, 3), X(5, 3), Y(5, -2); k = 5

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given,
V(-2, -2), W(-2, 3), X(5, 3), Y(5, -2), scaling factor = 5
Dilating the figure by scale factor of 5
V(-2, -2) = V'(-2 × 5, -2 × 5) = V'(-10, -10)
W(-2, 3) = W'(-2 × 5, 3 × 5) = W'(-10, 15)
X(5, 3) = X'(5 × 5, 3 × 5) = X'(25, 15)
Y(5, -2) = Y'(5 × 5, -2 × 5) = Y'(25, -10)
Hence the coordinate of the image are V'(-10, -10),W'(-10, 15), X'(25, 15), Y'(25, -10)
Big Ideas Math Grade 8 ch 2 answer key img_32

Question 21.
YOU BE THE TEACHER
Your friend finds the coordinates of the image of △ABC after a dilation with a scale factor of 2. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 101

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given,
The points of triangle: A(2, 5), B(2, 0), C(4, 0)
scale factor = 2
Dilating the figure by scale factor of 2
A(2, 5) = A'(2 × 2, 5 × 2) = A'(4, 10)
B(2, 0) = B'(2 × 2, 0 × 2) = B'(4, 0)
C(4, 0) = C'(4 × 2, 0 × 2) = C'(8, 0)
Hence the coordinate of the image are A'(4, 10), B'(4, 0), C'(8, 0)
By this, we can say that my friend is correct.

FINDING A SCALE FACTOR
The blue figure is a dilation of the red figure. Identify the type of dilation and find the scale factor.

Question 22.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 102

Answer:
Scale factor = side length of image/side length of original figure
Scale factor = A’B’/AB = 6/3 = 2
Scale factor = 2
Hence, type of dilation is enlargement with scale factor of 2.

Question 23.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 103

Answer:
Scale factor = side length of image/side length of original figure
Scale factor = X’Y’/XY= 2/8 = 1/4
Scale factor = 1/4
Hence, the type of dilation is reduction with the scale factor of 1/4

Question 24.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 104

Answer:
Scale factor = side length of image/side length of the original figure
Scale factor = J’K’/JK = 15/10 = 3/2
Scale factor = 3/2
Hence, the type of dilation is reduction with the scale factor of 3/2

Question 25.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 105

Answer:
Scale factor = side length of image/side length of original figure
Scale factor = Q’R’/QR = 4/8 = 1/2
Scale factor = 1/2
Hence, type of dilation is reduction with scale factor of 1/2

USING MORE THAN ONE TRANSFORMATION
The vertices of a figure are given. Find the coordinates of the image after the transformations given.

Question 26.
A(-5, 3), B(-2, 3), C(-2, 1), D(-5, 1)
Reflect in the y-axis. Then dilate using a scale factor of 2.

Answer:
We know that when a point is reflected about y-axis then is x-coordinate becomes opposite.
A(-5, 3), B(-2, 3), C(-2, 1), D(-5, 1)
A(x, y) = A'(-x, y)
A(-5, 3) = A'(5, 3)
B(-2, 3) = B'(2, 3)
C(-2, 1) = C'(2, 1)
D(-5, 1) = D'(5, 1)
Coordinate of the image are A'(5, 3), B'(2, 3), C'(2, 1), D'(5, 1)
A(-5, 3), B(-2, 3), C(-2, 1), D(-5, 1)
Reflect in the y-axis. Then dilate using a scale factor of 2
A'(5, 3) = A”(5 × 2, 3 × 2) = A”(10, 6)
B'(2, 3) = B”(2 × 2, 3 × 2) = B”(4, 6)
C'(2, 1) = C”(2 × 2, 1 × 2) = C”(4, 2)
D'(5, 1) = D”(5 × 2, 1 × 2) = D”(10, 2)
Coordinate of the image are A”(10, 6), B”(4, 6), C”(4, 2), D”(10, 2)
Big ideas math grade 8 chapter 2 solution key img_33

Question 27.
F(-9, -9), G(-3, -6), H(-3, -9).
Dilate using a scale factor of \(\frac{2}{3}\). Then translate 6 units up.

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given,
F(-9, -9), G(-3, -6), H(-3, -9) and scale factor of \(\frac{2}{3}\)
Dilating the figure by scale factor of \(\frac{2}{3}\)
F(-9, -9) = F'(-9 × 2/3, -9 × 2/3) = F'(-6, -6)
G(-3, -6) = G'(-3 × 2/3, -6 × 2/3) = G'(-2, -4)
H(-3, -9) = H'(-3 × 2/3, -9 × 2/3) = H'(-2, -6)
Coordinate of the image are F'(-6, -6), G'(-2, -4), H'(-2, -6)
Now translating above image 6 units up
F'(-6, -6), G'(-2, -4), H'(-2, -6) and a = 0, b = 6
F”(-6 + a, -6 + b) = F”(-6 + 0, -6 + 6) = F”(-6, 0)
G”(-2 + a, -4 + b) = G”(-2 + 0, -4 + 6) = G”(-2, 2)
H”(-2 + a, -6 + b) = H”(-2 + 0, -6 + 6) = H”(-2, 0)
Coordinate of the image are F”(-6, 0), G”(-2, 2), H”(-2, 0)
BIM Grade 8 Answers Chapter 2 img_34

Question 28.
J(1, 1), K(3, 4), L(5, 1)
Rotate 90° clockwise about the origin. Then dilate using a scale factor of 3.

Answer:
The rotation of an object 90 degrees clockwise is equal to the rotation of 270 degrees counterclockwise.
When a point is rotated 270 degrees counterclockwise about the origin then both x and y-coordinates gets interchanged and the x-coordinate becomes the opposite.
A(x, y) = A'(y, -x)
J(1, 1), K(3, 4), L(5, 1)
Rotate 90° clockwise about the origin.
J(1, 1) = J'(1, -1)
K(3, 4) = K'(4, -3)
L(5, 1) = L'(1, -5)
Coordinate of the image are J'(1, -1), K'(4, -3), L'(1, -5)
Now dilate using a scale factor of 3.
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
J'(1, -1) = J”(1 . 3, -1 . 3) = J”(3, -3)
K'(4, -3) = K”(4 . 3, -3 . 3) = K”(12, -9)
L'(1, -5) = L”(1 . 3, -5 . 3) = L”(3, -15)
Coordinate of the image are J”(3, -3), K”(12, -9), L”(3, -15)
BIM 8th Grade Answers Ch 2 transformations img_35

Question 29.
LOGIC
You can use a flashlight and a shadow puppet (your hands) to project shadows on the wall.
a. Identify the type of dilation.p

Answer: The type of dilation is an enlargement

b. What does the flashlight represent?

Answer: Flashlight represents center of dilation because all the line connecting shadow and hand meet at the flashlight.
c. The length of the ears on the shadow puppet is 3 inches. The length of the ears on the shadow is 4 inches. What is the scale factor?

Answer: Scale factor = length of ears on shadow/length of ears on puppet
Scale factor = 4/3
d. Describe what happens as the shadow puppet moves closer to the flashlight. How does this affect the scale factor?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 106

Answer:
As the flashlight will move closer the shadow will become larger. Also, the scale factor will increase.

Question 30.
REASONING
A triangle is dilated using a scale factor of 3. The image is then dilated using a scale factor of \(\frac{1}{2}\). What scale factor can you use to dilate the original triangle to obtain the final image? Explain.

Answer:
Given the first scale factor of triangle S1 = 3
Given second scale factor of triangle S2 = 1/2
We know that the final scale factor S = S1 × S2
Final Scale factor S = 3 × 1/2 = 3/2
Hence, the scale factor of the final image will be the multiplication of the first and second dilation scale factor and the final scale factor will be 3/2.

CRITICAL THINKING
The coordinate notation shows how the coordinates of a figure are related to the coordinates of its image after transformations. What are the transformations? Are the figure and its image congruent? Explain.

Question 31.
(x, y) → (2x + 4, 2y – 3)

Answer:
Given, (x, y) → (2x + 4, 2y – 3)
We can see that both x-coordinate and y-coordinate has been multiplied by 2 this means that the point has been dilated by the scale factor of 2.
Also, 4 has been added to x-coordinate while 3 is added to y-coordinate which means that obtained after the dilation has been translated 4 unit Right and 3 units Down.
The final image will not be congruent because after the dilation the size of the image either increases or decreases that depend on the type of dilation.

Question 32.
(x, y) → (-x – 1, y – 2)

Answer:
Given, (x, y) → (-x – 1, y – 2)
We can see that 1 has been subtracted from x-coordinate while 2 is subtracted from y-coordinate which means that will image has translated 1 unit left and 2 units down. And also x-coordinate is opposite which means the image has been reflected about the y-axis.
Hence, transforms translation of 1 unit left and 2 units down followed by reflection about y-axis.

Question 33.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 107

Answer:
Given, (x, y) → (1/3x, -1/3y)
We can see that both x-coordinate and y-coordinate has been multiplied by 2 this means that the point has been dilated by the scale factor of 1/3. Also, y-coordinate is opposite which means that image obtained after the dilation has been reflected about the x-axis.
Thus transforms are dilation with the scale factor of 1/3 followed by reflection about the x-axis.

STRUCTURE
The blue figure is a transformation of the red figure. Use coordinate notation to describe the transformation. Explain your reasoning.

Question 34.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 108

Answer:
Coordinates of original figure A(1, 1) B(1, 2), C(2, 1)
Coordinates of red figure A'(2, 3) B'(2, 6), C'(4, 3)
Scale factor of x-coordinate = x-coordinate of image/x-coordinate of image = 2/1 = 2
Scale factor of y-coordinate = y-coordinate of image/y-coordinate of image = 3/1 = 3
Thus to transfer the red-figure into the blue figure x-coordinate of all the points has been multiplied by 2 and the y-coordinate of all the points has been multiplied by 3.

Question 35.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 109

Answer:
Coordinates of original figure A(4, 4) B(4, 8), C(8, 8), D(8, 4)
Coordinates of red figure A'(1, 2) B'(1, 4), C'(2, 4), D'(2. 2)
Scale factor of x-coordinate = x-coordinate of image/x-coordinate of image = 1/4 = 0.25
Scale factor of y-coordinate = y-coordinate of image/y-coordinate of image = 2/4 = 1/2 = 0.5
Thus to transfer the red figure into the blue figure x-coordinate of all the points has been multiplied by 0.25 and the y-coordinate of all the points has been multiplied by 0.50

Question 36.
NUMBER SENSE
You dilate a figure using a scale factor of 2, and then translate it 3 units right. Your friend translates the same figure 3 units right and then dilates it using a scale factor of 2. Are the images congruent? Explain.

Answer:
Blue the final image in both the case will be of the same shape and size.
Yes, the image in both cases will be the same.

Question 37.
PROBLEM SOLVING
The vertices of a trapezoid are A(-2, 3), B(2, 3), C(5, -2), and D(-2, -2). Dilate the trapezoid with respect to vertex A using a scale factor of 2. What are the coordinates of the image? Explain the method you used.

Answer:
When the points of a given figure are dilated about a point we simply multiply the distance of each side by the given scale factor. The coordinate of one point remains the same about which dilation occurs.
The vertices of a trapezoid are A(-2, 3), B(2, 3), C(5, -2), and D(-2, -2).
Scale factor = 2
So, here the coordinate of point A(-2, 3) will remains the same but all the other coordinates of points B’, C’, D’ will change according to the distance between each side of the trapezoid.
Big Ideas Math Key Grade 8 Chapter 2 transformations img_36
Image of the figure after dilating by a scale factor of 2
A(-2, 3) = A'(-2, 3)
B(2, 3) = B'(6, 3)
C(5, -2) = C'(12, -7)
D(-2, -2) = D'(-2, -7)
Thus the coordinate of the image are A'(-2, 3), B'(6, 3), C'(12, -7), D'(-2, -7)

Question 38.
DIG DEEPER!
A figure is dilated using a scale factor of -1. How can you obtain the image without using a dilation? Explain your reasoning.

Answer:
When a figure is dilated using a scale factor of -1 then both the x and y-coordinate of the image will become opposite.
Example:
A'(x . -1, y . -1) = A'(-x, -y)
But there are two ways to get the same image:
1. By rotating the figure 180 degrees clockwise or anticlockwise
A(x, y) rotating 180 degrees about the origin = A'(-x, -y)
2. By rotating the figure about the x-axis and y axis
A(x, y) reflecting about the origin = A'(-x, -y)

Lesson 2.6 Similar Figures

EXPLORATION 1

Work with a partner. Use geometry software.
a. For each pair of figures whose vertices are given below, draw the figures in a coordinate plane. Use dilations and rigid motions to try to obtain one of the figures from the other figure.

  • A(-3, 6), B(0, -3), C(3, 6) and G(-1, 2), H(0, 1), J(1, 2)
  • D(0, 0), E(3, 0), F(3, 3) and L(0, 0), M(0, 6), N(-6, 6)
  • P(1, 0), Q(4, 2), R(7, 0) and X(-1, 0), Y(-4, 6), Z(-7, 0)
  • A(-3, 2), B(-1, 2), C(-1, -1), D(-3, -1) and F(6, 4), G(2, 4), H(2, -2), J(6, -2)
  • P(-2, 2), Q(-1, -1), R(1, -1), S(2, 2) and W(2, 8), X(3, 3), Y(7, 3), Z(8, 8)

Big Ideas Math Answers 8th Grade Chapter 2 Transformations 110
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 111
b. Is a scale drawing represented by any of the pairs of figures in part(a)? Explain your reasoning.
c. Figure A is a scale drawing of Figure B. Do you think there must be a sequence of transformations that obtains Figure A from Figure B? Explain your reasoning.

2.6 Lesson

Try It

Question 1.
A triangle has vertices D(0, 4), E(5, 4), and F(5, 0). Is △DEF similar to △ABC and △JKL in Example 1? Explain.

Answer:
Given coordinate of the triangle ABC: A(0, 3), B(3, 3), C(3, 0)
Given coordinate of the triangle DEF: D(0, 4), E(5, 4), F(5, 0)
Given coordinate of the triangle JKL: J(0, 6), K(6, 6), L(6, 0)
Here we can see that there is no fixed relation between the coordinate between triangle ABC and DEF or triangle ABC and JKL. So no triangle is dilation with the triangle ABC.
Hence, triangle ABC is not similar △DEF and △JKL.

Try It

Question 2.
Can you reflect the red figure first, and then perform the dilation to obtain the blue figure? Explain.

Answer:
Because the final image will not depend on the order of transformation. When we will first reflect red figure then the image will be of the same size and after the dilation of the image obtained after reflection we will get the same image so we can use any two method but the final image will be the same

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
IDENTIFYING SIMILAR FIGURES
In the coordinate plane at the left, determine whether Rectangle ABCD is similar to Rectangle EFGH. Explain your reasoning.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 112

Answer: No rectangle ABCD is not similar to rectangle EFGH

Explanation:
Because the orientation of rectangle ABCD is not the same as the rectangle EFGH. Also, rectangle ABCD is not dilated with rectangle EFGH. So there is no similarity transformation between rectangle ABCD and rectangle EFGH.

Question 4.
SIMILARITY TRANSFORMATION
The red triangle is similar to the blue triangle. Describe a similarity transformation between the figures.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 113

Answer:
Coordinate of left vertex of the red triangle: A(5, 5)
Coordinate of the same vertex after dilation: A'(10, 10)
Now coordinate left vertex of the blue triangle: A”(0, 2)
So, the value of a = 0 – 10 = -10 and b = 2 – 10 = -8
It is given that the red triangle is similar to the blue triangle so the steps of transformation:
First, dilate the figure by the scale factor of 2 and then translate the image 10 units left and 8 units down.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
A medical supplier sells gauze in large and small rectangular sheets. A large sheet has a length of 9 inches and an area of 45 square inches. A small sheet has a length of 4 inches and a width of 3 inches. Are the sheets similar? Justify your answer.

Answer:
Condition for the rectangular sheets to be similar is that all the corresponding sides of bigger and smaller rectangular sheets should be in proportional.
Area of larger rectangular sheets a = 45 sq in
length of larger rectangular sheets l1 = 9 in
Width = a/l = 45/9 = 5 in
length of smaller rectangular sheets l2 = 4 in
width of smaller rectangular sheets b2 = 3 in
condition for similarity l1/l2 = b1/b2
9/4 ≠ 5/3
These sheets are not similar

Question 6.
The sail on a souvenir boat is similar in shape to the sail on a sailboat. The sail on the sailboat is in the shape of a right triangle with a base of 9 feet and a height of 24 feet. The height of the souvenir’s sail is 3 inches. What is the base of the souvenir’s sail?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 114

Answer:
Given,
The base of sail on sailboat b1 = 9 ft
height of sail on sailboat h1 = 24 ft
Given height of sail on souvenir boat: h2 = 3in = 0.25 ft
h1/h2 = b1/b2
24/0.25 = 9/h2
h2 = (9 × 0.25)/24 = 0.9375 ft = 1.125 in
Thus the height of sail of a souvenir boat is 1.125 in

Question 7.
DIG DEEPER!
A coordinate plane is used to represent a cheerleading formation. The vertices of the formation are A(4, 4), B(0, 8), C(4, 4), and D(0, 6). A choreographer creates a new formation similar to the original formation. Three vertices of the new formation are J(-2, -2), K(0, -4), and L(2, -2). What is the location of the fourth vertex? Explain.

Answer:
The vertices of the formation are A(4, 4), B(0, 8), C(4, 4), and D(0, 6). A choreographer creates a new formation similar to the original formation.
We observe the image point carefully that both the x and y coordinate of the image point is just half of the original point and each y-coordinate is opposite.
A(4, 4) = J(-2, -2)
B(0, 8) = K(0, -4)
C(4, 4) = L(2, -2)
D(0, 6) = M(x, y)
This means that the point A, B, C are dilated by using a scale factor of 0.5 and the image obtained from the dilation is reflected about the x-axis.
So, the image point is D(0, 6) = M(0, -3)
Big Ideas Math Answers Grade 8 Chapter 2 Transformations img_37

Similar Figures Homework & Practice 2.6

Review & Refresh

Tell whether the blue figure is a dilation of the red figure.

Question 1.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 115

Answer: No

Explanation:
Because the shape and size of both red and blue figure are the same which is not the property of dilation. The blue figure is the result of the reflection of the red figure 180 degrees in the clockwise or counterclockwise direction.

Question 2.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 116

Answer: Yes

Explanation:
When we see both red and blue figures closely we observe that all the angles of the red figure are congruent to the blue figure. Also, there will be the lines connecting corresponding vertices meeting at a point. This means that the blue figure is a dilation of red figure.

Question 3.
You solve the equation S = lw + 2wh for w. Which equation is correct?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 117

Answer: Option C

Explanation:
S = lw + 2wh
lw + 2wh = S
Taking w as a common factor
w(l + 2h) = s
w = s/(l + 2h)
Thus the correct answer is option C.

Concepts, Skills, &Problem Solving
TRANSFORMING FIGURES
The vertices of a pair of figures are given. Determine whether a scale drawing is represented by the pair of figures. (See Exploration 1, p. 77.)

Question 4.
A(-8, -2), B(-4, 2), C(-4, -2) and G(2, -1), H(4, -1), J(2, -3)

Answer: Yes

Explanation:
BIM Grade 8 Answer Key Chapter 2 Transformations img_38
After plotting both triangles we see that the original figure is exactly double of the image. Each side of the original triangle is double the length of the image triangle. So given vertices pair represent a scale drawing.
Scale factor = 2

Question 5.
A(0, 3), B(3, 4), C(5, 3), D(3, 2) and F(-4, 4), G(-1, 5), H(5, 3), J(3, 2)

Answer: No

Explanation:
BIM Grade 8 Answers Ch 2 img_39
After plotting both given figures we can see that there is no transformation relation between the original figure and the image figure. So, the given vertices pair does not represent a scale drawing.

IDENTIFYING SIMILAR FIGURES
Determine whether the figures are similar. Explain your reasoning.(See Exploration 1, p. 77.)

Question 6.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 118

Answer: Rectangle ABCD is similar to rectangle EFGH.

Explanation:
Because when you see both the given figure we can see that all the corresponding angles of rectangle ABCD and rectangle EFGH are equal. And also the corresponding sides of both rectangle are in proportional.
∠A = ∠E, ∠B = ∠F, ∠C = ∠G, ∠D = ∠H
AB/EF = BC/FG = GH/CD = DA/HE = 1/2
Hence rectangle ABCD is similar to rectangle EFGH.

Question 7.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 119

Answer: Both the triangle are not similar

Explanation:
Because when you see both the given figure we can see that all the corresponding angles of triangle ABC and triangle JKL are equal. And also the corresponding sides of both triangles are not proportional.
AB/JK ≠ KL/BC ≠ CA/LJ
Hence triangle ABC is not similar to triangle JKL

IDENTIFYING SIMILAR FIGURES
Draw the figures with the given vertices in a coordinate plane. Which figures are similar? Explain your reasoning.

Question 8.
Rectangle A: (0, 0), (4, 0), (4, 2), (0, 2)
Rectangle B: (0, 0), (6, 0), (6, 3), (0, 3)
Rectangle C: (0, 0), (4, 0), (4, 2), (0, 2)

Answer: Rectangle A and B are similar

Explanation:
Rectangle A: (0, 0), (4, 0), (4, 2), (0, 2)
Big Ideas Math Grade 8 Chapter 2 Transformations img_39(i)
Rectangle B: (0, 0), (6, 0), (6, 3), (0, 3)
Big Ideas Math Grade 8 Chapter 2 Transformations img_39(ii)
Rectangle C: (0, 0), (4, 0), (4, 2), (0, 2)
Big Ideas Math Grade 8 Chapter 2 Transformations img_39(iii)
By seeing the above figure we can say that rectangle A and rectangle B are similar and rectangle A and Rectangle C are congruent.

Question 9.
FigureA: (4, 2), (2, 2), (2, 0), (4, 0)
Figure B: (1, 4), (4, 4), (4, 1), (1, 1)
Figure C: (2, 1), (5, 1), (5, 3), (2, 3)

Answer: Rectangle A and B are similar

Explanation:
FigureA: (4, 2), (2, 2), (2, 0), (4, 0)
Big Ideas Math Grade 8 Chapter 2 Transformations img_40(i)
Figure B: (1, 4), (4, 4), (4, 1), (1, 1)
Big Ideas Math Grade 8 Chapter 2 Transformations img_40(ii)
Figure C: (2, 1), (5, 1), (5, 3), (2, 3)
Big Ideas Math Grade 8 Chapter 2 Transformations img_40(iii)
Rectangles A and B are similar because in rectangle A and B all the corresponding angles are equal and also all the corresponding sides are equal.

DESCRIBING A SIMILARITY TRANSFORMATION
The red figure is similar to the blue figure. Describe a similarity transformation between the figures.

Question 10.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 120

Answer:
1. First rotate the red figure 90 degrees anticlockwise because the red figure is in the first quadrant and the blue figure is in the second quadrant.
2. Then dilate the image obtained after the rotation by the scale factor of 2 because the blue figure is double the size of the red figure.
Scale factor = side of the image/side of the original figure = 4/2 = 2

Question 11.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 121

Answer:
First, dilate the red figure by the scale figure of 3 because the blue figure is triple the size of red figure.
scale factor = side of image/side of original figure = 6/2 = 3

Question 12.
MODELING REAL LIFE
A barrier in the shape of a rectangle is used to retain oil spills. On a blueprint, a similar barrier is 9 inches long and 2 inches wide. The width of the actual barrier is 1.2 miles. What is the length of the actual barrier?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 122

Answer:
Given,
Width of the actual barrier = 1.2 miles
Width of the barrier in the blueprint = 2 inches
2 inch dimension of blueprint = 1.2 miles of original
So, 1 inch dimension of blueprint = 1.2/2 = 0.6 miles.
Since the length of the barrier in the blueprint = 9 inches,
Thus the length of the actual barrier = 9(0.6) = 5.4 miles.

Question 13.
LOGIC
Are the following figures always, sometimes, or never similar? Explain.
a. two triangles
b. two squares
c. two rectangles

Answer:
a. Two triangles sometimes two triangles are similar when all the corresponding angles are equal and all the corresponding sides lengths are in proportion.
b. Two squares always two square are similar only when all the sides are proportional and all the angles are equal.
c. Two rectangles are similar when all the corresponding angles are equal but the lengths of the corresponding sides are not always in proportion.

Question 14.
CRITICAL THINKING
Can you draw two quadrilaterals each having two 130° angles and two 50° angles that are not similar? Justify your answer.

Answer:
Quadrilateral 1: 50°, 50°, 130°, 130° (trapezoid)
Quadrilateral 2: 50°, 130°, 50°, 130° (parallelogram)
Big Ideas Math Grade 8 ch 2 transformations answer key img_41

Question 15.
REASONING
The sign is rectangular.
a. You increase each side length by 20%. Is the new sign similar to the original? Explain your reasoning.

Answer:
Given,
You increase each side length by 20%
Scale factor = 1 + percentage increase/100 = 1 + 20/100 = 1.2

b. You increase each side length by 6 inches. Is the new sign similar to the original? Explain your reasoning.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 123

Answer:
No, because when length and width are of a different size then adding 6 inches on each side will not increase the figure in a fixed proportion. So the corresponding length will be not proportional. This means that both the figure will be not in dilation, hence not similar.

Question 16.
DIG DEEPER!
A person standing 20 feet from a streetlight casts a shadow as shown. How many times taller is the streetlight than the person? Assume the triangles are similar.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 124

Answer:
Length of shadow l1 = 10 ft
height of man b1 = 6 ft
Total length of bigger triangle l2 = 20 + 10 = 30 ft
l1/l2 = b1/b2
10/30 = 6/b2
b2 = 180/10
b2 = 18 ft
The ratio of the height of streetlight and man is: 18/6 = 3
Hence, the streetlight is 3 times taller than that person.

Question 17.
GEOMETRY
Use a ruler to draw two different isosceles triangles similar to the one shown. Measure the heights of each triangle.
a. Are the ratios of the corresponding heights equivalent to the ratios of the corresponding side lengths?

Answer:
b1/b2 = h1/h2
6/3 = 4/2 = 2
Hence the ratio of corresponding heights is equivalent to the ratio of corresponding side lengths.

b. Do you think this is true for all similar triangles? Explain.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 125

Answer:
Yes, this will be true for all the similar triangles because the heights of the two similar triangles are multiplied by the same amount as the sides.

Question 18.
CRITICAL THINKING
Given △ABC ∼ △DEF and △DEF ∼ △JKL, is △ABC ∼ △JKL? Justify your answer.

Answer:
When △ABC is similar to △DEF and △DEF is similar to △JKL, then △ABC is similar to △JKL.

Lesson 2.7 Perimeters and Areas of Similar Figures

EXPLORATION 1

Work with a partner. Draw a rectangle in the coordinate plane.
a. Dilate your rectangle using each indicated scale factor k. Then complete the table for the perimeter P of each rectangle. Describe the pattern.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 126
b. Compare the ratios of the perimeters to the ratios of the corresponding side lengths. What do you notice?
c. Repeat part(a) to complete the table for the area A of each rectangle. Describe the pattern.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 127
d. Compare the ratios of the areas to the ratios of the corresponding side lengths. What do you notice?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 128
e. The rectangles shown are similar. You know the perimeter and the area of the red rectangle and a pair of corresponding side lengths. How can you find the perimeter of the blue rectangle? the area of the blue rectangle?
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 500

2.7 Lesson

Try It

Question 1.
The height of Figure A is 9 feet. The height of a similar Figure B is 15 feet. What is the value of the ratio of the perimeter of A to the perimeter of B?

Answer: The ratio of the perimeter of A to B is 3/5

Explanation:
We know that when two figures are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of figure B = Height of figure A/Height of figure B
Perimeter of figure A/Perimeter of figure B = 9/15 = 3/5
Thus the ratio of the perimeter of A to B is 3/5

Try It

Question 2.
The base of Triangle P is 8 meters. The base of a similar Triangle Q is 7 meters. What is the value of the ratio of the area of P to the area of Q?

Answer:
We know that when two figures are similar then the value of the ratio of their area is equal to the square of the value of the ratio of their corresponding side lengths.
The base of Triangle P is 8 meters. The base of a similar Triangle Q is 7 meters.
b1 = 8 m
b2 = 7 m
Area of triangle P/Area of triangle Q = base of triangle P/base of triangle Q

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

COMPARING PERIMETERS OF SIMILAR FIGURES
Find the value of the ratio (red to blue) of the perimeters of the similar figures.

Question 3.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 130

Answer:
We know that when two figures is similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of figure B = Side length of figure A/Side length of figure B
l1 = 9
l2 = 7
Perimeter of red figure/Perimeter of blue figure = Side length of red figure/Side length of blue figure
Perimeter of red figure/Perimeter of blue figure = 9/7
Thus the ratio of the perimeter of red to blue figure is 9/7

Question 4.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 131

Answer:
We know that when two figures is similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of figure B = Side length of figure A/Side length of figure B
b1 = 8
b2 = 10
Perimeter of red figure/Perimeter of blue figure = base length of red figure/base length of blue figure
Perimeter of red figure/Perimeter of blue figure = 8/10 = 4/5
Thus the ratio of the perimeter of red to blue triangle is 4/5

COMPARING AREAS OF SIMILAR FIGURES
Find the value of the ratio (red to blue) of the areas of the similar figures.

Question 5.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 132

Answer:
We know that when two figures is similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of figure B = (Side length of figure A/Side length of figure B)²
l1 = 12
l2 = 8
Perimeter of red figure/Perimeter of blue figure = side length of red figure/side length of blue figure
Perimeter of red figure/Perimeter of blue figure = (12/8)² = (3/2)² = 9/4
Thus the ratio of the perimeter of red to blue figure is 9/4

Question 6.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 133

Answer:
We know that when two figures is similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Area of figure A/Area of figure B = (Side length of A/Side length of B)²
l1 = 12
l2 = 8
Area of red figure/Area of blue figure = side length of red figure/side length of blue figure
Area of red figure/Area of blue figure = (4/5)² = 16/225
Thus the ratio of the perimeter of red to blue triangle is 16/225

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
Two similar triangular regions are prepared for development.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 134
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 136
a. It costs $6 per foot to install fencing. How much does it cost to surround the forest with a fence?

Answer:
Given,
It costs $6 per foot to install fencing.
Perimeter of grassland/perimeter of forest = Height of grassland/Height of forest
h1 = 60 yards
The perimeter of grassland = 240 yards
Height of forest h2 = 45 yards
Perimeter of grassland/perimeter of forest = 60/45
240/ perimeter of forest = 60/45
the perimeter of forest = 180 yards
Convert from yards to feet
180 yards = 540 feet
Thus the cost of fencing forest = 6 × 540 = $3,240

b. The cost to prepare 1 square yard of grassland is $15 and the cost to prepare 1 square yard of forest is $25. Which region costs more to prepare? Justify your answer.

Answer:
Perimeter of grassland/perimeter of forest = (Height of grassland/Height of forest)²
Height of grassland h1 = 60 yard
Height of forest h2 = 45 yards
Area of grassland = 2400 yd²
Cost to prepare 1 sq yd of grassland = $15
Cost to prepare 1 sq yd of forest = $25
Area of forest = (2400 × 9)/16
Thus the area of forest is 1350 yd²
Cost to prepare grassland = $15 × 2400 = $36,000
Cost to prepare of forest = $25 × 1350 = $33,750
Thus the grassland will cost more to prepare.

Question 8.
DIG DEEPER!
You buy a new television with a screen similar in shape to your old television screen, but with an area four times greater. The size of a television screen is often described using the distance between opposite corners of the screen. Your old television has a 30-inch screen. What is the size of your new television screen? Explain.

Answer:
Area of ΔABC/Area of ΔDEF = (Side length of AB/Side length of DE)²
Let the area of the screen of old television be x
Let the area of the screen of new television be 4x
l1 = 30 in
Area of the screen of new television/Area of the screen of new television= (distance of the screen of new television/distance of the screen of old television)²
4x/x = (distance of the screen of new television/30)²
distance of the screen of new television = 30 × 2 = 60 inch
Hence the distance of the screen of the new television is 60 inches.

Perimeters and Areas of Similar Figures Homework & Practice 2.7

Review & Refresh

The red figure is similar to the blue figure. Describe a similarity transformation between the figures.

Question 1.
Big Ideas Math Solutions Grade 8 Chapter 2 Transformations 501

Answer:
First, dilate the red figure using the scale factor of 3 because the side lengths of the blue figure are 3 times the side length of the red figure.
Scale factor = 6/2 = 3
Now reflect the image obtained after a dilation about the y-axis because both red and blue triangle is facing each other.

Question 2.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 137

Answer:
First, dilate the red figure using the scale factor of 0.5 because the side lengths of the blue figure are 3 times the side length of the red figure.
Scale factor = 2/4 = 0.5
Then rotate the image obtained after dilation in direction 90 degrees clockwise about the origin.

Find the area of the figure.

Question 3.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 138

Answer:
We know that the formula for the area of trapezoid = Base × height
h = 16 cm
b = 9 cm
Area of figure = 16 × 9 = 144 sq. cm
Hence the area of the given figure is 144 sq. cm

Question 4.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 139

Answer:
h = 5 in
b = 3 in
We know that,
A = 1/2 × b × h
A = 1/2 × 5 × 3
A = 7.5 sq. cm
Thus the area of the given figure is 7.5 sq. cm

Question 5.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 140

Answer:
h = 5 km
b1 = 6 km
b2 = 8 km
We know that,
Area of trapezoid = 1/2 × h × (b1 + b2)
A = 1/2 × 5 × 14 = 35 sq. km
Hence the area of the trapezoid is 35 sq. km

Concepts, Skills, &Problem Solving
COMPARING SIMILAR FIGURES
Dilate the figure using the indicated scale factor k. What is the value of the ratio (new to original) of the perimeters? the areas? (See Exploration 1, p. 83.)

Question 6.
a triangle with vertices (0, 0), (0, 2), and (2, 0); k = 3

Answer:
When the points of a given figure are dilated we simply multiply each x-coordinate and y-coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
where a is the scale factor
Given a triangle with vertices (0, 0), (0, 2), and (2, 0); k = 3
A(0, 0) = A'(0 . 3, 0 . 3) = A'(0, 0)
B(0, 2) = B'(0 . 3, 2 . 3) = B'(0, 6)
C(2, 0) = C'(2 . 3, 0 . 3) = C'(6, 0)
The coordinates of the image are A'(0, 0), B'(0, 6), C'(6, 0)
AB = √(2 – 0)² – (0 – 0)² = 2
A’B’ = √(6 – 0)² – (0 – 0)² = 6
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of new triangle/Perimeter of the original triangle = Distance of A’B’/Distance of AB = 6/2 = 3
Area of new triangle/Area of the original triangle = (Distance of A’B’/Distance of AB)² = (6/2)² = 3² = 9

Question 7.
a square with vertices (0, 0), (0, 4), (4, 4), and (4, 0); k = 0.5

Answer:
When the points of a given figure are dilated we simply multiply each x-coordinate and y-coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
where a is the scale factor
a square with vertices (0, 0), (0, 4), (4, 4), and (4, 0); k = 0.5
A(0, 0) = A'(0 . 0.5, 0 . 0.5) = A'(0, 0)
B(0, 4) = B'(0 . 0.5, 4 . 0.5) = B'(0, 2)
C(4, 4) = C'(4 . 0.5, 4 . 0.5) = C'(2, 2)
D(4, 0) = D'(4 . 0.5, 0 . 0.5) = D'(2, 0)
Coordinates of the image are A'(0, 0), B'(0, 2), C'(2, 2), D'(2, 0)
AB = √(4 – 0)² + (0 – 0)² = 4
A’B’ = √(2 – 0)² + (0 – 0)² = 2
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of new square/Perimeter of the original square = Distance of A’B’/Distance of AB = 2/4 = 1/2
Area of new square /Area of the original square = (Distance of A’B’/Distance of AB)² = (2/4)² = 1/4

PERIMETERS AND AREAS OF SIMILAR FIGURES
Find the values of the ratios (red to blue) of the perimeters and areas of the similar figures.

Question 8.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 141

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of red figure/Perimeter of the blue figure = Distance of red figure/Distance of blue figure= 11/6
Area of red figure /Area of the blue figure = (Distance of red figure/Distance of blue figure)² = (11/6)² = 121/36

Question 9.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 142

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of red figure/Perimeter of the blue figure = Distance of red figure/Distance of blue figure= 5/8
Area of red figure /Area of the blue figure = (Distance of red figure/Distance of blue figure)² = (5/8)² = 25/64

Question 10.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 143

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of red figure/Perimeter of the blue figure = Distance of red figure/Distance of blue figure= 4/7
Area of red figure /Area of the blue figure = (Distance of red figure/Distance of blue figure)² = (4/7)² = 16/49

Question 11.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 144

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of red figure/Perimeter of the blue figure = Distance of red figure/Distance of blue figure = 14/9
Area of red figure /Area of the blue figure = (Distance of red figure/Distance of blue figure)² = (14/9)² = 196/81

USING SIMILAR FIGURES
The figures are similar. Find x.

Question 12.
The ratio of the perimeters is 7 : 10.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 145

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of the figure B = Distance of figure A/Distance of figure B
7/10 = x/12
x = 84/10
x = 8.4
Thus the value of x is 8.4

Question 13.
The ratio of the perimeters is 8 : 5.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 146

Answer:
We know that when two figure are similar then the value of the ratio of their perimeter is equal to the value of the ratio of their corresponding side lengths.
Perimeter of figure A/Perimeter of the figure B = Distance of figure A/Distance of figure B
8/5 = x/16
x = 25.6
Thus the value of x is 25.6

Question 14.
COMPARING AREAS
The playing surfaces of two foosball tables are similar. The ratio of the corresponding side lengths is 10:7. What is the ratio of the areas?

Answer:
Area of figure A /Area of figure B = (Distance of figure A/Distance of figure B)²
Area of figure A /Area of figure B = (10/7)²
Area of figure A /Area of figure B = 100/49
Hence, the ratio of their areas is 100/49

Question 15.
CRITICAL THINKING
The ratio of the side length of Square A to the side length of Square B is 4:9. The side length of Square A is 12 yards. What is the perimeter of Square B?

Answer:
Given,
The ratio of the side length of Square A to the side length of Square B is 4/9.
The side length of Square A is 12 yards.
side length of Square A/side length of Square B = 4/9
12 /side length of Square B = 4/9
side length of Square B = 27 yards
We know that,
The perimeter of the square is = 4s
The perimeter of the square B = 4 × 27 = 108 yards

Question 16.
MODELING REAL LIFE
The cost of the piece of fabric shown is $1.31. What would you expect to pay for a similar piece of fabric that is 18 inches by 42 inches?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 147

Answer:
Given,
l = 21 in
w = 9 in
Area of the rectangle = l × w
A = 21 × 9 = 189 sq. in
The cost of the piece of fabric shown is $1.31
The cost of 1 sq. in of fabric = 1.31/189
l = 18 in
b = 42 in
Area of new fabric = 18 × 42 = 756 sq. in
Given the cost of new fabric = 1.31/189 × 756 = $5.24
Hence the cost of the new fabric is $5.24

Question 17.
PROBLEM SOLVING
A scale model of a merry-go-round and the actual merry-go-round are similar.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 148
a. How many times greater is the base area of the actual merry-go-round than the base area of the scale model? Explain.

Answer:
Radius of model merry go round = 6 in
Radius of actual merry go round = 10 ft = 120 in
Area of base of actual merry/Area of base of model merry = (Radius of actual merry/Radius of model merry)²
Area of base of actual merry/Area of base of model merry = (120/6)² = 400

b. What is the base area of the actual merry-go-round in square feet?

Answer:
The radius of model merry go round = 6 in
Radius of actual merry go round = 10 ft = 120 in
Area of base of actual merry = 450 sq. in
Area of base of actual merry/Area of base of model merry = (Radius of actual merry/Radius of model merry)²
Area of base of actual merry/450 = (120/6)² = 400
Area of base of actual merry = 400 × 450 = 180000 sq. in = 1250 ft²

Question 18.
STRUCTURE
The circumference of Circle K is π. The circumference of Circle L is 4π. What is the value of the ratio of their circumferences? of their radii? of their areas?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 149

Answer:
Given,
The circumference of Circle K is π
The circumference of Circle L is 4π.
circumference of Circle = 2πr
2πr = π
The radius of circle K r1 = 1/2
2πr = 4π
The radius of circle K r2 = 2
The ratio of their circumference = π/4π = 1/4
The ratio of radius of both circle = 1/4
The ratio of their area = π(r1)²/π(r2)² = 1/16

Question 19.
GEOMETRY
A triangle with an area of 10 square meters has a base of 4 meters. A similar triangle has an area of 90 square meters. What is the height of the larger triangle?

Answer:
Given,
A triangle with an area of 10 square meters has a base of 4 meters.
A similar triangle has an area of 90 square meters.
Area of the triangle = bh/2
h = 2a/b
h = (2 × 10)/4
h = 5 meters
Area of larger triangle/Area of smaller triangle = (height of larger triangle/height of smaller triangle)²
90/10 = (height of larger triangle/5)²
3 = (height of larger triangle/5)
Thus the height of larger triangle = 3 × 5 = 15 meters

Question 20.
PROBLEM SOLVING
You need two bottles of fertilizer to treat the flower garden shown. How many bottles do you need to treat a similar garden with a perimeter of 105 feet?
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 150

Answer:
The sides of the above figure are 4ft, 15 ft, 18 ft, 5 ft
Perimeter = 4ft + 15 ft + 18 ft + 5 ft = 42 ft
Number of bottle of fertilizer used in the above garden = 2
Number of bottle of fertilizer used in 1 ft = 2/42
Fertilizer used for 105 ft = 2/42 × 105 = 5
Thus 5 bottles will be used for 105 feet

Question 21.
REPEATED REASONING
Three square mirrors are used for a light reflection experiment. The ratio of the side length of Mirror A to the side length of Mirror B is 5 : 6. The ratio of the area of Mirror B to the area of Mirror C is 16 : 25. The perimeter of Mirror C is 280 centimeters. What is the area of Mirror A? Justify your answer.

Answer:
Given,
Three square mirrors are used for a light reflection experiment.
The ratio of the side length of Mirror A to the side length of Mirror B is 5 : 6 = 5/6
The ratio of the area of Mirror B to the area of Mirror C is 16 : 25 = 16/25
The perimeter of Mirror C is 280 centimeters
Side length of mirror C = Perimeter/4 = 280/4 = 70 cm
(Side length of mirror B/Side length of mirror C)² = Area of mirror A/Area of mirror B
(Side length of mirror B/Side length of mirror C)² = 16/25
(Side length of mirror B/Side length of mirror C) = 4/5
The side length of mirror B = 4/5 × Side length of mirror C
Side length of mirror B = 4/5 × 70 = 56 cm
A = s × s
A = 56 cm × 56 cm = 3136 sq. cm
Area of mirror A/Area of mirror B = (Side length of mirror A/Side length of mirror B)²
Area of mirror A/3136 = (5/6)²
Area of mirror A = 25/36 × 3136
Area of mirror A = 2177.7 sq. cm

Transformations Connecting Concepts

2 Connecting Concepts

Using the Problem-Solving Plan

Question 1.
A scale drawing of a helipad uses a scale of 1 ft : 20 ft. The scale drawing has an area of 6.25 square feet. What is the area of the actual helipad?
Understand the problem.
You know the scale of the drawing and the area of the helipad in the drawing. You are asked to find the area of the actual helipad.
Make a plan.
A scale drawing is similar to the actual object. So, use the scale 1 ft : 20 ft and the ratio 6.25 ft2 : A ft2 to write and solve a proportion that represents the area A of the actual helipad.
Solve and Check.
Use the plan to solve the problem. Then check your solution.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 150.1

Answer: 125 sq. ft

Question 2.
The locations of three cargo ships are shown in the coordinate plane. Each ship travels at the same speed in the same direction. After 1 hour, the x- and y-coordinates of Ship A increase 80%. Use a translation to describe the change in the locations of the ships. Then find the new coordinates of each ship.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 151
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 152

Question 3.
All circles are similar. A circle with a radius of 2 inches is dilated, resulting in a circle with a circumference of 22π inches. What is the scale factor? Justify your answer.

Answer:
Given,
A circle with a radius of 2 inches is dilated, resulting in a circle with a circumference of 22π inches.
C = 2π . r
22π = 2π . 2
p = 2π . 2
Thus the scale factor is 2.

Performance Task

Master Puppeteer

At the beginning of this chapter, you watched a STEAM Video called “Shadow Puppets.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 153

Transformations Connecting Concepts

2 Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 154

Graphic Organizers
You can use a Summary Triangle to explain a concept. Here is an example of Summary Triangle for translating a figure.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 155

Choose and complete a graphic organizer to help you study the concept.

  1. reflecting a figure
  2. rotating a figure
  3. congruent figures
  4. dilating a figure
  5. similar figures
  6. perimeters of similar figures
  7. areas of similar figures

Big Ideas Math Answers 8th Grade Chapter 2 Transformations 156

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 157

2.1 Translations (pp. 43–48)

Tell whether the blue figure is a translation of the red figure.

Question 1.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 158

Answer: No

Explanation:
The answer is no because in the case of translation the size of the figure does not change, only the position of the figure changes. But here the size of the blue figure is larger as compared to the red figure so this is not the translation. Here the blue figure is the result of the dilation of red figure.

Question 2.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 159

Answer: Yes

Explanation:
Yes, because in the case of translation the size of the figure does not change, only the position of the figure changes. Here the size of both blue figure and the red figure is the same but there is only a change in the position of the red figure to get blue figure. Here the blue figure is the result of the translation of the red figure.

Question 3.
The vertices of a quadrilateral are W(1, 2), X(1, 4), Y(4, 4), and Z(4, 2). Draw the figure and its image after a translation 3 units left and 2 units down.

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: W(1, 2), X(1, 4), Y(4, 4), and Z(4, 2) and a = -3, b = -2
W'(1+a, 2+b) = W'(1-3, 2-2) = W'(-2,0)
X'(1+a, 4+b) = X'(1-3, 4-2) = X'(-2,2)
Y'(4+a, 4+b) = Y'(4-3, 4-2) = Y'(1, 2)
Z'(4+a, 2+b) = C'(4-3, 2-2) = Z'(1,0)
Hence the coordinate of image are W'(-2,0), X'(-2,2), Y'(1, 2), Z'(1,0)
big ideas math answers grade 8 chapter 2 img_41

Question 4.
The vertices of a triangle are A(-1, -2), B(-2, 2), and C(-3, 0). Draw the figure and its image after a translation 5 units right and 1 unit up.

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: A(-1, -2), B(-2, 2), and C(-3, 0) and a = 5, b = 1
A'(-1+a, -2+b) = A'(-1-5, -2+1) = A'(4,-1)
B'(-2+a, 2+b) = B'(-2+5, 2+1) = B'(3,3)
C'(-3+a, 0+b) = C'(-3+5, 0+1) = C'(2, 1)
Hence the coordinate of image are A'(4,-1), B'(3,3), C'(2, 1)
BIM Grade 8 Chapter 2 Answer Key img_42

Question 5.
Your locker number is 20 and your friend’s locker number is 33. Describe the location of your friend’s locker relative to the location of your locker.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 159.1

Answer:
The location of my friend’s locker is first 1 locker Down and then 3 locker Right.

Question 6.
Translate the triangle 4 units left and 1 unit down. What are the coordinates of the image?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 160

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: A(3, 5), B(6, 3), and C(4, 1) and a = -4, b = -1
A'(3+a, 5+b) = A'(3-4, 5-1) = A'(-1,4)
B'(6+a, 3+b) = B'(6-4, 3-1) = B'(2,2)
C'(4+a, 1+b) = C'(4-4, 1-1) = C'(0, 0)
Hence the coordinate of image are A'(-1,4), B'(2,2), C'(0, 0)
BIM 8th Grade Answer Key Chapter 2 img_42

Question 7.
Describe a translation of the airplane from point A to point B.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 161

Answer:
First, move the aeroplane 6 units right from point A and then 4 units down.

2.2 Reflections (pp. 49 – 54)

Tell whether the blue figure is a reflection of the red figure.

Question 8.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 162

Answer:
No, because in the above figure the plane of reflection is inclined at 45 degrees with the horizontal line. So, the reflected figure will be perpendicular to the original figure. But in the given figure both are facing each other. This means that the blue figure is not the reflection of red figure.

Question 9.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 163

Answer:
The answer is no because the blue figure is not the mirror image of the red figure. The side of the red figure is not facing the side of the blue figure this means that the blue figure is not the reflection of red figure.

Question 10.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 163.1

Answer:
The answer is yes because the blue figure is the mirror image of the red figure. The side of the red figure is facing the side of the blue figure this means that the blue figure is the reflection of red figure.

Draw the figure and its reflection in (a) the x-axis and (b) the y-axis. Identify the coordinates of the image.

Question 11.
A(2, 0), B(1, 5), C(4, 3)

Answer:
A(x, y) = A'(x, -y)
Given: A(2, 0), B(1, 5), C(4, 3)
Reflection about x-axis:
A(2, 0) = A'(2, 0)
B(1, 5) = B'(1, -5)
C(4, 3) = C'(4, -3)
BIM Grade 8 Answers Chapter 2 img_43
Reflection about y-axis:
A(x, y) = A'(-x, y)
A(2, 0) = A'(-2, 0)
B(1, 5) = B'(-1, 5)
C(4, 3) = C'(-4, 3)
BIM Grade 8 Answers Chapter 2 img_44

Question 12.
D(-5, -5), E(-5, 0), F(-2, -2), G(-2, -5)

Answer:
Given, D(-5, -5), E(-5, 0), F(-2, -2), G(-2, -5)
Reflection about x-axis:
A(x, y) = A'(x, -y)
D(-5, -5) = D'(-5, 5)
E(-5, 0) = E'(-5, 0)
F(-2, -2) = F'(-2, 2)
G(-2, -5) = G'(-2, 5)
BIM Grade 8 Answers Chapter 2 img_45
Reflection about y-axis:
A(x, y) = A'(-x, y)
D(-5, -5) = D(5, -5)
E(-5, 0) = E'(5, 0)
F(-2, -2) = F'(2, -2)
G(-2, -5) = G'(2, -5)
BIM Grade 8 Answers Chapter 2 img_46

Question 13.
The vertices of a rectangle are E(-1, 1), F(-1, 3), G(-5, 3), and H(-5, 1). Find the coordinates of the figure after reflecting in the x-axis, and then translating 3 units right.

Answer:
We know that when a point is reflected about x-axis then y-coordinate becomes the opposite.
A(x, y) = A'(x, -y)
The vertices of a rectangle are E(-1, 1), F(-1, 3), G(-5, 3), and H(-5, 1).
Reflection about x-axis:
E(-1, 1) = E'(-1, -1)
F(-1, 3) = F'(-1, -3)
G(-5, 3) = G'(-5, 3)
H(-5, 1) = H'(-5, 1)
Thus the coordinates of the image are E'(-1, -1), F'(-1, -3), G'(-5, 3), H'(-5, 1)
Now translating the image 3 units Right.
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if shift is Right and Vertical Up and the value of a and b will be negative if shift is left and vertical Down.
Given: E'(-1, -1), F'(-1, -3), G'(-5, 3), H'(-5, 1) a = 3, b = 0
E”(-1 + a, -1 + b) = E”(-1 + 3, -1 + 0) = E”(2, -1)
F”(-1 + a, -3 + b) = F”(-1 + 3, -3 + 0) = F”(2, -3)
G”(-5 + a, 3 + b) = G”(-5 + 3, 3 + 0) = G”(-2, 3)
H”(-5 + a, 1 + b) = H”(-5 + 3, 1 + 0) = H”(-2, 1)
Thus the coordinates of the image are E”(2, -1), F”(2, -3), G”(-2, 3), H”(-2, 1)

The coordinates of a point and its image after a reflection are given. Identify the line of reflection.

Question 14.
(-1, -3) → (1, -3)

Answer:
Given,
(-1, -3) → (1, -3)
We can see that the y-coordinate of both points and its image are the same but the x-coordinate of the image is the opposite of its points.
Hence, Y-axis is the line of reflection.

Question 15.
(2, 1) → (2, -1)

Answer:
Given,
(2, 1) → (2, -1)
We can see that the x-coordinate of both points and its image are the same but the y-coordinate of the image is the opposite of its points.
Hence, X-axis is the line of reflection.

Question 16.
You perform an experiment involving angles of refraction with a laser pen. You point a laser pen from point L at a mirror along the red path and the image is a reflection in the y-axis.
a. Does the light reach a cat at point C? Explain.

Answer:
Yes, the light will reach at point C.
Because the coordinate of point L is (4, 3) and the coordinate of point C is (-4, 3) and it is given problems that laser is reflected about the y-axis.
So when point L(4, 3) is reflected about the y-axis its x-coordinates become opposite and y-coordinates remain the same.
So when point L(4, 3) is reflected about the y-axis its image will be point C(-4, -3)

b. You bounce the light off the mirror so its path is a reflection. What line of reflection is needed for the light to reach the cat?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 164

Answer: The line of reflection will be y-axis.

2.3 Rotations (pp. 55–62)

Tell whether the blue figure is a rotation of the red figure about the origin. If so, give the angle and the direction of rotation.

Question 17.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 165

Answer:
The answer is no because the blue figure is the mirror image of the red figure. The blue figure is the result of the reflection of red figure about the y-axis. Also, both red and blue figure are facing each other with the y-axis in the center of both which remains that it is not the case of rotation.

Question 18.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 166

Answer:
The answer is yes, because the red figure is in the 1st quadrant and the blue figure in third quadrant. Also, both blue and red figure are facing each other in opposite directions which blue figure is the result of the rotation of red figure.
When red figure is rotated 180 degrees counterclockwise it will result in a blue figure.

The vertices of a triangle are A(-4, 2), B(-2, 2), and (-3, 4). Rotate the triangle as described. Find the coordinates of the image.

Question 19.
180° about the origin

Answer:
We know that when a point is rotated 180 degrees about origin then both x coordinate and y coordinate becomes opposite.
A(x, y) = A'(-x, -y)
Given points: A(-4, 2), B(-2, 2), and C(-3, 4)
Rotated 180 degrees about origin:
A(-4, 2) = A'(4, -2)
B(-2, 2) = B'(2, -2)
C(-3, 4) = C'(3, -4)
The coordinate of the image are A'(4, -2), B'(2, -2), C'(3, -4)

Question 20.
270° clockwise about the origin

Answer:
We know that when a point is rotated 90 degrees counterclockwise about origin then both x coordinate and y coordinate becomes opposite.
P(x, y) = P'(-y, x)
Given points: A(-4, 2), B(-2, 2), and C(-3, 4)
Rotating 270 degrees clockwise about the origin:
A(-4, 2) = A'(-2, -4)
B(-2, 2) = B'(-2, -2)
C(-3, 4) = C'(-4, -3)
The coordinate of the image are A'(-2, -4), B'(-2, -2), C'(-4, -3)

Question 21.
A bicycle wheel is represented in a coordinate plane with the center of the wheel at the origin. Reflectors are placed on the bicycle wheel at points (7, 4) and (-5, -6). After a bike ride, the reflectors have rotated 90° counterclockwise about the origin. What are the locations of the reflectors at the end of the bike ride?

Answer:
We know that when a point is rotated 90 degrees counterclockwise about origin then both x coordinate and y coordinate becomes opposite.
P(x, y) = P'(-y, x)
Reflectors are placed on the bicycle wheel at points (7, 4) and (-5, -6)
A(7, 4) = A'(-4, 7)
B(-5, -6) = B'(6, -5)
Hence the new coordinate of the reflector are A'(-4, 7), B'(6, -5)

2.4 Congruent Figures (pp. 63–68)

Identify any congruent figures in the coordinate plane.

Question 22.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 167

Answer:
AB = ED
BC = DC
CA = CE
∠A = ∠E
∠B = ∠D
∠C = ∠C
When we see both triangles ABC and EDC closely we observe that both the triangles are the mirror images of each other with the y-axis as the line of reflection. So all the corresponding sides are equal and also all the corresponding angles, this means that both the triangle are congruent.
Hence ΔABC is congruent to ΔEDC
GH = JK
HF = KI
FG = IJ
∠G = ∠J
∠H = ∠K
∠F = ∠I
So we can see that all the corresponding sides are equal and also all the corresponding angles. this means that both the triangles are congruent.
Hence ΔGHF is congruent to ΔJKI

Question 23.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 168

Answer:
When we observe square ABCD and square EFGH we can see that
AB = EF
BC = FG
CD = GH
DA = HE
∠A = ∠E
∠B = ∠F
∠C = ∠G
∠D = ∠H
We can see that all the corresponding sides are equal and also all the corresponding angles, this means that both are congruent.
RS = IJ
ST = JK
TU = KL
UR = LI
∠R = ∠I
∠S = ∠J
∠T = ∠K
∠L = ∠U
Hence rectangle RSTU is congruent to rectangle IJKL

The red figure is congruent to the blue figure. Describe a sequence of rigid motions between the figures.

Question 24.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 169

Answer:
First, rotate the blue figure 90° clockwise because the blue figure is in a vertical position but the red figure is in the horizontal position.
Translate the image 5 units right because the first image formed after the rotation will in the second quadrant but the red figure in the first quadrant.

Question 25.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 170

Answer:
First, reflect the blue figure about the y-axis because both red and blue figure is facing each other and they are the mirror image of each other.
Then translate the image 2 units up because the first image formed after reflection will be at the same distance from the x-axis but the red figure is touchung the x-axis.

Question 26.
The figures are congruent. Name the corresponding angles and the corresponding sides.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 171

Answer:
Corresponding sides are
AB = KL
BC = LM
CA = MK
Corresponding angles
∠A = ∠K
∠B = ∠L
∠C = ∠M

Question 27.
Trapezoids EFGH and QRST are congruent.
a. What is the length of side QR ?

Answer:
Length of side:
QR = EF = 3 feet

b. Which angle in QRST corresponds to ∠H?
Answer:
The angle that corresponds to ∠H is ∠T

c. What is the perimeter of QRST ?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 172

Answer:
Perimeter of QSRT = QR + RS + ST + TQ
= EF + FG + GH + HE
= 3 + 5 + 4 + 8
= 20 ft

2.5 Dilations (pp. 69–76)

Tell whether the blue figure is a dilation of the red figure.

Question 28.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 173

Answer:
The answer is no because dilation the size of the image either increases or decreases that depend on the type of dilation. We can see that both the red and blue figure are of the same size which means that the blue figure is not dilation of the red figure.

Question 29.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 174

Answer:
The answer is yes because both red and blue figures are of the same shape and all the corresponding angles are equal but the blue figure is larger than the red figure. And in dilation, the size of the image is either increases or decreases but the shape always remains the same. So blue figure is the result of dilation of red figure.

The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation.

Question 30.
P(-3, -2), Q(-3, 0), R(0, 0); k = 4

Answer:
A(x, y) = A'(x . a, y . a)
Where a is the scaling factor
Given,
P(-3, -2), Q(-3, 0), R(0, 0); k = 4
P(-3, -2) = P'(-3 . 4, -2 . 4) = P'(-12, -8)
Q(-3, 0) = Q'(-3 . 4, 0 . 4) = Q'(-12, 0)
R(0, 0) = R'(0 . 4, 0 . 4) = R'(0, 0)
Thus the coordinates of the image: P'(-12, -8), Q'(-12, 0), R'(0, 0)
Bigideas Math Answer Key for Grade 8 Chapter 2 img_45

Question 31.
B(3, 3), C(3, 6), D(6, 6), E(6, 3); k = \(\frac{1}{3}\)

Answer:
A(x, y) = A'(x . a, y . a)
Where a is the scaling factor
Given,
B(3, 3), C(3, 6), D(6, 6), E(6, 3); k = \(\frac{1}{3}\)
B(3, 3) = B'(3 . \(\frac{1}{3}\), 3 . \(\frac{1}{3}\)) = B'(1, 1)
C(3, 6) = C'(3 . \(\frac{1}{3}\), 6 . \(\frac{1}{3}\)) = C'(1, 2)
D(6, 6) = D'(6 . \(\frac{1}{3}\), 6 . \(\frac{1}{3}\)) = D'(2, 2)
E(6, 3) = E'(6 . \(\frac{1}{3}\), 3 . \(\frac{1}{3}\)) = E'(2, 1)
Thus the coordinates of the image: B'(1, 1), C'(1, 2), D'(2, 2), E'(2, 1)
BIm Grade 8 Chapter 2 Answers img_46

Question 32.
The blue figure is a dilation of the red figure. Identify the type of dilation and find the scale factor.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 175

Answer:
AB = 1 unit
A’B’ = 2 units
scale factor = size of image figure/size of actual figure
scale factor = 2/1 = 2
We can see from the above figure that the size of the image figure are larger as compared to the size of the original image so it is the Enlargement dilation.

Question 33.
The vertices of a rectangle are Q(-6, 2), R(6, 2), S(6, -4), and T(-6, -4). Dilate the rectangle with respect to the origin using a scale factor of \(\frac{3}{2}\). Then translate it 5 units right and 1 unit down. What are the coordinates of the image?

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the rectangle: Q(-6, 2), R(6, 2), S(6, -4), and T(-6, -4), scale factor = \(\frac{3}{2}\)
Q(-6, 2) = Q'(-6 . \(\frac{3}{2}\), 2 . \(\frac{3}{2}\)) = Q'(-9, 3)
R(6, 2) = R'(6 . \(\frac{3}{2}\), 2 . \(\frac{3}{2}\)) = R'(9, 3)
S(6, -4) = S'(6 . \(\frac{3}{2}\), -4 . \(\frac{3}{2}\)) = S'(9, 6)
T(-6, -4) = T'(-6 . \(\frac{3}{2}\), -4 . \(\frac{3}{2}\)) = T'(-9, -6)
Thus the coordinates of the image: Q'(-9, 3), R'(9, 3), S'(9, 6), T'(-9, -6)
Q'(-9, 3), R'(9, 3), S'(9, 6), T'(-9, -6) a = 5, b = -1
Q”(-9 + a, 3 + b) = Q”(-9 + 5, 3 – 1) = Q”(-4, 2)
R”(9 + a, 3 + b) = R”(9 + 5, 3 – 1) = R”(14, 2)
S”(9 + a, -6 + b) = S”(9 + 5, -6 – 1) = S”(14, -7)
T”(-9 + a, -6 + b) = T”(-9 + 5, -6 – 1) = T”(-4, -7)
Thus the coordinates of the image: Q”(-4, 2), R”(14, 2), S”(14, -7), T”(-4, -7)

2.6 Similar Figures

Question 34.
Determine whether the two figures are similar. Explain your reasoning.

Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 175.1

Answer:
No, the above two figures are not similar.

Question 35.
Draw figures with the given vertices in a coordinate plane. Which figures are similar? Explain your reasoning.
Triangle A: (-4, 4), (-2, 4), (-2, 0)
Triangle B: (-2, 2), (-1, 2), (-1, 0)
Triangle C: (6, 6), (3, 6), (3, 0)

Answer:
Triangle A: (-4, 4), (-2, 4), (-2, 0)
BIM Grade 8 Solution Key Ch 2 img_48
Triangle B: (-2, 2), (-1, 2), (-1, 0)
Bigideas Math Answers Grade 8 Ch 2 img_49
Triangle C: (6, 6), (3, 6), (3, 0)
BIM Answers for Grade 8 Chapter 2 img_50

The figures are similar. Find x.

Question 36.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 177

Answer:
Ratio of sides of larger triangle = Ratio of sides of smaller triangle
20/14 = x/7
x = 10
Thus the value of x is 7 inches.

Question 37.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 178

Answer:
Ratio of sides of larger parallelogram= Ratio of sides of smaller parallelogram
x/6 = 6/4
x = 9
Thus the value of x is 9 cm

2.7 Perimeters and Areas of Similar Figures (pp. 83-88)

Find the values of the ratios (red to blue) of the perimeters and areas of the similar figures.

Question 38.
Big Ideas Math Answers 8th Grade Chapter 2 Transformations 179

Answer:
Perimeter of figure A/Perimeter of figure B = Side length of figure A/Side length of figure B
Perimeter of red figure/Perimeter of blue figure = 6/8 = 3/4
Hence the ratio of perimeter of red triangle to blue is 3/4
Area of figure A/Area of figure B = (Side length of A/Side length of B)²
Area of red figure/Area of blue figure = (6/8)² = 9/16
Thus the ratio of the perimeter of the red to the blue triangle is 9/16

Question 39.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 180

Answer:
Perimeter of figure A/Perimeter of figure B = Side length of figure A/Side length of figure B
Perimeter of red figure/Perimeter of blue figure = 28/16 = 7/4
Hence the ratio of the perimeter of red rectangle to blue is 7/4
Area of figure A/Area of figure B = (Side length of A/Side length of B)²
Area of red figure/Area of blue figure = (28/16)² = 49/16
Thus the ratio of the perimeter of the red to the blue rectangle is 49/16

The figures are similar. Find x.

Question 40.
The ratio of the perimeters is 5 :7.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 181

Answer:
5/7 = 12/x
x =(12 × 7)/5
x = 16.8 cm
Thus the value of x is 16.8 cm

Question 41.
The ratio of the perimeters is 6 : 5.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 182

Answer:
6/5 = x/6
x = 36/5
x = 7.2
Thus the value of x is 7.2

Question 42.
Two photos are similar. The ratio of the corresponding side lengths is 3 : 4. What is the ratio of the areas?

Answer:
Area of photo A/Area of photo B = (Side length of photo A/Side length of photo B)²
Area of photo A/Area of photo B = (3/4)² = 9/16
Thus the ratio of the area of two photos is 9/16

Question 43.
The ratio of side lengths of Square A to Square B is 2 : 3. The perimeter of Square A is 16 inches. What is the area of Square B?

Answer:
Given,
The ratio of side lengths of Square A to Square B is 2 : 3.
The perimeter of Square A is 16 inches.
Perimeter of Square A/Perimeter of Square B = Side length of Square A/Side length of Square B
The perimeter of Square B = (16 × 3)/2
The perimeter of Square B = 24 inches
Side length of square B = Perimeter/4 = 24/4 = 6 inch
Area of square B = s × s = 6 × 6 = 36 sq. in

Question 44.
The TV screen is similar to the computer screen. What is the area of the TV screen?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 183

Answer:
Area of computer screen = 108 sq. in
The side length of the computer screen = 12 in
The side length of the TV screen = 20 in
Area of TV screen = (25 × 108)/9
Area of TV screen = 300 sq. in
Hence the area of the TV screen is 300 sq. in

Transformations Practice Test

2 Practice Test

Triangles ABC and DEF are congruent.

Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 183.1

Question 1.
Which angle of DEF corresponds to ∠C ?

Answer:
Corresponding sides
Side AB = Side DE
Side BC = Side EF
Side CA = Side ED
Corresponding angles
∠A = ∠D
∠B = ∠E
∠C = ∠F
Thus the angle correspond to ∠C is ∠F

Question 2.
What is the perimeter of DEF ?

Answer:
Corresponding sides
Side AB = Side DE
Side BC = Side EF
Side CA = Side ED
Corresponding angles
∠A = ∠D
∠B = ∠E
∠C = ∠F
Perimeter of DEF = DE + EF + FD
= AB + BC + CA
= 5 + 4 + 6
= 15 cm
Thus the perimeter of the ΔDEF is 15 cm

Tell whether the blue figure is a translation, reflection, rotation, or dilation of the red figure.

Question 3.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 184

Answer: The scale factor of a dilation is greater than 1 because the shape of the blue figure is larger than the red figure.

Question 4.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 185

Answer:
The blue figure is the reflection of the red figure. Because the blue figure is the mirror image of red figure. Also, the shape and size of both red and blue figures are the same.

Question 5.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 186

Answer:
The blue figure is the translation of the red figure. Because the shape and size of both red and blue figures are the same. And both the figure is not facing to each other. This means that the blue figure is the result of the translation of the red figure.

Question 6.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 187

Answer:
The blue figure is the result of the rotation of the red figure. Because the shape and size of both the red and blue figure are the same but the figure is horizontal and the blue figure is vertical. This means that the blue figure is the result of the rotation of the term figure.

The vertices of a triangle are A(2, 5), B(1, 2), and C(3, 1). Find the coordinates of the image after the transformations given.

Question 7.
Reflect in the y-axis.

Answer:
A(x, y) = A'(-x, y)
A(2, 5), B(1, 2), C(3, 1)
Reflection about the y-axis:
A(2, 5) = A'(-2, 5)
B(1, 2) = B'(-1, 2)
C(3, 1) = C'(-3, 1)
Thus the coordinates of the image: A'(-2, 5), B'(-1, 2), C'(-3, 1)

Question 8.
Rotate 90° clockwise about the origin.

Answer:
A(x, y) = A'(x, -y)
Given, A(2, 5), B(1, 2), C(3, 1)
A(2, 5) = A'(2, -5)
B(1, 2) = B'(1, -2)
C(3, 1) = C'(3, -1)
Thus the coordinates of the image: A'(2, -5), B'(1, -2), C'(3, -1)

Question 9.
Reflect in the x-axis, and then rotate 90° counterclockwise about the origin.

Answer:
A(x, y) = A'(x, -y)
Given, A(2, 5), B(1, 2), C(3, 1)
A(2, 5) = A'(2, -5)
B(1, 2) = B'(1, -2)
C(3, 1) = C'(3, -1)
Thus the coordinates of the image: A'(2, -5), B'(1, -2), C'(3, -1)
A'(2, -5) = A”(2, 5)
B'(1, -2) = B”(1, 2)
C'(3, -1) = C”(3, 1)
Thus the coordinates of the image: A”(2, 5), B”(1, 2), C”(3, 1)

Question 10.
Dilate with respect to the origin using a scale factor of 2. Then translate 2 units left and 1 unit up.

Answer:
When the points of given figure are dilated we simply multiply each x and y coordinate by the given scale factor.
P(x, y) = P'(x . a, y . a)
Given points of the triangle: A(2, 5), B(1, 2), C(3, 1) and scale factor = 2
Dilating the figure by scale factor of 2
A (2, 5) = A'(2 . 2, 5 . 2) = A'(4, 10)
B (1, 2) = B'(1 . 2, 2 . 2) = B'(2, 4)
C (3, 1) = C'(3 . 2, 1 . 2) =  C'(6, 2)
Hence the coordinates of the image are A'(4, 10), B'(2, 4),  C'(6, 2)
Now translating image 2 unit left and 1 unit up.
Given: A'(4, 10), B'(2, 4),  C'(6, 2) a = -2, b = 1
A”(4 + a, 10 + b) = A”(4 – 2, 10 + 1) = A”(2, 11)
B”(2 + a, 4 + b) = B”(2 – 2, 4 + 1) = B”(0, 5)
C”(6 + a, 2 + b) = C”(6 – 2, 2 + 1) = C”(4, 3)
Hence the coordinates of the image are A”(2, 11), B”(0, 5), C”(4, 3)

Question 11.
In a coordinate plane, draw Rectangle A: (-4, 4), (0, 4), (0, 2), (-4, 2); Rectangle B: (-2, 2), (0, 2), (0, 1), (-2, 1); and Rectangle C:(-6, 6), (0, 6), (0, 3), (-6, 3). Which figures are similar? Explain your reasoning.

Answer:
Rectangle A: (-4, 4), (0, 4), (0, 2), (-4, 2)
BIM Grade 8 Chapter 2 Solutions img_51
Rectangle B: (-2, 2), (0, 2), (0, 1), (-2, 1)
BIM Grade 8 Chapter 2 Solutions img_52
Rectangle C:(-6, 6), (0, 6), (0, 3), (-6, 3)
BIM Grade 8 Chapter 2 Solutions img_53

Question 12.
Translate a point (x, y) 3 units left and 5 units up. Then translate the image 5 units right and 2 units up. What are the coordinates of the point after the translations?

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if the shift is Right and Vertical Up and the value of a and b will be negative if the shift is left and vertical Down.
Given: A(x,y) and a = -3, b = 5
A'(x+a, y+b) = A'(x – 3, y + 5)
Now translating image 5 units right and 2 units up.
Image after first translation: A'(x – 3, y + 5) and a = 5, b = 2
A”(x – 3 + a, y + 5 + b) = A”(x – 3 + 5, y + 5 + 2) = A”(x + 2, y + 7)
Thus the final image will be A”(x + 2, y + 7)

Question 13.
The two figures are similar.
(a) Find the value of x.

Answer:
Ratio of sides of red figure = Ratio of sides of blue figure
x/14 = 10/8
x = (10 × 14)/8
x = 17.5

(b) Find the values of the ratios (red to blue) of the perimeters and of the areas.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 188

Answer:
Perimeter of red figure/Perimeter of blue figure = Side length of red figure/Side length of blue figure
Perimeter of red figure/Perimeter of blue figure = 14/8 = 7/4
Thus the ratio of the perimeter of red to blue figure is 7/4
Area of red figure/Area of blue figure = (side length of red figure/side length of blue figure)²
Area of red figure/Area of blue figure = (14/8)² = 49/16
Thus the ratio of the perimeter of red to blue triangle is 49/16

Question 14.
A wide-screen television measures 36 inches by 54 inches. A movie theater screen measures 42 feet by 63 feet. Are the screens similar? Explain.

Answer:
Given,
A wide-screen television measures 36/54 = 2/3
A movie theater screen measures 42/63 = 2/3
We can see that the ratio of corresponding sides of the television screen is equal to the ratio of corresponding sides of the movie theatre. So television screens and movie theatres are similar.

Question 15.
You want to use the rectangular piece of fabric shown to make a pair of curtains for your window. Name the types of congruent shapes you can make with one straight cut. Draw an example of each type.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 189

Answer:
The types of congruent shapes that can be made with one straight cut
2 right triangles
2 rectangles
2 right trapezoid

Transformations Cumulative Practice

Cumulative Practice

Question 1.
A clockwise rotation of 90° is equivalent to a counterclockwise rotation of how many degrees?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 190

Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 191

Answer:
90° of clockwise rotation = (360 – 90)° of counterclockwise rotation
= 270° of counterclockwise rotation

Question 2.
The formula K = C + 273.15 converts temperatures from degrees Celsius C to Kelvin K. Which of the following formulas is not correct?
A. K – C = 273.
B. C = K – 273.15
C. C – K = -273.15
D. C = K + 273.15

Answer: C = K + 273.15

Question 3.
You want to solve the equation -3(x + 2) = 12x. What should you do first?
F. Subtract 2 from each side.
G. Add 3 to each side.
H. Multiply each side by -3.
I. Divide each side by -3.

Answer: I. Divide each side by -3.

Explanation:
-3(x + 2) = 12x
x + 2 = -4x
x = -4x – 2
x + 4x = -2
5x = -2
x = –\(\frac{2}{5}\)
Thus the correct answer is option I.

Question 4.
Which value of x makes the equation \(\frac{3}{4} x\) = 12 true?
A. 9
B. 11\(\frac{1}{4}\)
C. 16
D. 48

Answer: C. 16

Explanation:
\(\frac{3}{4} x\) = 12
3x = 12 × 4
3x = 48
x = \(\frac{48}{3}\)
x = 16
Thus the correct answer is option C.

Question 5.
A triangle is graphed in the coordinate plane. What are the coordinates of the image after a translation 3 units right and 2 units down?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 191.1
F. A'(1, 4), B'(1, 1), C'(3, 1)
G. A'(1, 2), B'(1, -1), C'(3, -1)
H. A'(-2, 2), B'(-2, -1), C'(0, -1)
I. A'(0, 1), B'(0, -2), C'(2, -2)

Answer:
We know that to translate a figure ‘a’ units horizontally and ‘b’ units vertically in the coordinate plane, ‘a’ is added to x-coordinate and ‘b’ is added to the y-coordinate of the vertices.
A(x,y) = A'(x+a, y+b)
The value a and b will be positive if the shift is Right and Vertical Up and the value of a and b will be negative if the shift is left and vertical Down.
A(-2, 4), B(-2, 1), C(0, 1) and a = 3, b = -2
A'(-2+a, 4+b) = A'(-2 + 3, 4 – 2) = A'(1, 2)
B'(-2+a, 1+b) = B'(-2 + 3, 1 – 2) = B'(1,-1)
C'(0+a, 1+b) = C'(0 + 3, 1 – 2) = C'(3, -1)
Coordinate of the image are: A'(1, 2), B'(1,-1), C'(3, -1)
Thus the correct answer is option G.

Question 6.
Your friend solved the equation in the box shown. What should your friend do to correct the error that he made?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 192
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 193

Answer:
–\(\frac{x}{3}\) + \(\frac{2}{5}\) = –\(\frac{7}{15}\)
–\(\frac{x}{3}\) = –\(\frac{13}{15}\)
x = 2\(\frac{3}{5}\)
Thus the correct answer is option C.

Question 7.
Your teacher dilates the rectangle using a scale factor of \(\frac{1}{2}\).
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 194
What is the area of the dilated rectangle in square inches?

Answer:
l = 10 in
b = 6 in
scale factor = \(\frac{1}{2}\)
New length after dilation = 10 × \(\frac{1}{2}\) = 5
New breadth after dilation = 6 × \(\frac{1}{2}\) = 3
Area of rectangle = l × b
A = 5 × 3 = 15 sq. in
The area of the dilated rectangle will be 5 in²

Question 8.
Your cousin earns $9.25 an hour at work. Last week she earned $222.00 How many hours did she work last week?
F. \(\frac{1}{24}\)
G. 22 hours
H. 24 hours
I. 212.75 hours

Answer: H. 24 hours

Explanation:
Given,
Your cousin earns $9.25 an hour at work.
Last week she earned $222.00
Total no. of working hour = total earning of week/rate of one hour
= \(\frac{222}{9.25}\)
= 24 hours
Thus the correct answer is option H.

Question 9.
Triangle EFG is a dilation of Triangle HIJ. Which proportion is not true for Triangle EFG and Triangle HIJ ?
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 195
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 196

Answer: \(\frac{EG}{HI}\) = \(\frac{FG}{IJ}\)
The correct answer is option B.

Question 10.
The red figure is congruent to the blue figure. Which of the following is a sequence of rigid motions between the figures?
F. Reflect the red triangle in the x-axis, and then translate it 3 units left.
G. Reflect the red triangle in the x-axis, and then translate it 3 units right.
H. Reflect the red triangle in the y-axis, and then translate it 3 units left.
I. Rotate the red triangle 90° clockwise about the origin.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 197

Answer:
1. First red triangle is reflected about the x-axis because both red and blue triangles are the mirror image of each other and also the red triangle is in the 1st quadrant and the blue triangle is in 4th quadrant.
2. Then translate the image 3 unit left because the base of both red and blue triangles is not opposite to each other.
Thus the correct answer is option F.

Question 11.
Several transformations are used to create the pattern.
Part A
Describe the transformation of Triangle GLM to Triangle DGH

Answer:
Both ΔGLM and Δ DGH are of the same shape and size but their position are different so the transformation will be translated.

Part B
Describe the transformation of Triangle ALQ to Triangle GLM.

Answer:
The size of the triangle ALQ is four times the size of triangle GLM and the shape of both triangles is the same so the transformation will be dilation.

Part C
Triangle DFN is a dilation of Triangle GHM. Find the scale factor.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 198
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 198.1

Answer:
The size of the triangle DFN is double the size of triangle GHM. So the scale factor of dilation will be 2.

Question 12.
A rectangle is graphed in the coordinate plane.
Big Ideas Math Answer Key Grade 8 Chapter 2 Transformations 199
What are the coordinates of the image after a reflection in the y-axis?
A. J'(4, 1), K'(4, 3), L'( 1, 3), M'(-1, 1)
B. J'(-4, 1), K(-4, -3), L'(1, -3), M'(1, 1)
C. J'(1, 4), K'(3, 4), L'(3, -1), M'(1, -1)
D. J'(-4, 1), K'(-4, 3), L'(1, 3), M'(1, 1)

Answer:
We know that when a point is reflected about the y-axis then the x-coordinate becomes the opposite.
A(x, y) = A'(-x, y)
J(-4, 1), K(-4, 3), L(1, 3), M(1, 1)
Reflection about the y-axis:
J(-4, 1) = J'(4, 1)
K(-4, 3) = K'(4, 3)
L(1, 3) = L'(-1, 3)
M(1, 1) = M'(-1, 1)
Coordinate of image are: J'(4, 1), K'(4, 3), L'(-1, 3), M'(-1, 1)
Thus the correct answer is option A.

Conclusion:

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Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations

Big Ideas Math Answers Grade 8 Chapter 4

Are you searching everywhere regarding the Big Ideas Math 8th Grade Answer Key Chapter 4 Graphing and Writing Linear Equations? If so, halt your search as this is the one-stop destination for all your needs. Practice using the Graphing and Writing Linear Equations Big Ideas Math Grade 8 Answers and understand the concepts easily. Begin your preparation right away and seek the homework help needed right after class in a matter of seconds.

Big Ideas Math Book 8th Grade Answer Key Chapter 4 Graphing and Writing Linear Equations

Make the most out of the handy resources available for Big Ideas Math Ch 4 Graphing and Writing Linear Equations and stand out from the rest of the crowd. BIM Book 8th Grade Chapter 4 Solutions include questions belonging to Lessons 4.1 to 4.7, Cumulative Practice, Assessment Tests, Review Tests, etc. Big Ideas Math 8th Grade Chapter 4 Solution Key is given by subject experts after extensive research. Access the quick links over here during your preparation and get the assistance needed at the comfort of your home.

Performance Task

Lesson: 1 Graphing Linear Equations

Lesson: 2 Slope of a Line

Lesson: 3 Graphing Proportional Relationships

Lesson: 4 Graphing Linear Equations in Slope-Intercept Form

Lesson: 5 Graphing Linear Equations in Standard Form

Lesson: 6 Writing Equations in Slope-Intercept Form

Lesson: 7 Writing Equations in Point-Slope Form

Chapter: 4 – Graphing and Writing Linear Equations

Graphing and Writing Linear Equations STEAM Video/Performance Task

STEAM Video

“Hurricane
A hurricane is a storm with violent winds. How can you prepare your home for a hurricane?
Watch the STEAM Video “Hurricane!” Then answer the following questions.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 1
1. Robert says that the closer you are to the eye of a hurricane, the stronger the winds become. The wind speed on an island is 50 miles per hour when the eye of a hurricane is 140 miles away.
a. Describe the wind speed on the island when the eye of the hurricane is 100 miles away.
b. Describe the distance of the island from the eye of the hurricane when the wind speed on the island is 25 miles per hour.
c. Sketch a line that could represent the wind speed y (in miles per hour) on the island when the eye of x the hurricane is miles away from the island. Wind speed
2. A storm dissipates as it travels over land. What does this mean?

Performance Task

Anatomy of a Hurricane
After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given information about the atmospheric pressure inside a hurricane.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 2
You will be asked to use a model to find the strength of a hurricane after x hours of monitoring. Why is it helpful to predict how strong the winds of a hurricane will become?

Graphing and Writing Linear Equations Getting Ready for Chapter 4

Chapter Exploration
1. Work with a partner.
a. Use the equation y = \(\frac{1}{2}\)x + 1 to complete the table. (Choose any two x-values and find the y-values.)
b. Write the two ordered pairs given by the table. These are called solutions of the equation.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 3.1
c. PRECISION Plot the two solutions. Draw a line exactly through the points.
d. Find a different point on the line. Check that this point is a solution of the equation y = \(\frac{1}{2}\)x + 1.
e. LOGIC Do you think it is true that any point on the line is a solution of the equation y = \(\frac{1}{2}\)x + 1? Explain.
f. Choose five additional x-values for the table below. (Choose both positive and negative x-values.) Plot the five corresponding solutions. Does each point lie on the line?
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 3
g. LOGIC Do you think it is true that any solution of the equation y = \(\frac{1}{2}\)x + 1 is a point on the line? Explain.
h. Why do you think y = ax + b is called a linear equation?

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
linear equation
slope
y-intercept
solution of a linear equation
x-intercept

Lesson 4.1 Graphing Linear Equations

EXPLORATION 1

Creating Graphs
Work with a partner. It starts snowing at midnight in Town A and Town B. The snow falls at a rate of 1.5 inche sper hour.
a. In Town A, there is no snow on the ground at midnight. How deep is the snow at each hour between midnight and 6 A.M.? Make a graph that represents this situation.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 1
b. Repeat part(a) for TownB, which has 4 inches of snow on the ground at midnight.
c. The equations below represent the depth y(in inches) of snow x hours after midnight in Town C and Town D. Graph each equation.
Town C y = 2x + 3
Town D y = 8
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 2
d. Use your graphs to compare the snowfall in each town.
Answer:

Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 3

Try It

Graph the linear equation.
Question 1.
y = 3x
Answer:
Make to table of values
Replace x with a number and find the value of y
Big ideas math answers grade 8 chapter 4 img_1
Plot the values of x and y obtained above, on the graph
Big ideas math answers grade 8 chapter 4 img_1.1

Draw the line through the points

Big ideas math answers grade 8 chapter 4 img_1.2

Question 2.
y = – 2x – 1
Answer:
Big ideas math answers grade 8 chapter 4 img_2.1
Plot the values of x and y
Big ideas math answers grade 8 chapter 4 img_2.2
Now the line through the points
Big ideas math answers grade 8 chapter 4 img_2.3

Question 3.
y = –\(\frac{1}{2}\)x + 2
Answer:
Big ideas math answers grade 8 chapter 4 img_3.1
Plot the ordered pairs
Big ideas math answers grade 8 chapter 4 img_3.2

Graph the linear equation.
Question 4.
y = 3
Answer:
The graph of y = 3 is a horizontal like passing through (0, 3)
Draw a horizontal line through this point.
Big ideas math answer key grade 8 chapter 4 img_4

Question 5.
y = – 1.5
Answer:
The graph of y = -1.5 is a horizontal line passing through (0, -1.5)
Draw a horizontal line through this point.
Big ideas math answer key grade 8 chapter 4 img_5

Question 6.
x = – 4
Answer:
The graph of x = – 4 is a vertical line passing through (-4, 0)
Draw a vertical line through this point.
BIM Grade 8 Answers Chapter 4 img_6

Question 7.
x = \(\frac{1}{2}\)
Answer:
The graph of x = \(\frac{1}{2}\) is a vertical line passing through (\(\frac{1}{2}\), 0)
Draw a vertical line through this point.
Big Ideas Math 8th Grade Solution Key Chapter 4 img_7

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 8.
y = – x + 1
Answer:
Make of a table of values
Replace x with a number and find the value of y
Big Ideas Math 8th Grade Solution Key Chapter 4 img_8.1
Plot the values of x and y obtained, on the graph,
Big Ideas Math 8th Grade Solution Key Chapter 4 img_8.2

Question 9.
y = 0.8x – 2
Answer:
Replace x with a number and find the value of y
Big Ideas Math 8th Grade Solution Key Chapter 4 img_9.1
Big Ideas Math 8th Grade Solution Key Chapter 4 img_9.2

Question 10.
x = 2.5
Answer:
The graph of x = 2.5 is a vertical line passing through (2.5, 0)
Draw a vertical line through this point.
Big Ideas Math Answers Grade 8 Chapter 4 img_10.1

Question 11.
y = \(\frac{2}{3}\)
Answer:
The graph of y = \(\frac{2}{3}\) is a horizontal line passing through (0, \(\frac{2}{3}\))
Draw a horizontal line through this point.
BIM Grade 8 Answer Key Chapter 4 img_11

Question 12.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 4
Answer:
y = x – 2
4x + 3 = y
y = x² + 6
x + 5 = y

Self-Assessment for Problem Solving
Solve each exercise. The rate your understanding of the success criteria in your journal.

Question 13.
A game show contestant earns y dollars for completing a puzzle in x minutes. This situation is represented by the equation y = – 250x + 5000. How long did a contestant who earned $500 take to complete the puzzle? Justify your answer.
Answer:
Given,
A game show contestant earns y dollars for completing a puzzle in x minutes.
This situation is represented by the equation y = – 250x + 5000.
y = -250x + 5000
500 = -250x + 5000
500 – 5000 = -250x + 5000 – 5000
-4500 = -250x
x = 18

Question 14.
The total cost y (in dollars) to join a cheerleading team and attend x competitions is represented by the equation y = 10x + 50.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 5
a. Graph the linear equation.

Answer:
Big Ideas Math Answers 8th Grade Chapter 4 img_14

b. You have $75 to spend. How many competitions can you attend?
Answer:
75 ≤ 10x + 50
75 – 50 ≤ 10x
25 ≤ 10x
2.5 ≥ x
By this I can say that I can attend 2 competitions if I have $75 to spend.

Question 15.
The seating capacity for a banquet hall is represented by y = 8x + 56, where x is the number of extra tables you need. How many extra tables do you need to double the original seating capacity?
Answer:
Given,
The seating capacity for a banquet hall is represented by y = 8x + 56, where x is the number of extra tables you need.
y = 8x + 56
2 × 56 = 8x + 56
112 = 8x + 56
8x = 112 – 56
8x = 56
x = 7 tables

Graphing Linear Equations Homework & Practice 4.1

Review & Refresh

Tell whether the triangles are similar. Explain.
Question 1.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 6
Answer:
x° + 46° + 95° = 180°
x° + 141° = 180°
x° = 180° – 141°
x° = 39°
Thus the angles of the triangle are 39°, 46°, 95°
y° + 39° + 46° = 180°
y° + 75° = 180°
y° = 180° – 75°
y° = 95°
Three angles of the triangle are 39°, 46°, 95°
The triangles have two pairs of congruent angles.

Question 2.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 7
Answer:
x° + 40° + 51° = 180°
x° + 91° = 180°
x° = 180° – 91°
x° = 89°
Three angles of the triangle are 40°, 51°, 89°
y° + 40° + 79° = 180°
y° + 119° = 180°
y° = 180° – 119°
y° = 61°

Describe the translation of the point to its image.
Question 3.
(1, – 4) → (3, 0)
Answer:
A(1, -4) = A'(1 + 2, -4) = (3, -4)
A'(3, 4) = B(3, -4 + 4) = (3, 0)
Translate 2 units right and 4 units up.

Question 4.
(6, 4) → (- 4, – 6)
Answer:
We are given the points
(6, 4) → (- 4, – 6)
A(6, 4) = A'(6 – 10, 4) = (-4, 4)
A'(-4, -4) = B(-4, 4 – 10) = (-4, -6)

Question 5.
(4, – 2) → (- 9, 3)
Answer:
We are given the points
A(4, -2)
B(-9, 3)
A(4, -2) = A'(4 – 13, -2) = (-9, -2)
A'(-9, -2) = B(-9, -2 + 4) = (-9, 3)

Concepts, Skills, & Problem Solving

CREATING GRAPHS Make a graph of the situation. (See Exploration 1, p. 141.)
Question 6.
The equation y = – 2x + 8 represents the amount (in fluid ounces) of dish detergent in a bottle after x days of use.
Answer:
Bigideas math answers grade 8 chapter 4 img_15

Question 7.
The equation y = 15x + 20 represents the cost (in dollars) of a gym membership after x months.
Answer:
Bigideas math answers grade 8 chapter 4 img_16

PRECISION Copy and complete the table with two solutions. Plot the ordered pairs and draw the graph of the linear equation. Use the graph to find a third solution of the equation.
Question 8.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 8
Answer:
Bigideas math answers grade 8 chapter 4 img_17
(x, y) = (2, 5)
Bigideas math answers grade 8 chapter 4 img_18

Question 9.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 9
Answer:
Bigideas math answers grade 8 chapter 4 img_19
(x, y) = (3, 3)
Bigideas math answers grade 8 chapter 4 img_20

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 10.
y = – 5x
Answer:
Bigideas math answers grade 8 chapter 4 img_21

Question 11.
y = 9x
Answer:
Bigideas math answers grade 8 chapter 4 img_22

Question 12.
y = 5
Answer:
The graph of y = 5 is a horizontal line passing through (0, 5)
Draw a horizontal line through this point.
Bigideas math answers grade 8 chapter 4 img_23

Question 13.
x = – 6
Answer:
Bigideas math answers grade 8 chapter 4 img_24

Question 14.
y = x – 3
Answer:
Bigideas math answers grade 8 chapter 4 img_25

Question 15.
y = – 7x – 1
Answer:
Bigideas math answers grade 8 chapter 4 img_26

Question 16.
y = – \(\frac{x}{8}\) + 4
Answer:
Bigideas math answers grade 8 chapter 4 img_27

Question 17.
y = 0.75x – 0.5
Answer:
Bigideas math answers grade 8 chapter 4 img_28

Question 18.
y = – \(\frac{2}{3}\)
Answer:
Bigideas math answers grade 8 chapter 4 img_29

Question 19.
y = 6.75
Answer:
Bigideas math answers grade 8 chapter 4 img_30

Question 20.
x = – 0.5
Answer:
The graph of x = -0.5 is a vertical line passing through (-0.5, 0)
Draw a vertical line through this point.
Big Ideas Math Answers Grade 8 Ch 4 img_26

Question 21.
x = \(\frac{1}{4}\)
Answer:
The graph of x = \(\frac{1}{4}\) is a vertical line passing through (\(\frac{1}{4}\), 0)
Draw a vertical line through this point.
Big Ideas Math Answers Grade 8 Ch 4 img_27

Question 22.
YOU BE THE TEACHER
Your friend graphs the equation y = 4. Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Answers Grade 8 Ch 4 img_28
No my friend is not correct because the graph for the equation y = 4 is a  horizontal line not a vertical line, and it passes through the point (0, 4) not (4, 0)

Question 23.
MODELING REAL LIFE
The equation y = 20 represents the cost y (in dollars) for sending x text messages in a month. Graph the linear equation. What does the graph tell you about your texting plan?
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 10
Answer:
Big Ideas Math Answers Grade 8 Ch 4 img_29

Question 24.
MODELING REAL LIFE
The equation y = 2x + 3 represents the cost y (in dollars) of mailing a package that weighs x pounds.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 11
a. Use a graph to estimate how much it costs to mail the package.
b. Use the equation to find exactly how much it costs to mail the package.
Answer:
Given the equation y = 2x + 3
The ordered pairs will be (0, 3), (2,7), (4, 11)
Now plot the ordered pairs
Big Ideas Math Answers Grade 8 Ch 4 img_30
y = 2(1.126) + 3
= 5.252 ≈ 5.25

SOLVING A LINEAR EQUATION Solve for y. Then graph the linear equation.
Question 25.
y – 3x = 1
Answer:
y – 3x = 1
y = 3x + 1
Big Ideas Math Answers Grade 8 Ch 4 img_31
Draw a line through the points
Big Ideas Math Answers Grade 8 Ch 4 img_32

Question 26.
5x + 2y = 4
Answer:
5x + 2y = 4
2y = 4 – 5x
y = – \(\frac{5}{2}\)x + 2
BIM 8th Grade Solution Key Chapter 4 img_33

Question 27.
– \(\frac{1}{3}\)y + 4x = 3
Answer:
– \(\frac{1}{3}\)y + 4x = 3
– \(\frac{1}{3}\)y = 3 – 4x
y = 12x – 9
BIM 8th Grade Solution Key Chapter 4 img_34

Question 28.
x + 0.5y = 1.5
Answer:
x + 0.5y = 1.5
0.5y = -x + 1.5
y = -2x + 3
Big Ideas Math Grade 8 Answers Chapter 4 img_35

Question 29.
MODELING REAL LIFE
The depth y (in inches) of a lake after x years is represented by the equation y = 0.2x + 42. How much does the depth of the lake increase in four years? Use a graph to justify your answer.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 12
Answer:
y = 0.2x + 42
Depth of the lake now: y = 0.2(0) + 42 = 42
Depth of the lake after 4 years: y = 0.2(4) + 42 = 42.8
Big Ideas Math Grade 8 Answers Chapter 4 img_36
42.8 – 42 = 0.8 inches

Question 30.
MODELING REAL LIFE
The amount y (in dollars) of money in your savings account after x months is represented by the equation y = 12.5x + 100.
a. Graph the linear equation.

Answer:
Big Ideas Math Grade 8 Answers Chapter 4 img_37
b. How many months will it take you to save a total of $237.50?
Answer:
y = 12.5x + 100
237.5 = x + 100
237.5 – 100 = 12.5x + 100 – 100
12.5x = 137.5
x = 11

Question 31.
PROBLEM SOLVING
The radius y (in millimeters) of a chemical spill after x days is represented by the equation y = 6x + 50.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.1 13
a. Graph the linear equation.

Answer:
Big Ideas Math Grade 8 Answers Chapter 4 img_38
b. The leak is noticed after two weeks. What is the area of the leak when it is noticed? Justify your answer.
Answer:
y = 6(14) + 50
y = 84 + 50
y = 134 mm
2πr = 2π = 841.95 sq. mm

Question 32.
GEOMETRY
The sum S of the interior angle measures of a polygon with n sides is S = (n – 2) • 180°.
a. Plot four points (n, S) that satisfy the equation. Is the equation a linear equation? Explain your reasoning.

Answer:
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_39
b. Does the value n = 3.5 make sense in the context of the problem? Explain your reasoning.
Answer:
The value n = 3.5 does not make sense because the number of angles cannot be other than integer greater or equal to 2.

Question 33.
DIG DEEPER!
One second of video on your cell phone uses the same amount of memory as two pictures. Your cell phone can store 2500 pictures.
a. Create a graph that represents the number y of pictures your cell phone can store when you take x seconds of video.

Answer:
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_40
b. How many pictures can your cell phone store in addition to a video that is one minute and thirty seconds long?
Answer:
Determine the number of pictures you can store in addition to a video of 1 min 30 seconds.
1 min 30 seconds = (60 + 90) 3 seconds = 90 seconds
2500 – (2 . 90)
2500 – 180 = 2320 pictures

Lesson 4.2 Slope of a Line

EXPLORATION 1

Measuring the Steepness of a Line
Work with a partner. Draw any nonvertical line in a coordinate plane.
a. Develop a way to measure the steepness of the line. Compare your method with other pairs.
b. Draw a line that is parallel to your line. What can you determine about the steepness of each line? Explain your reasoning.
Answer:

EXPLORATION 2

Using Right Triangles
Work with a partner. Use the figure shown.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 1
a. △ABC is a right triangle formed by drawing a horizontal line segment from point A and a vertical line segment from point B. Use this method to draw another right triangle, △DEF, with its longest side on the line
b. What can you conclude about the two triangles in part(a)? Justify your conclusion. Compare your results with other pairs.
c. Based on your conclusions in part(b), what is true about \(\frac{BC}{AC}\) and the corresponding measure in △DEF? Explain your reasoning. What do these values tell you about the line?
Answer:

Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 2

Try It

Find the slope of the line.
Question 1.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 3
Answer:
(x1, y1) = (-2, 3)
(x2, y2) = (3, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 -3)/(3 – (-2))
m = -1/5
Thus slope = -1/5

Question 2.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 4
Answer:
(x1, y1) = (-4, -1)
(x2, y2) = (2, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – (-1))/(2 – (-4))
m = 2/6
Thus slope = 1/3

Find the slope of the line through the given points.
Question 3.
(1, -2), (7, -2)
Answer:
(x1, y1) = (1, -2)
(x2, y2) = (7, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – (-2))/(7 – 1)
m = 0/6
Thus slope = 0

Question 4.
(-3, -3), (-3, -5)
Answer:
(x1, y1) = (-3, -3)
(x2, y2) = (-3, -5)
m = (y2 – y1)/(x2 – x1)
m = (-5 + 3)/(-3 + 3)
m = -2/0
Thus slope = undefined

Question 5.
WHAT IF
The blue line passes through (-4, -3) and (-3, 2). Are any of the lines parallel? Explain.
Answer:
(x1, y1) = (-4, -3)
(x2, y2) = (-3, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 + 3)/(-3 + 4)
m = 5/1
m = 5
The slpe of the blue line is 5 and the slope of the red line is also 5.
The blue lines and red lines have same slopes so they are parallel.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
VOCABULARY
What does it mean for a line to have a slope of 4?
Answer:
If a line have a slope of 4 it means that the line rises 4 units for every 1 units it runs.

FINDING THE SLOPE OF A LINE Find the slope of the line through the given points.
Question 7.
(1, -1), (6, 2)
Answer:
(x1, y1) = (1, -1)
(x2, y2) = (6, 2)
m = (y2 – y1)/(x2 – x1)
m = (2 – (-1))/(6 – 1)
m = 3/5

Question 8.
(2, -3), (5, -3)
Answer:
(x1, y1) = (2, -3)
(x2, y2) = (5, -3)
m = (y2 – y1)/(x2 – x1)
m = (5 – 2)/(-3 + 3)
m = 3/0
m = undefined

Question 9.
FINDING SLOPE
Are the lines parallel? Explain your reasoning.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 5
Answer:
Red line:
(x1, y1) = (-1, 0)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 0)/1 – (-1))
m = -2/2
m = -1
Blue Line:
(x1, y1) = (-1, 3)
(x2, y2) = (1, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 3)/(1 – (-1))
m = -4/2
m = -2
The slope of the blue line and red line are not the same. So they are not parallel.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
The table shows the lengths (in inches) of your hair months after your last haircut. The points in the table lie on a line. Find and interpret the slope of the line. After how many months is your hair 4 inches long?
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 6
Answer:
Determine the slope of the line using two points from the table:
(2, 1), (4, 2)
m = (2 – 1)/4 – 2
m = 1/2
m = 0.5
This means that each month the hair grows 0.5 inches
As the hair grows 0.5 inches/ month, it will be 4 inches long after 4/0.5 = 8 months.

Question 11.
A customer pays an initial fee and a daily fee to rent a snowmobile. The total payment for 3 days is 92 dollars. The total payment for 5 days is 120 dollars. What is the daily fee? Justify your answer.
Answer:
Given,
A customer pays an initial fee and a daily fee to rent a snowmobile.
The total payment for 3 days is 92 dollars. The total payment for 5 days is 120 dollars.
m = (120 – 92)/5 – 3
m = 28/2
m = 14

Question 12.
You in-line skate from an elevation of 720 feet to an elevation of 750 feet in 30 minutes. Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour. Compare your rates of change in elevation.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 7
Answer:
Given,
You in-line skate from an elevation of 720 feet to an elevation of 750 feet in 30 minutes.
Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour.
(750 – 720)/30 = 30/30 = 1 ft/min
(690 – 600)/60 = 90/60 = 1.5 ft/min

Slope of a Line Homework & Practice 4.2

Review & Refresh

Graph the linear equation.
Question 1.
y = 4x – 3
Answer:
Big Ideas Math Grade 8 Chapter 4 Solution Key img_41

Question 2.
x = -3
Answer:
Big Ideas Math Grade 8 Chapter 4 Solution Key img_42

Question 3.
y = 2
Answer:
Big Ideas Math Grade 8 Chapter 4 Solution Key img_43

Question 4.
y = \(\frac{3}{2}\)x – \(\frac{1}{2}\)
Answer:
Big Ideas Math Grade 8 Chapter 4 Solution Key img_44

Find the missing values in the ratio table.
Question 5.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 8
Answer:
Big-Ideas-Math-Answers-8th-Grade-Chapter-4-Graphing-and-Writing-Linear-Equations-4.2-8
x/10 = 1/3
x = 10/3
x = 3.33
1/3 = 5/y
y = 5 × 3
y = 15
1/3 = 7/z
z = 3 × 7
z = 21

Question 6.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 9
Answer:
Big-Ideas-Math-Answers-8th-Grade-Chapter-4-Graphing-and-Writing-Linear-Equations-4.2-9

Concepts, Skills, &Problem Solving

USING RIGHT TRIANGLES Use the figure shown (See Exploration 2, p. 147.)
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 10
Question 7.
Find the slope of the line.
Answer:
(x1, y1) = B(-4, 2)
(x2, y2) = A(-2, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 2)/(-2 – (-4))
m = -1/2
Thus the slope m = -1/2

Question 8.
Let point D be at (-4, 1). Use the sides of △BDA to find the slope of the line.
Answer:
Big Ideas Math Grade 8 Chapter 4 Answers img_45
m = -BD/DA = -1/2

FINDING THE SLOPE OF A LINE Find the slope of the line.
Question 9.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 11
Answer:
(x1, y1) = (-2, 0)
(x2, y2) = (2, 3)
m = (y2 – y1)/(x2 – x1)
m = (3 – 0)/(2 – (-2))
m = 3/4

Question 10.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 11
Answer:
(x1, y1) = (-2, 5)
(x2, y2) = (2, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – 5)/(2 – (-2))
m = -5/4

Question 11.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 13
Answer:
(x1, y1) = (-4, 1)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(1 + 2)
m = -3/5

Question 12.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 14
Answer:
(x1, y1) = (-5, -4)
(x2, y2) = (1, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – (-4))/(1 – (-5))
m = 1/6

Question 13.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 15
Answer:
(x1, y1) = (-1, 3)
(x2, y2) = (3, 3)
m = (y2 – y1)/(x2 – x1)
m = (3 – 3)/(3 – (-1))
m = 0/4
m = 0

Question 14.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 16
Answer:
(x1, y1) = (1, 3)
(x2, y2) = (1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 3)/(1 – 1)
m = -5/0
m = undefined

FINDING THE SLOPE OF A LINE Find the slope of the line through the given points.
Question 15.
(4, -1), (-2, -1)
Answer:
(x1, y1) = (4, -1)
(x2, y2) = (-2, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-1))/(-2 – 4)
m = 0/-6
m = 0

Question 16.
(5, -3), (5, 8)
Answer:
(x1, y1) = (5, -3)
(x2, y2) = (5, 8)
m = (y2 – y1)/(x2 – x1)
m = (8 – 3)/(5 – 5)
m = 5/0
m = undefined

Question 17.
(-7, 0), (-7, -6)
Answer:
(x1, y1) = (-7, 0)
(x2, y2) = (-7, -6)
m = (y2 – y1)/(x2 – x1)
m = (-6 – 0)/(-7 – (-7))
m = -6/0
m = undefined

Question 18.
(-3, 1), (-1, 5)
Answer:
(x1, y1) = (-3, 1)
(x2, y2) = (-1, 5)
m = (y2 – y1)/(x2 – x1)
m = (5 – 1)/(-1 + 3)
m = 4/2
m = 2

Question 19.
(10, 4), (4, 15)
Answer:
(x1, y1) = (10, 4)
(x2, y2) = (4, 15)
m = (y2 – y1)/(x2 – x1)
m = (15 – 4)/(4 – 10)
m = 11/-6
m = -11/6

Question 20.
(-3, 6), (2, 6)
Answer:
(x1, y1) = (-3, 6)
(x2, y2) = (2, 6)
m = (y2 – y1)/(x2 – x1)
m = (6 – 6)/(2 – (-3))
m = 0/5
m = 0

Question 21.
REASONING
Draw a line through each point using slope of m = \(\frac{1}{4}\). Do the lines intersect? Explain.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 17
Answer:
Big Ideas math Grade 8 Answer Key Chapter 4 img_46
The 2 lines are parallel because they have the same slope and they do not intersect.

Question 22.
YOU BE THE TEACHER
Your friend finds the slope of the line shown. Is your friend correct? Explain your reasoning.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 18
Answer:
Big Ideas math Grade 8 Answer Key Chapter 4 img_47
No my friend is not correct because the denominator should be 2 – 4
(x1, y1) = (2, 3)
(x2, y2) = (4, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 3)/(4 – 2)
m = -2/2
m = -1

IDENTIFYING PARALLEL LINES Which lines are parallel? How do you know?
Question 23.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 19
Answer:
Blue line:
(x1, y1) = (-5, 2)
(x2, y2) = (-4, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 2)/(-4 – (-5))
m = -3/1
m = -3
Red line:
(x1, y1) = (-2, 1)
(x2, y2) = (-1, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(-1 – (-2))
m = -3/1
m = -3
Green Line:
(x1, y1) = (1, 3)
(x2, y2) = (2, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – 3)/(2 – 1)
m = -4/1
m = -4
Blue line and red line have slope of -3, so they are parallel.

Question 24.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 20
Answer:
Blue line:
(x1, y1) = (-2, 3)
(x2, y2) = (-5, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 3)/(-5 – (-2))
m = -5/-3
m = 5/3
Red line:
(x1, y1) = (1, 2)
(x2, y2) = (-2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 2)/(-2 – 1)
m = -4/-3
m = 4/3
Green Line:
(x1, y1) = (4, 1)
(x2, y2) = (1, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – 1)/(1 – 4)
m = -4/-3
m = 4/3
Red line and green line have slope of 4/3 by this we can say that they are parallel.

IDENTIFYING PARALLEL LINES Are the given lines parallel? Explain your reasoning.
Question 25.
y = -5, y = 3
Answer:
8th Grade Big Ideas Math Answers Chapter 4 img_47
Both lines are horizontal and have slope = 0

Question 26.
y = 0, x = 0
Answer:
8th Grade Big Ideas Math Answers Chapter 4 img_48
The line y = 0 have slope = 0 and are horizontal lines.
The line x = 0 have slope = undefined and are vertical lines.
So, they are not parallel.

Question 27.
x = -4, x = 1
Answer:
8th Grade Big Ideas Math Answers Chapter 4 img_49
Both lines are vertical and have an undefined slope.

FINDING SLOPE The points in the table lie on a line. Find the slope of the line.
Question 28.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 21
Answer:
m = (y2 – y1)/(x2 – x1)
m = (10 – 2)/(3 – 1) = (18 – 10)/(5 – 3) = (26 – 18)/(7 – 5)
m = 8/2 = 8/2 = 8/2
m = 4 = 4 = 4
Slope = 4

Question 9.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 22
Answer:
m = (y2 – y1)/(x2 – x1)
m = (2 – 0)/(2 – (-3)) = (4 – 2)/(7 – 2) = (6 – 4)/(12 – 7)
m = 2/5 = 2/5 = 2/5
m = 2/5

Question 30.
MODELING REAL LIFE
Carpenters refer to the slope of a roof as the pitch of the roof. Find the pitch of the roof.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 23
Answer:
Pitch of the roof = rise/run
= 4/12 = 1/3

Question 31.
PROJECT
The guidelines for a wheelchair ramp suggest that the ratio of the rise to the run be no greater than 1 : 12.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 24
a. CHOOSE TOOLS Find a wheelchair ramp in your school or neighborhood. Measure its slope. Does the ramp follow the guidelines?

Answer:
rise/run < 1/12
m = 0.06
1/12 = 0.0833
0.06 < 0.0833
As m < 1/12 the wheelchair ramp follows the guides.

b. Design a wheelchair ramp that provides access to a building with a front door that is 2.5 feet above the sidewalk. Illustrate your design.
Answer:
AC/AB = 1/12
2.5/AB = 1/12
AB = 2.5 × 12
AB = 30
So the end of the ramp should be placed at least 30 feet from the front door.

USING AN EQUATION Use an equation to find the value of k so that the line that passes through the given points has the given slope.
Question 32.
(1, 3), (5, k); m = 2
Answer:
A(1, 3)
B(5, k)
m = 2
2 = (k – 3)/(5 – 1)
2 × 4 = k – 3
8 = k – 3
k = 8 + 3
k = 11

Question 33.
(-2, k), (2, 0); m = -1
Answer:
Given,
A(-2, k)
B(2, 0)
m = -1
-1 = (0 – k)/2 – (-2)
-1 = -k/4
-4 = -k
k = 4

Question 34.
(-4, k), (6, -7); m = –\(\frac{1}{5}\)
Answer:
Given,
A(-4, k)
B(6, -7)
m = –\(\frac{1}{5}\)
–\(\frac{1}{5}\) = (-7 – k)/6 – (-4)
-2 = -7 – k
-2 + 7 = -k
5 = -k
k = -5

Question 35.
(4, -4), (k, -1); m = \(\frac{3}{4}\)
Answer:
\(\frac{3}{4}\) = (-1 – (-4))/(k – 4)
4 = k – 4
k = 4 + 4
k = 8

Question 36.
MODELING REAL LIFE
The graph shows the numbers of prescriptions filled over time by a pharmacy.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 25
a. Find the slope of the line.
Answer:
(0, 0), (20, 5)
m = (5 – 0)/(20 – 0)
m = 5/20
m = 1/4
b. Explain the meaning of the slope as a rate of change.
Answer:
This means that every 4 minutes a prescription is filled.

Question 37.
CRITICAL THINKING
Which is steeper: the boatramp, or a road with a 12% grade? Note: Explain. (Road grade is the vertical increase divided by the horizontal distance.)
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 26
Answer:
Mramp = rise/run = 6/36 = 1/6
Mroad = 12% = 12/100 = 0.12
0.16 = 0.166… > 0.12
Mramp > Mroad
Therefore the slope of the ramp is steeper than the slope of the road.

Question 38.
REASONING
Do the points A(-2, -1), B(1, 5), and C(4, 11) lie on the same line? Without using a graph, how do you know?
Answer:
Given,
A(-2, -1), B(1, 5), and C(4, 11)
mAB = (5 – (-1))/(1 – (-2)) = 6/3 = 2
mBC = (11 – 5)/(4 – 1) = 6/3 = 2
By seeing the slopes we can say that the points A, B, C lie on the same line.

Question 39.
PROBLEM SOLVING
A small business earns a profit of $6500 in January and $17,500 in May. What is the rate of change in profit for this time period? Justify your answer.
Answer:
Pjan = 6500
Pmay = 17,500
Pmay – Pjan/5 – 1
= (17,500 – 6500)/4
= 11,000/4 = 2750

Question 40.
STRUCTURE
Choose two points in the coordinate plane. Use the slope formula to find the slope of the line that passes through the two points. Then find the slope using the formula \(\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\). Compare your results.
Answer:
P1(2, 5)
P2(3, 10)
m1 = (10 – 5)/(3 – 2) = 5/1 = 5
m2 = (5 – 10)/(1 – 3) = -5/-1 = 5
m1 = m2

Question 41.
DIG DEEPER!
The top and the bottom of the slide are level with the ground, which has a slope of 0.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.2 27
a. What is the slope of the main portion of the slide?
b. Describe the change in the slope when the bottom of the slide is only 12 inches above the ground. Explain your reasoning.
Answer:
18 inches = 1.5 feet
mMC = rise/run = (8 – 1.5)/(12 – 1 – 1) = 6.5/10 = 0.65
AD = 1
mMC = CR/MR
= (8 – 1)/(12 – 1 – 1) = 7/10 = 0.7
The slope increases from 0.65 to 0.70 because the rise increasses, while the run stays the same.

Lesson 4.3 Graphing Proportional Relationships

EXPLORATION 1

Using a Ratio Table to Find Slope
Work with a partner. The graph shows amounts of vinegar and water that can be used to make a cleaning product.
a. Use the graph to make a ratio table relating the quantities. Explain how the slope of the line is represented in the table.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 1
b. Make a ratio table that represents a different ratio of vinegar to water. Use the table to describe the slope of the graph of the new relationship.
Answer:

EXPLORATION 2

Deriving an Equation
Work with a partner. Let (x, y) represent any point on the graph of a proportional relationship.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 2
a. Describe the relationship between the corresponding side lengths of the triangles shown in the graph. Explain your reasoning.
b. Use the relationship in part(a) to write an equation relating y, m, and x. Then solve the equation for y. How can you find the side lengths of the triangles in the graph?
c. What does your equation in part(b) describe? What does represent? Explain your reasoning.
Answer:

Try It

Question 1.
WHAT IF
The cost of frozen yogurt is represented by y = 0.75x. Graph the equation and interpret the slope.
Answer:
The equation shows that the slope m is 0.75. So the graph passes through the points (0, 0) and (1, 0.75).
Plot the ordered pairs and draw the graph.
Big Ideas Math Grade 8 Answer Key Chapter 4 img_48
The slope indicates that the unit cost is $0.75 per ounce.

Question 2.
How much would a spacecraft that weighs 3500 kilograms on Earth weigh on Titan?
Answer:
y = 1/7 x
y = 1/7 × 3500
y = 500 kg
So a spacecraft would weigh 500 kg on Titan.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

GRAPHING A PROPORTIONAL RELATIONSHIP Graph the equation.
Question 3.
y = 4x
Answer:
Big Ideas Math Grade 8 Answer Key Chapter 4 img_49

Question 4.
y = -3x
Answer:
Big Ideas Math Grade 8 Answer Key Chapter 4 img_50

Question 5.
y = 8x
Answer:
Big Ideas Math Grade 8 Answer Key Chapter 4 img_51

Question 6.
WRITING AND USING AN EQUATION
The number of objects a x machine produces is proportional to the time (in minutes) that the machine runs. The machine produces five objects in four minutes.
a. Write an equation that represents the situation.

Answer:
As 5 objects are produced in 4 minutes, the slope of the line is m = 5/4.
The equation that represents the situation is
y = 5/4 x
y = 1.25 x

b. Graph the equation in part (a) and interpret the slope.

Answer:
Use the slope. The equation shows that the slope m is 1.25. So the graph passes through the points (0, 0) and (1, 1.25)

c. How many objects does the machine produce in one hour?
Answer:
Big Ideas Math Grade 8 Answer Key Chapter 4 img_52

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.
The amount y (in liters) of water that flows over a natural waterfall in x seconds is represented by the equation y = 500x. The graph shows the number of liters of water that flow over an artificial waterfall. Which waterfall has a greater flow? Justify your answer.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 3
Answer:
Given the equation y = 500x
15000 – 3000 = 12000
12000/4 = 3000
Mnatural = 500
3000 > 500
Therefore the artificial waterfall has greater flow.

Question 8.
The speed of sound in air is 343 meters per second. You see lightning and hear thunder 12 seconds later.
a. Is there a proportional relationship between the amount of time that passes and your distance from a lightning strike? Explain.

Answer:
y = kx
where k is the speed of sound, x the time and y the distance.
Yes, there is a proportional relationship between the amount of time that passes and your distance from the lightning strike as the further you are, the more time will pass until the sound reaches you.

b. Estimate your distance from the lightning strike.
Answer:
y = 343 × 12
= 4116 meters

Graphing Proportional Relationships Homework & Practice 4.3

Review & Refresh

Find the slope of the line.
Question 1.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 4
Answer:
(x1, y1) = (0, -3)
(x2, y2) = (3, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – (-3))/(3 – 0)
m = (0 + 3)/(3 – 0)
m = 3/3
m = 1

Question 2.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 5
Answer:
(x1, y1) = (0, 1)
(x2, y2) = (3, -5)
m = (y2 – y1)/(x2 – x1)
m = (-5 – 1)/(3 – 0)
m = -6/3
m = -2

Question 3.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 6
Answer:
(x1, y1) = (0, 0)
(x2, y2) = (2, 8)
m = (y2 – y1)/(x2 – x1)
m = (8 – 0)/(2 – 0)
m = 8/2
m = 4

Solve the equation. Check your solution.
Question 4.
2x + 3x = 10
Answer:
Given the equation
2x + 3x = 10
5x = 10
x = 10/5
x = 2

Question 5.
x + \(\frac{1}{6}\) = 4 – 2x
Answer:
Given the equation
x + \(\frac{1}{6}\) = 4 – 2x
x + 2x = 4 – \(\frac{1}{6}\)
3x = 4 – \(\frac{1}{6}\)
3x = \(\frac{23}{6}\)
x = \(\frac{23}{18}\)

Question 6.
2(1 – x) = 11
Answer:
2(1 – x) = 11
2 – 2x = 11
2 – 11 = 2x
2x = -9
x = -9/2

Concepts, Skills, & Problem Solving

USING EQUIVALENT RATIOS The graph shows amounts of water and flour that can be used to make dough. (See Exploration 1, p. 155.)
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 7
Question 7.
Use the graph to make a ratio table relating the quantities. Explain how the slope of the line is represented in the table.
Answer:
BIM Grade 8 Solution Key Chapter 4 img_53
m = rise/run
= (10 – 5)/(6 – 3)
= 5/3
That means to every 5 cups of flour there is an increase of 3 cups of water.
The slope m is 5/3.

Question 8.
Make a ratio table that represents a different ratio of flour to water. Use the table to describe the slope of the graph of the new relationship.
Answer:
BIM Grade 8 Solution Key Chapter 4 img_54
From the table we find that for every increase of 7 cups of flour there is an increase of 4 cups of water.
The slope is 7/4.

Question 9.
GRAPHING AN EQUATION
The amount y(in dollars) that you raise by selling fundraiser tickets is represented by the equation y = 5x. Graph the equation and interpret the slope.
Answer:
BIM Grade 8 Solution Key Chapter 4 img_55
The slope indicates that the unit cost is $5 per ticket.

IDENTIFYING PROPORTIONAL RELATIONSHIPS Tell whether and are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 10.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 8
Answer:
The graph doesn’t represent a proportional relationship because it doesn’t pass through the point (0, 0).

Question 11.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 9
Answer:
The graph represents a proportional relationship because it is linear and passes through the point (0, 0)
(0, 0), (2, 8)
m = (8 – 0)/(2 – 0)
m = 8/2
m = 4
The equation is y = 4x

Question 12.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 10
Answer:
(2 – 1)/(6 – 3) = 1/3
(3 – 2)/(9 – 6) = 1/3
(4 – 3)/(12 – 9) = 1/3
As the rate of change is constant, it means that the graph is a line.
(1 – y)/(3 – 0) = 1/3
(1 – y)/3 = 1/3
1 – y = 1
y = 1 – 1
y = 0
Therefore the point (0, 0) belomgs to the graph.
So the table represents a proportional relationship
y = 1/3 x

Question 13.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 11
Answer:
(8 – 4)/(5 – 2) = 4/3
(13 – 8)/(8 – 5) = 5/3
(23 – 13)/10 – 8 = 10/2 = 5

Question 14.
MODELING REAL LIFE
The cost y (in dollars) to rent a kayak is proportional to the number x of hours that you rent the kayak. It costs $27 to rent the kayak for 3 hours.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 12
a. Write an equation that represents the situation.
b. Interpret the slope of the graph of the equation.
c. How much does it cost to rent the kayak for 5 hours? Justify your answer.
Answer:
y = kx
27 = k × 3
k = 27/3
k = 9
The equation is k = 9x
b. The slope k = 3 shows that the cost of renting the kayak per hour is $9.
c. y = 9 × 5
y = 45

Question 15.
MODELING REAL LIFE
The distance y (in miles) that a truck travels on x gallons of gasoline is represented by the equation y = 18x. The graph shows the distance that a car travels.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 13
a. Which vehicle gets better gas mileage? Explain how you found your answer.

Answer:
y = 18x
(0, 0), (2, 50)
m = (50 – 0)/(2 – 0)
m = 50/2
m = 25
25 > 18
Therefore the car has better mileage.

b. How much farther can the vehicle you chose in part(a) travel on 8 gallons of gasoline?
Answer:
y = 25 × 8 – 18 × 8
= 200 – 144
= 56 miles

Question 16.
PROBLEM SOLVING
Toenails grow about 13 millimeters per year. The table shows fingernail growth.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 14
a. Do fingernails or toenails grow faster? Explain.

Answer:
y = 0.25x
m = (1.4 – 0.7)/(2 – 1)
m = 0.7
y = 0.7x
Because 0.7 > 0.25, the fingernails grow faster.

b. In the same coordinate plane, graph equations that represent the growth rates of toenails and fingernails. Compare and interpret the steepness of each graph.
Answer:
BIM Answer Key Grade 8 Chapter 4 img_57

Question 17.
REASONING
The quantities and are in a proportional relationship. What do you know about the ratio of y to x for any point (x, y) on the graph of x and y?
Answer:
y = kx
where k is constant
y/x = k
This means the ratio of y to x is constant.

Question 18.
DIG DEEPER!
The graph relates the temperature change y (in degrees Fahrenheit) to the altitude change x (in thousands of feet).
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.3 15
a. Is the relationship proportional? Explain.

Answer: The relationship is proportional because the graph is linear and passes through the origin.

b. Write an equation of the line. Interpret the slope.

Answer:
(0,0), (10, -35)
m = (-35 – 0)/(10 – 0)
= -35/10
= -3.5
y = -3.5x

c. You are at the bottom of a mountain where the temperature is 74°F. The top of the mountain is 5500 feet above you. What is the temperature at the top of the mountain? Justify your answer.
Answer:
x = 5.5 – 0 = 5.5 thousand feet
y = -3.5x = -3.5(5.5) = -19.25
74 – 19.25 = 54.75°F

Question 19.
CRITICAL THINKING
Consider the distance equation d = rt, where d is the distance (in feet), r is the rate (in feet per second), and t is the time (in seconds). You run for 50 seconds. Are the distance you run and the rate you run at proportional? Use a graph to justify your answer.
Answer:
d = rt
d = 50r
Having the form y = kx the equation represents a proportional relationship.
BIM Answer Key Grade 8 Chapter 4 img_58

Lesson 4.4 Graphing Linear Equations in Slope-Intercept Form

EXPLORATION 1

Deriving an Equation
Work with a partner. In the previous section, you learned that the graph of a proportional relationship can be represented by the equation y = mx, where m is the constant of proportionality.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 1
a. You translate the graph of a proportional relationship 3 units up as shown below. Let (x, y) represent any point on the graph. Make a conjecture about the equation of the line. Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 2
b. Describe the relationship between the corresponding side lengths of the triangles. Explain your reasoning.
c. Use the relationship in part(b) to write an equation relating y, m, and x. Does your equation support your conjecture in part(a)? Explain.
d. You translate the graph of a proportional relationship b units up. Write an equation relating y, m, x, and b. Justify your answer.
Answer:

Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 3

Try It

Find the slope and the y-intercept of the graph of the linear equation.
Question 1.
y = 3x – 7
Answer:
Given the equation
y = 3x – 7
Write the equation in slope – intercept form: y = mx + b
The slope of the line is m and the y – intercept of the line is b.
y = 3x – 7
Slope = 3 and y – intercept = -7

Question 2.
y – 1 = –\(\frac{2}{3}\)x
Answer:
Write the equation in slope – intercept form: y = mx + b
The slope of the line is m and the y – intercept of the line is b.
y – 1 = –\(\frac{2}{3}\)x
y = –\(\frac{2}{3}\)x + 1
Slope = –\(\frac{2}{3}\) and y – intercept = 1

Graph the linear equation. Identify the x-intercept.
Question 3.
y = x – 4
Answer:
y = x – 4
Comparing the above equation with slope – intercept equation.
slope = 1, y-intercept = -4
Ploy y – intercept and slope
slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, -4) = (1, -3)
Grade 8 BIM Answers Chapter 4 img_59
Thus the intercept is 4.

Question 4.
y = –\(\frac{1}{2}\)x + 1
Answer:
y = –\(\frac{1}{2}\)x + 1
Comparing the above equation with slope – intercept equation.
Slope = –\(\frac{1}{2}\), y-intercept = 1
y-intercept = 1. So plot (0, 1)
Slope = rise/run = -1/2
Plot the point that is 2 units right and 1 unit down from (0, -4) = (2, 0)
Grade 8 BIM Answers Chapter 4 img_60
So, the x-intercept is 2.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
IN YOUR OWN WORDS
Consider the graph of the equation y = mx + b.
a. How does changing the value of m affect the graph of the equation?

Answer:
The value of m is the slope of the graph. If the value of m changes it means the slope of the graph is changing, whether it will rise or fall from left or right is dependent on the value of m.

b. How does changing the value of b affect the graph of the equation?
Answer:
The value of b is the y-intercept of the graph. If the value of b changes it means it affects where the graph crosses the y – axis.

IDENTIFYING SLOPE AND y-INTERCEPT Find the slope and the y-intercept of the graph of the linear equation.
Question 6.
y = -x + 0.25
Answer:
y = mx + c
slope = -1 and y – intercept = 0.25

Question 7.
y – 2 = –\(\frac{3}{4}\)x
Answer:
Given the equation
y – 2 = –\(\frac{3}{4}\)x
y = –\(\frac{3}{4}\)x + 2
slope = –\(\frac{3}{4}\) and y – intercept = 2

GRAPHING A LINEAR EQUATION Graph the linear equation. Identify the x-intercept.
Question 8.
y = x – 7
Answer:
Grade 8 BIM Answers Chapter 4 img_61
The line crosses the x-axis at (7, 0)
So, the x – intercept is 7.

Question 9.
y = 2x + 8
Answer:
Grade 8 BIM Answers Chapter 4 img_62
The line crosses the x – axis at (-4, 0)
So, the x – intercept is -4.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 10.
The height y (in feet) of a movable bridge after rising for seconds is represented by the equation y = 3x + 6. Graph the equation. Interpret the y-intercept and slope. How many seconds does it take the bridge to reach a height of 76 feet? Justify your answer.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 4
Answer:
Given,
y = 3x + 6.
slope = 3, y – intercept = 16
Grade 8 BIM Answers Chapter 4 img_63
The y – intercept is 16. So, the initial height of the bridge is 16 feet.
The slope is 3. So, the bridge rises 3 feet per second.
The bridge will reach a height of 76 feet in 20 seconds.

Question 11.
The number of perfume bottles in storage after x months is represented by the equation y = -20x + 460. Graph the equation. Interpret the y-intercept and the slope. In how many months will there be no perfume bottles left in storage? Justify your answer.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 5.1
Answer:
Given the equation
y = -20x + 460
Slope = -20, y-intercept = 460
Grade 8 BIM Answers Chapter 4 img_64
The y-intercept is 460. So, the initial number of perfume in the storage is 460.
The slope is -20. So, the number of perfume bottle decrease with 20 bottles per months.
There will be no perfume bottle left in the storage in 23 months.

Graphing Linear Equations in Slope-Intercept Form Homework & Practice 4.4

Review & Refresh

Tell whether x and y are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 1.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 5
Answer:
(8 – 6)/(2 – 1) = 2/1 = 2
(10 – 8)/(3 – 2) = 2/1 = 2
(12 – 10)/(4 – 3) = 2/1 = 2
The rate of change in the table is constant.
(6 – y)/(1 – 0) = 2
6 – y = 2
y = 6 – 2
y = 4
Therefore the graph does not pass through the origin.
So x and y are not proportional.

Question 2.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 6
Answer:
(4 – 0)/(-8 – 0) = 4/-8 = -1/2 = -0.5
(2 – 4)/(-4 – (-8)) = -2/4 = -1/2 = -0.5
(-2 – 2)/(4 – (-4)) = -4/8 = -1/2 = -0.5
(-4 – (-2))/(8 – 4) = -2/4 = -1/2 = -0.5
As the rate of change is constant, x and y are in a proportional relationship.
y = -0.5x

Solve the equation for y.
Question 3.
x = 4y – 2
Answer:
Given the equation
x = 4y – 2
x – 2 = 4y
y = x/4 + 1/2

Question 4.
3y = -6x + 1
Answer:
Given the equation
3y = -6x + 1
y = -2x + 1/3

Question 5.
1 + y = –\(\frac{4}{5}\)x – 2
Answer:
Given the equation
1 + y = –\(\frac{4}{5}\)x – 2
y = –\(\frac{4}{5}\)x – 3

Question 6.
2.5y = 5x – 5
Answer:
Given the equation
2.5y = 5x – 5
y = 2x – 2

Question 7.
1.3y + 5.2 = -3.9x
Answer:
Given the equation
1.3y + 5.2 = -3.9x
1.3y = -3.9x – 5.2
y = -3x – 4

Question 8.
y – \(\frac{2}{3}\)x = -6
Answer:
Given the equation
y – \(\frac{2}{3}\)x = -6
y = \(\frac{2}{3}\)x -6

Concepts, Skills, &Problem Solving

GRAPHING A LINEAR EQUATION Graph the equation. (See Exploration 1, p. 161.)
Question 9.
The graph of y = 3.5x is translated up 2 units.
Answer:
Given the equation
y = 3.5x
The line obtained by translating the graph of the line y = 3.5x up 2 units has the same slope (3.5) and y – intercept 2 units greater, which means b = 0 + 2 = 2
Big Ideas Math Grade 8 Answer Key Chapter 4 img_60

Question 10.
The graph of y = -5x is translated down 3 units.
Answer:
y = -5x
The line obtained by translating the graph of the line y = -5x down 3 units has the same slope and the y – intercept 3 units smaller, which means b = 0 – 3 = -3
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_61

MATCHING EQUATIONS AND GRAPHS Match the equation with its graph. Identify the slope and the y-intercept.
Question 11.
y = 2x + 1
Answer:
Given the eqation
y = 2x + 1
slope = 2 and y – intercept = 1
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_62

Question 12.
y = \(\frac{1}{3}\)x – 2
Answer:
slope = 1/3 and y – intercept = -2
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_63

Question 13.
y = –\(\frac{2}{3}\)x + 1
Answer:

Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 7

Answer:
Slope = -2/3 and y – intercept = 1
The graph which passes through the point (0, 1) and has a negative slope is the matching graph of the given equation.

IDENTIFYING SLOPES AND y-INTERCEPTS Find the slope and the y-intercept of the graph of the linear equation.
Question 14.
y = 4x – 5
Answer:
y = mx + b
slope = 4 and y — intercept = -5

Question 15.
y = -7x + 12
Answer:
y = -7x + 12
y = mx + b
slpoe = -7 and y – intercept = 12

Question 16.
y = –\(\frac{4}{5}\)x – 2
Answer:
y = mx + b
slope = -4/5
y – intercept = -2

Question 17.
y = 2.25x + 3
Answer:
y = mx + b
slope = 2.25 and y – intercept = 3

Question 18.
y + 1 = \(\frac{4}{3}\)x
Answer:
y = mx + b
y + 1 = \(\frac{4}{3}\)x
y = \(\frac{4}{3}\)x – 1
slope = \(\frac{4}{3}\), y – intercept = -1

Question 19.
y – 6 = \(\frac{3}{5}\)x
Answer:
y = mx + b
y – 6 = \(\frac{3}{5}\)x
y = \(\frac{3}{5}\)x + 6
slope = 3/8 and y – intercept = 6

Question 20.
y – 3.5 = -2x
Answer:
y = mx + b
y – 3.5 = -2x
y = -2x + 3.5
slope = -2 and y – intercept = 3.5

Question 21.
y = -5 – \(\frac{1}{2}\)x
Answer:
y = mx + b
y = -5 – \(\frac{1}{2}\)x
y =- \(\frac{1}{2}\)x – 5
slope = – \(\frac{1}{2}\) and y – intercept = -5

Question 22.
y = 11 + 1.5x
Answer:
y = mx + b
y = 1.5x + 11
slope = 1.5 and y – intercept = 11

Question 23.
YOU BE THE TEACHER
Your friend finds the slope and y-intercept of the graph of the equation y = 4x – 3. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 8
Answer:
y = 4x – 3
No my friend is not correct because the y – intercept is -3.

Question 24.
MODELING REAL LIFE
The number y of seasonal allergy shots available at a facility x days after receiving a shipment is represented by y = -15x + 375.
a. Graph the linear equation.
b. Interpret the slope and the y-intercept.
Answer:
y = -15x + 375
x = 0
y = -15(0) + 375 = 375
y = 0
0 = -15x + 375
15x = 375
x = 375/15
x = 25
BIM Grade 8 Solution Key Chapter 4 img_64
The slope shows that the number of seasonal allergy shots decrease by 15 shots each day.
The y – intercept shows that the number of shots immediately after receiving a shipment is 375.

GRAPHING AN EQUATION Graph the linear equation. Identify the x-intercept.
Question 25.
y = x + 3
Answer:
Given the equation
y = x + 3
slope = 1 and y – intercept = 3
Slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, 3) = (1, 4)
BIM Grade 8 Solution Key Chapter 4 img_65
So, the x – intercept is -3.

Question 26.
y = 4x – 8
Answer:
y = 4x – 8
Comparing the above equation with slope – intercept equation.
slope = 4 and y – intercept = -8
Slope = rise/run = 4/1 = 4
Plot the point that is 1 unit right and 4 unit up from (0, -8) = (1, -4)
BIM Grade 8 Solution Key Chapter 4 img_66

Question 27.
y = -3x + 9
Answer:
y = -3x + 9
slope = -3 and y – intercept = 9
slope rise/run = -3/1 = -3
BIM Grade 8 Solution Key Chapter 4 img_67

So, the intercept is 3.

Question 28.
y = -5x – 5
Answer:
y = -5x – 5
slope = -5 and y – intercept = -5
slope = rise/run = -5/1
Plot the point that is 1 unit right and 5 unit up from (0, -5) = (1, -10)
BIM Grade 8 Solution Key Chapter 4 img_68
So, the x – intercept is -1.

Question 29.
y + 14 = -7x
Answer:
y + 14 = -7x
y = -7x – 14
slope = -7 and y – intercept = -14
Slope = rise/run = -7/1
Plot the point that is 1 unit right and 7 unit down from (0, -14) = (1, -21)
BIM Grade 8 Solution Key Chapter 4 img_69
So, the x – intercept is -2.

Question 30.
y = 8 – 2x
Answer:
Given the equation
y = 8 – 2x
y = -2x + 8
slope = -2 and y – intercept = 8
slope = rise/run = -2/1
Plot the point 1 unit right and 2 units down from (0, 8) = (1, 6)
BIM Grade 8 Solution Key Chapter 4 img_70
So, the x – intercept is 4.

Question 31.
PRECISION
You go to a harvest festival and pick apples.
a. Which equation represents the cost (in dollars) of going to the festival and picking x pounds of apples? Explain.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 9
b. Graph the equation you chose in part(a).
Answer:
Picking a pound of apples costs $0.75, therefore x pounds cost 0.75 × x = 0.75x
y = 0.75x + 5
BIM Grade 8 Solution Key Chapter 4 img_71

Question 32.
REASONING
Without graphing, identify the equations of the lines that are parallel. Explain your reasoning.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 10
Answer:
The lines which area parallel are those having the same slope.
y = 2x + 4
y = 2x – 3
y = 2x + 1
y = 1/2x + 1
y = 1/2x + 2

Question 33.
PROBLEM SOLVING
A skydiver parachutes to the ground. The height y (in feet) of the skydiver after x seconds is y = -10x + 3000.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation 4.4 11
a. Graph the linear equation.
b. Interpret the slope, y-intercept, and x-intercept.
Answer:
y = -10x + 3000
x = 0
y = -10(0) + 3000 = 3000
y = 0
0 = -10 + 3000
10x = 3000
x = 3000/10 = 300
BIM Grade 8 Solution Key Chapter 4 img_72
b. The slope shows that each second the skydiver descends 10 feet.
The y – intercept shows that the skydiver begins its dive from 3000 feet.
The x – intercept shows that he reaches the ground after 300 seconds.

Question 34.
DIG DEEPER!
Six friends create a website. The website earns money by selling banner ads. It costs $120 a month to operate the website.
a. A banner ad earns $0.005 per click. Write a linear equation that represents the monthly profit after paying operating costs.
b. Graph the equation in part(a). On the graph, label the number of clicks needed for the friends to start making a profit. Explain.
Answer:
y = 0.005x – 120
x = 0
y = 0.005(0) – 120
y = -120
y = 0
0 = 0.005x – 120
0.005x = 120
x = 24000
BIM Grade 8 Solution Key Chapter 4 img_73
x > 24,000

Lesson 4.5 Graphing Linear Equations in Standard Form

EXPLORATION 1

Using Intercepts
Work with a partner. You spend $150 on fruit trays and vegetable trays for a party.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 1
a. You buy x fruit trays and y vegetable trays. Complete the verbal model. Then use the verbal model to write an equation that relates x and y.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 2
b. What is the greatest number of fruit trays that you can buy? vegetable trays? Can you use these numbers to graph your equation from part (a) in the coordinate plane? Explain.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 3
c.Use a graph to determine the different combinations of fruit trays and vegetable trays that you can buy. Justify your answers algebraically.
d. You are given an extra $50 to spend. How does this affect the intercepts of your graph in part(c)? Explain your reasoning.
Answer:

Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 4

Try It

Graph the linear equation.
Question 1.
x + y = -2
Answer:
Given the equation
y = mx + b
x + y = -2
y = -x – 2
Comparing the value of b and m from y = mx + b
m = -1 and b = -2
Plot y – intercept = (0, b) = (0, -2)
Slope = -1
run/rise = -1/1
Plot the point 1 unit down and 1 unit to the right = (1, -3)
Now plot the points and draw the graph
BIM Grade 8 Solution Key Chapter 4 img_74

Question 2.
–\(\frac{1}{2}\)x + 2y = 6
Answer:
–\(\frac{1}{2}\)x + 2y = 6
2y = 6 + \(\frac{1}{2}\)x
y = 0.25x + 3
Comparing the value of b and m from y = mx + b
m = 0.25 and b = 3
Plot y – intercept = (0, b) = (0, 3)
Slope = 0.25
run/rise = 0.25/1
Plot the point 0.25 unit up and 1 unit to the right = (1, 3.25)
Now plot the points and draw the graph

BIM Grade 8 Solution Key Chapter 4 img_75

Question 3.
–\(\frac{2}{3}\)x + y = 0
Answer:
–\(\frac{2}{3}\)x + y = 0
y = \(\frac{2}{3}\)x
Comparing the value of b and m from y = mx + b
m = \(\frac{2}{3}\) and b = 0
Plot y – intercept = (0, b) = (0, 0)
Slope =\(\frac{2}{3}\)
run/rise = \(\frac{2}{3}\)
Plot the point 0.25 unit up and 1 unit to the right = (3, 2)
Now plot the points and draw the graph
BIM Grade 8 Solution Key Chapter 4 img_76

Question 4.
2x + y = 5
Answer:
2x + y = 5
y = -2x + 5
Comparing the value of b and m from y = mx + b
m = -2 and b = 5
Plot y – intercept = (0, b) = (0, 5)
Slope = -2
run/rise = \(\frac{-2}{1}\)
Plot the point 0.25 unit up and 1 unit to the right = (1, 3)
Now plot the points and draw the graph
BIM Grade 8 Solution Key Chapter 4 img_77

Graph the linear equation using intercepts.
Question 5.
2x – y = 8
Answer:
y = 0
2x – y = 8
2x – 0 = 8
2x = 8
x = 4
The x – intercept is (4, 0)
Y – intercept :
x = 0
2x – y = 8
2(0) – y = 8
y = -8
BIM Grade 8 Solution Key Chapter 4 img_78

Question 6.
x + 3y = 6
Answer:
X-intercept:
y = 0
x + 3y = 6
x + 3(0) = 6
x + 0 = 6
x = 6
The x – intercept is (6, 0)
Y – intercept:
x = 0
x + 3y = 6
0 + 3y = 6
y = 2
The y – intercept is (0, 2)
BIM Grade 8 Solution Key Chapter 4 img_79

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

STRUCTURE Determine whether the equation is in standard form. If not, rewrite the equation in standard form.
Question 7.
y = x – 6
Answer:
y = x – 6
The standard form of equation is: Ax + By = C
The given equation is not in the standard form.
y = x – 6
x – y = 6

Question 8.
y – \(\frac{1}{6}\)x + 5 = 0
Answer:
The standard form of equation is: Ax + By = C
The given equation is not in the standard form.
y – \(\frac{1}{6}\)x + 5 = 0
\(\frac{1}{6}\)x – y = 5

Question 9.
4x + y = 5
Answer:
The standard form of equation is: Ax + By = C
The given equation is in the form of the standard form.

Question 10.
WRITING
Describe two ways to graph the equation 4x + 2y = 6.
Answer:
The two ways to graph the equation:
1. Graph the equation using standard form
2. Graph the equation using intercept.

GRAPHING A LINEAR EQUATION Graph the linear equation.
Question 11.
4x + y = 5
Answer:
Given the equation
4x + y = 5
y = -4x + 5
Comparing the value of b and m from y = mx + b
m = -4 and b = 5
Plot y – intercept = (0, b) = (0, 5)
Slope = -4
run/rise = \(\frac{-4}{1}\)
Plot the point 4 unit down and 1 unit to the right = (1, 1)
Now plot the points and draw the graph
BIM Grade 8 Solution Key Chapter 4 img_81

Question 12.
\(\frac{1}{3}\)x + 2y = 8
Answer:
X – intercept:
y = 0
\(\frac{1}{3}\)x + 2y = 8
\(\frac{1}{3}\)x + 2(0) = 8
\(\frac{1}{3}\)x = 8
x = 24
The x – intercept is (24, 0)
Y – intercept:
x = 0
\(\frac{1}{3}\)x + 2y = 8
\(\frac{1}{3}\)(0) + 2y = 8
2y = 8
y = 4
The y – intercept is (0, 4)
BIM Grade 8 Solution Key Chapter 4 img_82

Question 13.
5x – y = 10
Answer:
X – intercept:
y = 0
5x – 0 = 10
5x = 10
x = 2
The x-intercept is (2, 0)
Y – intercept:
x = 0
5x – y = 10
5(0) – y = 10
-y = 10
y = -10
The y – intercept is (0, -10)
BIM Grade 8 Solution Key Chapter 4 img_83

Question 14.
x – 3y = 9
Answer:
X – intercept:
y = 0
x – 3(0) = 9
x = 9
The x – intercept is (9, 0)
Y – intercept:
x = 0
0 – 3y = 9
-3y = 9
y = -3
The y – intercept is (0, -3)
BIM Grade 8 Solution Key Chapter 4 img_84

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 15.
You have $30 to spend on paint and clay. The equation 2x + 6y = 30 represents this situation, where x is the number of paint bottles and y is the number of tubs of clay. Graph the equation. Interpret the intercepts. How many bottles of paint can you buy if you buy 3 tubs of clay? Justify your answer.
Answer:
Given,
You have $30 to spend on paint and clay.
The equation 2x + 6y = 30 represents this situation,
where x is the number of paint bottles and y is the number of tubs of clay.
X – intercept:
y = 0
2x + 6y = 30
2x + 6(0) = 30
2x = 30
x = 15
The x – intercept is (15, 0)
Y – intercept:
x = 0
2x + 6y = 30
2(0) + 6y = 30
6y = 30
y = 5
The y – intercept is (0, 5)
BIM Grade 8 Solution Key Chapter 4 img_85
From the graph, I can buy 6 bottles of point if I buy 3 tubs of clay.
BIM Grade 8 Solution Key Chapter 4 img_86

Question 16.
You complete two projects for a class in 60 minutes. The equation x + y = 60 represents this situation, where x is the time (in minutes) you spend assembling a birdhouse and y is the time (in minutes) you spend writing a paper.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 5
a. Graph the equation. Interpret the intercepts.

Answer:
x + y = 60
y = -x + 60
BIM Grade 8 Solution Key Chapter 4 img_87

b. You spend twice as much time assembling the birdhouse as you do writing the paper. How much time do you spend writing the paper? Justify your answer.
Answer:
We are given,
y = 2x
2x = -x + 60
2x + x = 60
3x = 60
x = 20
y = 2 (20)
y = 40

Graphing Linear Equations in Standard Form Homework & Practice 4.5

Review & Refresh

Find the slope and the y-intercept of the graph of the linear equation.
Question 1.
y = x – 1
Answer:
y = mx + b
Slope = -1 and y – intercept = -1

Question 2.
y = -2x + 1
Answer:
y = -2x + 1
y = mx + b
Slope = -2 and y – intercept = 1

Question 3.
y = \(\frac{8}{9}\)x – 8
Answer:
y = \(\frac{8}{9}\)x – 8
y = mx + b
Slope = \(\frac{8}{9}\) and y – intercept = -8

Tell whether the blue figure is a reflection of the red figure.
Question 4.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 6
Answer:
The blue figure is not a reflection of the red figure because, for example the reflection of the upper leg of the upper leg of the red triangle across the y-axis is the top vertex of the blue triangle, not a point.

Question 5.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 7
Answer:
The blue figure is a reflection of the red figure because to each point in the red figure corresponds a symmetrical point in the blue figure.

Question 6.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 8
Answer:
The blue figure is a reflection of the red figure because to each point in the red figure corresponds a symmetrical point in the blue figure.

Concepts, Skills, &Problem Solving

USING INTERCEPTS Define two variables for the verbal model. Write an equation in slope-intercept form that relates the variables. Graph the equation using intercepts. (See Exploration 1, p. 167.)
Question 7.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 9
Answer:
x = amount of peaches (in pounds)
y = the amount of apples (in pounds)
2x + 1.5y = 15
y = 0 = 2x + 1.5(0) = 15
2x = 15
x = 7.5
x = 0
2(0) + 1.5y = 15
1.5y =15
y = 10
BIM Grade 8 Solution key Chapter 4 img_88

Question 8.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 10
Answer:
x = the biked distance (in miles)
y = the walked distance (in miles)
y = 0
16x + 2(0) = 32
16x = 32
x = 2
x = 0
16(0) + 2y = 32
2y = 32
y = 16
BIM Grade 8 Solution key Chapter 4 img_89

REWRITING AN EQUATION Write the linear equation in slope-intercept form.
Question 9.
2x + y = 17
Answer:
Given the equation
2x + y = 17
y = 17 – 2x
y = -2x + 17

Question 10.
5x – y = \(\frac{1}{4}\)
Answer:
Given the equation
5x – y = \(\frac{1}{4}\)
-y = \(\frac{1}{4}\) – 5x
y = 5x – \(\frac{1}{4}\)

Question 11.
–\(\frac{1}{2}\)x + y = 10
Answer:
Given the equation
–\(\frac{1}{2}\)x + y = 10
y = \(\frac{1}{2}\)x + 10

GRAPHING AN EQUATION Graph the linear equation.
Question 12.
-18x + 9y = 72
Answer:
Given the equation
-18x + 9y = 72
X – intercept:
y = 0
-18x + 9(0) = 72
-18x = 72
x = -4
The x – intercept is (-4, 0)
Y – intercept:
x = 0
-18x + 9y = 72
-18(0) + 9y = 72
9y = 72
y = 8
BIM Grade 8 Answers Chapter 4 img_90

Question 13.
16x – 4y = 2
Answer:
Given the equation
16x – 4y = 2
X – intercept:
y = 0
16x – 4y = 2
16x – 4(0) = 2
16x = 2
x = 0.125
The X – intercept is (0.125, 0)
Y – intercept:
x = 0
16(0) – 4y = 2
-4y = 2
y = -2
BIM Grade 8 Answers Chapter 4 img_91

Question 14.
\(\frac{1}{4}\)x + \(\frac{3}{4}\)y = 1
Answer:
Given the equation
\(\frac{1}{4}\)x + \(\frac{3}{4}\)y = 1
x + 3y = 4
y = 0
x + 3(0) = 4
x = 4
x = 0
0 + 3y = 4
3y = 4
y = 4/3
BIM Grade 8 Answers Chapter 4 img_93

MATCHING Match the equation with its graph.
Question 15.
15x – 12y = 60
Answer:
y = 0
15x – 12(0) = 60
15x = 60
x = 60/15
x = 4
x = 0
15(0) – 12y = 60
-12y = 60
y = -5
The graph having the x – intercept 4 and y – intercept -5

Question 16.
5x + 4y = 20
Answer:
Given the linear equation
5x + 4y = 20
y = 0
5x + 4(0) = 20
5x = 20
x = 4
x = 0
5(0) + 4y = 20
4y = 20
y = 5

Question 17.
10x + 8y = -40
Answer:

Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 11
10x + 8y = -40
y = 0
10x + 8(0) = -40
10x = -40
x = -4
x = 0
10(0) + 8y = -40
8y = -40
y = -5

Question 18
YOU BE THE TEACHER
Your friend finds the x-intercept of -2x + 3y = 12. Is your friend correct? Explain your reasoning.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 12
Answer:
-2x + 3y = 12
y = 0
-2x + 3(0) = 12
-2x = 12
x = -6
Your friend is not correct because the x – intercept is the value of x corresponding to y = 0.
Your friend computed the y – intercept.

Question 19.
MODELING REAL LIFE
A charm bracelet costs $65, plus $25 for each charm. The equation -25x + y = 65 represents the cost y (in dollars) of the bracelet, where x is the number of charms.
a. Graph the equation.
b. How much does a bracelet with three charms cost?
Answer:
BIM Grade 8 Answers Chapter 4 img_94
y = 25x + 65
Substitute the value of x in the equation
y = 25(3) + 65
y = 75 + 65
y = 140

USING INTERCEPTS TO GRAPH Graph the linear equation using intercepts.
Question 20.
3x – 4y = -12
Answer:
Given the equation
3x – 4y = -12
3x – 4(0) = -12
3x = -12
x = -4
The x – intercept is (-4, 0)
Y – intercept:
x = 0
3(0) – 4y = -12
-4y = -12
y = 3
The y – intercept is (0, 3)
BIM Grade 8 Answers Chapter 4 img_95

Question 21.
2x + y = 8
Answer:
X – intercept:
y = 0
2x + y = 8
2x + 0 = 8
2x = 8
x = 4
The x – intercept is (4, 0)
Y – intercept:
x = 0
2x + y = 8
2(0) + y = 8
y = 8
The y – intercept is (0, 8)
BIM Grade 8 Answers Chapter 4 img_96

Question 22.
\(\frac{1}{3}\)x – \(\frac{1}{6}\)y = –\(\frac{2}{3}\)
Answer:
X – intercept:
y = 0
\(\frac{1}{3}\)x – \(\frac{1}{6}\)(0) = –\(\frac{2}{3}\)
\(\frac{1}{3}\)x = –\(\frac{2}{3}\)
x = -2
The x – intercept is (-2, 0)
Y – intercept:
x = 0
\(\frac{1}{3}\)(0) – \(\frac{1}{6}\)y = –\(\frac{2}{3}\)
y = 4
The y – intercept is (0, 4)
BIM Grade 8 Answers Chapter 4 img_97

Question 23.
MODELING REAL LIFE
Your cousin has $90 to spend on video games and movies. The equation 30x + 15y = 90 represents this situation, where x is the number of video games purchased and y is the number of movies purchased. Graph the equation. Interpret the intercepts.
Answer:
30x + 15y = 90
x = 0
30(0) + 15y = 90
15y = 90
y = 6
y = 0
30x + 15(0) = 90
30x = 90
x = 3
BIM Grade 8 Answers Chapter 4 img_98
The x – intercept shows that 3 video games are purchased when no movies are purchased.
The y – intercept shows that 6 movies are purchased when no video games are purchased.

Question 24.
PROBLEM SOLVING
A group of friends go scuba diving. They rent a boat for x days and scuba gear for y people, represented by the equation 250x + 50y = 1000.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 13
a. Graph the equation and interpret the intercepts.
b. How many friends can go scuba diving if they rent the boat for 1 day? 2 days?
c. How much money is spent in total?
Answer:
250x + 50y = 1000
x = 0
250(0) + 50y = 1000
50y = 1000
y = 20
when y = 0
250x + 50(0) = 1000
250x = 1000
x = 4
BIM Grade 8 Answers Chapter 4 img_99
b.
250(1) + 50y = 1000
250 + 50y = 1000
50y = 1000 – 250
50y = 750
y = 15
when x = 2
250(2) + 50y = 1000
500 + 50y = 1000
50y = 1000 – 500
50y = 500
y = 500/50
y = 10

Question 25.
DIG DEEPER!
You work at a restaurant as a host and a server. You earn $9.45 for each hour you work as a host and $3.78 for each hour you work as a server.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 14
a. Write an equation in standard form that models your earnings.
b. Graph the equation.
Answer:
You earn $9.45 for each hour you work as a host and $3.78 for each hour you work as a server.
Number of hours worked as host + $3.78.
Number of hours worked as server = $113.40
9.45x + 3.78y = 113.40
x = 0
9.45(0) + 3.78y = 113.40
3.78y = 113.40
y = 30
when y = 0
9.45x + 3.78(0) = 113.40
9.45x = 113.40
x = 12
BIM Grade 8 Answers Chapter 4 img_100

Question 26.
LOGIC
Does the graph of every linear equation have an x-intercept? Justify your reasoning.
Answer:
y = mx + b
y = 0
0 = mx + b
mx = -b
x = -b/m for m ≠ 0
If m = 0 the equation has no solution. Therefore the equation y = b has no x – intercept.

Question 27.
CRITICAL THINKING
For a house call, a veterinarian charges $70, plus $40 per hour.
Big Ideas Math Answer Key Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.5 15
a. Write an equation that represents the total fee y (in dollars) the veterinarian charges for a visit lasting x hours.

b. Find the x-intercept. Does this value make sense in this context? Explain your reasoning.
c. Graph the equation.
Answer:
Total fee = fixed charge + number of hours . cost per hour
y = 70 + 40x
y = 0
0 = 70 + 40x
-70 = 40x
x = -1.75
x = 0
y = 70 + 40(0)
y = 70
BIM Grade 8 Solutions Chapter 4 img_101

Lesson 4.6 Writing Equations in Slope-Intercept Form

EXPLORATION 1

Writing Equations of Lines
Work with a partner.For each part, answer the following questions.

  • What are the slopes and the y-intercepts of the lines?
  • What are equations that represent the lines?
  • What do the lines have in common?

Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 1
Answer:

EXPLORATION 2

Interpreting the Slope and the y-Intercept
Work with a partner. The graph represents the distance y (in miles) of a car from Phoenix after t hours of a trip.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 2
a. Find the slope and the y-intercept of the line. What do they represent in this situation?
b. Write an equation that represents the graph.
c. How can you determine the distance of the car from Phoenix after 11 hours?
Answer:

Try It

Write an equation in slope-intercept form of the line that passes through the given points.
Question 1.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 3
Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 2)/(1 – 0)
= 2/1
= 2
Because the line crosses the y – axis at (0, 2)
y = mx + b
y = 2x + 2

Question 2.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 4
Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 – 3)/(0 – (-3))
= -4/3
Because y = -1 when x = 0, the y – intercept is -1
y = mx + b
y = -4/3 x – 1

Write an equation of the line that passes through the given points.
Question 3.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 5
Answer:
m = (y2 – y1)/(x2 – x1)
= (5 – 5)/(0 – (-4))
= 0/4
Because y = 5 when x = 0, the y – intercept is 5
y = mx + b
y = (0)x + 5
y = 5

Question 4.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 6
Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – 1)/(3 – 0)
= 0/3
= 0
Because the line crosses the y – axis at (0, 1) the y – intercept is 1
y = mx + b
y = (0)x + 1
y = 1

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING EQUATIONS IN SLOPE-INTERCEPT FORM Write an equation in slope-intercept form of the line that passes through the given points.
Question 5.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 7
Answer:
m = (y2 – y1)/(x2 – x1)
= (5 – 2)/(1 – 0)
= 3/1
= 3
Because y = 2 when x = 0, the y – intercept is 2
y = mx + b
y = (3)x + 2
y = 3x + 2

Question 6.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 8
Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 – 5)/(1 – (-1))
= -6/2
= -3
Because the line crosses the y – axis at (0, 2) the y – intercept is 2
y = mx + b
y = -3x + 2

Question 7.
WRITING AN EQUATION
Write an equation of the line that passes through (0, -5) and (2, -5).
Answer:
m = (y2 – y1)/(x2 – x1)
= (-5 – (-5))/(2 – 0)
= 0/2
= 0
Because y = -5 when x = 0, the y – intercept is -5
y = mx + b
y = (0)x + -5
y = -5

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.
You load boxes onto an empty truck at a constant rate. After 3 hours, there are 100 boxes on the truck. How much longer do you work if you load a total of 120 boxes? Justify your answer.
Answer:
Let x be the number of hours you work if you load a total of 120 boxes.
100/3 = 120/x
100x = 3 × 120
x = 360/100
x = 3.6 hours
3.6 – 3 = 0.6 hours

Question 9.
The table shows the amounts (in tons) of waste left in a landfill after x months of waste relocation. Interpret the slope and the y-intercept of the line that passes through the given points. How many months does it take to empty the landfill? Justify your answer.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 9
Answer:
m = (12 – 15)/ (6 – 0)
m = -3/6
m = -0.5
b = 15
The y – intercept shows that there are 150 tons of waste in the beginning.
y = -0.5x + 15
y = 0
0 = -0.5x + 15
x = 30
So the ladfill will be emptied after 30 months.

Question 10.
DIG DEEPER!
A lifetime subscription to a website costs $250. A monthly subscription to the website costs $10 to join and $15 per month. Write equations to represent the costs of each plan. If you want to be a member for one year, which plan is less expensive? Explain.
Answer:
Given,
A lifetime subscription to a website costs $250. A monthly subscription to the website costs $10 to join and $15 per month.
Total cost for plan 1 = the lifetime subscription
y = 250
Total cost for Plan 2 = Fixed tax + Number of months . monthly cost
y = 10 + 15x
Plan 1: y = 250
Plan 2: y = 10 + 15(12) = 190
As 190 < 250, plan 1 is less expensive.

Writing Equations in Slope-Intercept Form Homework & Practice 4.6

Review & Refresh

Write the linear equation in slope-intercept form.
Question 1.
4x + y = 1
Answer:
Given the equation
4x + y = 1
y = -4x + 1

Question 2.
x – y = \(\frac{1}{5}\)
Answer:
Given the equation
x – y = \(\frac{1}{5}\)
x – \(\frac{1}{5}\) = y

Question 3.
–\(\frac{2}{3}\)x + 2y = -7
Answer:
Given the equation
–\(\frac{2}{3}\)x + 2y = -7
2y = -7 + \(\frac{2}{3}\)x
y = \(\frac{1}{3}\)x – \(\frac{7}{2}\)

Plot the ordered pair in a coordinate plane.
Question 4.
(1, 4)
Answer:
8th Grade BIM Answers Chapter 4 img_102

Question 5.
(-1, -2)
Answer:
8th Grade BIM Answers Chapter 4 img_103

Question 6.
(0, 1)
Answer:
8th Grade BIM Answers Chapter 4 img_104

Question 7.
(2, 7)
Answer:
8th Grade BIM Answers Chapter 4 img_105

Concepts, Skills, & Problem Solving

INTERPRETING THE SLOPE AND THE y-INTERCEPT The graph y represents the cost (in dollars) to open an online gaming account and buy x games. (See Exploration 2, p. 173.)
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 10
Question 8.
Find the slope and the y-intercept of the line. What do they represent in this situation?
Answer:
(0, 15), (3, 45)
m = (45 – 15)/(3 – 0)
m = 30/3 10
Thus the slope of the line is m – 3.
b = 15
The slope represents the cost of one game, while the y – intercept is the cost of opening the gaming account.

Question 9.
Write an equation that represents the graph.
Answer:
m = 10
b = 15
y = mx + b
y = 10x + 15

Question 10.
How can you determine the total cost of opening an account and buying 6 games?
Answer:
y = 10x + 15
y = 10(6) + 15
y = 60 + 15
y = 75

WRITING EQUATIONS IN SLOPE-INTERCEPT FORM Write an equation in slope-intercept form of the line that passes through the given points.
Question 11.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 11
Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 3)/(0 – (-1))
= 1/1
= 1
Because the line crosses the y – axis at (0, 4) the y – intercept is 4
y = mx + b
y = (1)x + 4
y = x + 4

Question 12.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 12
Answer:
m = (y2 – y1)/(x2 – x1)
= (6 – 0)/(-3 – 0)
= 6/-3
= -2
Because the line crosses the y – axis at (0, 2) the y – intercept is 2
y = mx + b
y = -2x + 0
y = -2x

Question 13.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 13
Answer:
m = (y2 – y1)/(x2 – x1)
= (2 – 1)/(4 – 0)
= 1/4
Because the line crosses the y – axis at (0, 1) the y – intercept is 1
y = mx + b
y = 1/4 x + 1

Question 14.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 14
Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – 2)/(0 – (-2))
= -1/2
Because y = 1 when x = 0, the y – intercept is 1
y = mx + b
y = -1/2 x + 2

Question 15.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 15
Answer:
m = (y2 – y1)/(x2 – x1)
= (-3 – (-4))/(0 – (-3))
= 1/3
Because y = -3 when x = 0, the y – intercept is -3
y = mx + b
y = 1/3 x – 3

Question 16.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 16
Answer:
m = (y2 – y1)/(x2 – x1)
= (-1 -4)/(0 – (-2))
= -5/2
Because y = -1 when x = 0, the y – intercept is -1
y = mx + b
y = -5/2 x – 1

WRITING EQUATIONS Write an equation of the line that passes through the given points.
Question 17.
(-1, 4), (0, 2)
Answer:
m = (y2 – y1)/(x2 – x1)
= (2 – 4)/(0 – (-1))
= -2/1
= -2
Because y = 2 when x = 0, the y – intercept is 2
y = mx + b
y = -2x + 2

Question 18.
(-1, 0), (0, 0)
Answer:
m = (y2 – y1)/(x2 – x1)
= (0 – 0)/(0 – (-1))
= 0/1
= 0
Because y = 0 when x = 0, the y – intercept is 0
y = mx + b
y = 0

Question 19.
(0, 4), (0, -3)
Answer:
Both points belong to the y-axis. Therefore the equation of the line passing through them is
x = 0

Question 20.
YOU BE THE TEACHER
Your friend writes an equation of the line shown. Is your friend correct? Explain your reasoning.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 17
Answer:
Because in the given graph, y = -2 when x = 0, so the y – intercept is -2. The equation of the line should be: y = 1/2 x – 2
No my friend is NOT correct.

Question 21.
MODELING REAL LIFE
A boa constrictor is 18 inches long at birth and grows 8 inches per year. Write an equation in slope y-intercept form that represents the length (in feet) of a boa constrictor that is x years old.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 18
Answer:
Given,
A boa constrictor is 18 inches long at birth and grows 8 inches per year.
Length after x years = birth length + number of years . Growth per year
y = 18 + 8x
y = 8x + 18
Convert it into feet
y = 2/3 x + 3/2

Question 22.
MODELING REAL LIFE
The table shows the speeds y (in miles per hour) of a car after x seconds of braking. Write an equation of the line that passes through the points in the table. Interpret the slope and the y-intercept of the line.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 19
Answer:
m = (y2 – y1)/(x2 – x1)
= (60 – 70)/(1 – 0)
= -10/1
= -10
Because y = 70 when x = 0, the y – intercept is 70
y = mx + b
y = -10x + 70
Slope = -10 represents the decrease in the speed of the car each seconds after breaking.
The y – intercept of 70 represents the initial speed of the car.

Question 23.
MODELING REAL LIFE
A dentist charges a flat fee for an office visit, plus an additional fee for every tooth removed. The graph shows the total cost y (in dollars) for a patient when the dentist removes x teeth. Interpret the slope and the y-intercept.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 20
Answer:
(2, 500), (4, 900)
m = (900 – 500)/(4 – 2)
m = 400/2
m = 200
y = mx + b
500 = 200(2) + b
500 = 400 + b
b = 500 – 400
b = 100
The slope shows that the amount charged for each removed tooth is $200.
The y – intercept shows that the flat fee for an office visit is $100.

Question 24.
MODELING REAL LIFE
One of your friends gives you $10 for a charity walkathon. Another friend gives you an amount per mile. After 5 miles, you have raised $13.50 total. Write an equation that represents the amount y of money you have raised after x miles.
Answer:
Given,
One of your friends gives you $10 for a charity walkathon.
Another friend gives you an amount per mile. After 5 miles, you have raised $13.50 total.
y = mx + b
b = 10
13.50 = 5m + 10
13.50 – 10 = 5m
3.50 = 5m
m = 3.50/5
m = 0.7
y = 0.7x + 10

Question 25.
PROBLEM SOLVING
You have 500 sheets of notebook paper. After 1 week, you have 72% of the sheets left. You use the same number of sheets each week. Write an equation that represents the number y of sheets remaining after x weeks.
Answer:
y = mx + b
500 – 0.72 × 500 = 500 – 360 = 140 sheets
m = -140
b = 500
y = -140x + 500

Question 26.
DIG DEEPER!
The palm tree on the left is 10 years old. The palm tree on the right is 8 years old. The trees grow at the same rate.
Big Ideas Math Answers 8th Grade Chapter 4 Graphing and Writing Linear Equations 4.6 21
a. Estimate the height y (in feet) of each tree.
b. Plot the two points (x, y), where x is the age of each tree and y is the height of each tree.
c. What is the rate of growth of the trees?
d. Write an equation that represents the height of a palm tree in terms of its age.
Answer:
a. estimate
left: 18
right: 12
plot y = 1.8x

Lesson 4.7 Writing Equations in Point-Slope Form

EXPLORATION 1

Deriving an Equation
Work with a partner. Let (x1, y1) represent a specific point on a line. Let (x, y) represent any other point on the line.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 1
a. Write an equation that represents the slope m of the line. Explain your reasoning.
b. Multiply each side of your equation in part(a) by the expression in the denominator. What does the resulting equation represent? Explain your reasoning.
Answer:

EXPLORATION 2

Writing an Equation
Work with a partner.
For 4 months, you saved $25 a month. You now have $175 in your savings account.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 2
a. Draw a graph that shows the balance in your account after t months.
b.Use your result from Exploration 1 to write an equation that represents the balance A after t months.
Answer:

Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 3

Try It
Write an equation in point -slope form of the line that passes through the given point and has the given slope.
Question 1.
(1, 2); m = -4
Answer:
y – y1 = m(x – x1)
y – 2 = -4(x – (1))
y – 2 = -4(x – 1)

Question 2.
(7, 0); m = 1
Answer:
y – y1 = m(x – x1)
y – 0 = 1(x – (7))
y – 0 = 1(x – 7)

Question 3.
(-8, -5); m = –\(\frac{3}{4}\)
Answer:
y – y1 = m(x – x1)
y – (-5) = –\(\frac{3}{4}\)(x – (-8))
y + 5 = –\(\frac{3}{4}\)(x + 8)

Write an equation in slope-intercept form of the line that passes through the given points.
Question 4.
(-2, 1), (3, -4)
Answer:
Slope(m) = (-4 – 1)/(3 – (-2))
= -5/5
m = -1
y – y1 = m(x – x1)
y – 1 = -1(x – (-2))
y – 1 = -1(x + 2)
y – 1 = -x – 2
y = -x – 1

Question 5.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 4
Answer:
Slope(m) = (3 – 5)/(-3 – (-5))
= -2/2
m = -1
y – y1 = m(x – x1)
y – 1 = -1(x – (-1))
y – 1 = -1(x + 1)
y – 1 = -x – 1
y = -x – 1 + 1
y = -x

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING AN EQUATION Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 6.
(2, 0); m = 1
Answer:
y – y1 = m(x – x1)
y – 0 = 1(x – (2))
y – 0 = 1(x – 2)

Question 7.
(-3, -1); m = –\(\frac{1}{3}\)
Answer:
y – y1 = m(x – x1)
y – (-1) = –\(\frac{1}{3}\)(x – (-3))
y + 1 = –\(\frac{1}{3}\)(x + 3)

Question 8.
(5, 4); m = 3
Answer:
y – y1 = m(x – x1)
y – 4 = 3(x – (5))
y – 4 = 3(x – 5)

Question 9.
WRITING AN EQUATION
Write an equation of the line that passes through the points given in the table.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 5
Answer:
Slope(m) = (-2 – 1)/(5 – 3)
= -3/2
m = -1
y – y1 = m(x – x1)
y – (-5) = -3/2(x – 7)
y + 5 = -3/2(x – 7)
y + 5 = -3/2 x + 21/2
y = -3/2 x + 11/2

Question 10.
DIFFERENT WORDS, SAME QUESTION
Which is different? Sketch “both” graphs.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 6
Answer:
y – 7 = 4x – 4
y = 4x + -4 + 7
y = 4x + 3
Graph line passes through the points (4, 5) and (5, 9)

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 11.
A writer finishes a project that a coworker started at a rate of 3 pages per hour. After 3 hours,25% of the project is complete.
a. The project is 200 pages long. Write and graph an equation for the total number y of pages that have been finished after the writer works for x hours.
b. The writer has a total of 45 hours to finish the project. Will the writer meet the deadline? Explain your reasoning.
Answer:
m = 3
y = 3x + b
b + 9 = 25%(200)
b + 9 = 0.25(200)
b + 9 = 50
b = 50 – 9
b = 41
y = 3x + 41
BIM 8th Grade Solution Key Ch 4 img_106
y = 3x + 41
y = 3(45) + 41 = 176 pages
As 176 < 200, the writer will not meet the deadline.

Question 12.
DIG DEEPER!
You and your friend begin to run along a path at different constant speeds.After 1 minute,your friend is 45 meters ahead of you. After 3 minutes, your friend is 105 meters ahead of you.
a. Write and graph an equation for the distance y (in meters) your friend is ahead of you after x minutes. Justify your answer.

Answer:
y = mx + b
45 = m + b
105 = 3m + b
105 – 45 = (3m + b) – (m + b)
60 = 2m
m = 30
45 = 30 + b
b = 45 – 30
b = 15
y = 30x + 15
BIM 8th Grade Solution Key Ch 4 img_107

b. Did you and your friend start running from the same spot? Explain your reasoning.
Answer:
The distance between you and your friend in the initial moment is b = 15 meters. So you are ahead your friend by 15 meters at the starting point.

Writing Equations in Point-Slope Form Homework & Practice 4.7

Review & Refresh

Write an equation in slope-intercept form of the line that passes through the given points.
Question 1.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 7
Answer:
Slope(m) = (5 – 4)/(0 – (-2))
= 1/2
m = 1/2
Because y = 5 when x = 0, the y – intercept is 5.
y = mx + b
y = 1/2 x + 5

Question 2.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 8
Answer:
Slope(m) = (5 – (-1))/(2 – (-2))
= (5 + 1)/(2 + 2)
m = 6/4
m = 3/2
From the graph, the line crosses the y – axis at (0, 2)
y = mx + b
y = 3/2 x + 2

Solve the equation. Check your solution, if possible.
Question 3.
2x + 3 = 2x
Answer:
Given the equation
2x + 3 = 2x
3 = 2x – 2x
3 ≠ 0

Question 4.
6x – 7 = 1 – 3x
Answer:
Given the equation
6x – 7 = 1 – 3x
6x + 3x = 1 + 7
9x = 8
x = 8/3

Question 5.
0.1x – 1 = 1.2x – 5.4
Answer:
Given the equation
0.1x – 1 = 1.2x – 5.4
0.1x – 1.2x = 1 – 5.4
-1.1x = -4.4
x = 4

Concepts, Skills, &Problem Solving

WRITING AN EQUATION The value of a new car decreases $4000 each year. After 3 years, the car is worth $18,000. (See Exploration 2, p. 179.)
Question 6.
Draw a graph that shows the value of the car after t years.
Answer:
BIM 8th Grade Solution Key Chapter 4 img_111

Question 7.
Write an equation that represents the value V of the car after t years.
Answer:
y = -4000t + b
where b is the original price
18,000 = -4000(3) + b
18,000 + 12,000 = b
b = 30,000
y = -4000t + 30,000

WRITING AN EQUATION Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 8.
(3, 0); m = –\(\frac{2}{3}\)
Answer:
y – y1 = m(x – x1)
y – (0) = -2/3(x – 3)
y – 0 = -2/3(x – 3)

Question 9.
(4, 8); m = \(\frac{3}{4}\)
Answer:
y – y1 = m(x – x1)
y – (8) = 3/4(x – 4)
y – 8 = 3/4(x – 4)

Question 10.
(1, -3); m = 4
Answer:
y – y1 = m(x – x1)
y – (-3) = 4(x – 1)
y + 3 = 4(x – 1)

Question 11.
(7, -5); m = –\(\frac{1}{7}\)
Answer:
y – y1 = m(x – x1)
y – (-5) = –\(\frac{1}{7}\)(x – 7)
y + 5 = –\(\frac{1}{7}\)(x – 7)

Question 12.
(3, 3); m = \(\frac{5}{3}\)
Answer:
y – y1 = m(x – x1)
y – (3) = \(\frac{5}{3}\)(x – 3)
y – 3 = \(\frac{5}{3}\)(x – 3)

Question 13.
(-1, -4); m = -2
Answer:
y – y1 = m(x – x1)
y – (-4) = -2(x – (-1))
y + 4 = -2(x + 1)

WRITING AN EQUATION Write an equation in slope-intercept form of the line that passes through the given points.
Question 14.
(-1, -1), (1, 5)
Answer:
Slope(m) = (5 – (-1))/(2 – (-1))
= (5 + 1)/(1 + 1)
m = 6/2
m = 3
y – y1 = m(x – x1)
y – (5) = 3(x – (1))
y – 5 = 3x – 3
y = 3x + 2

Question 15.
(2, 4), (3, 6)
Answer:
Slope(m) = (6 – 4)/(3 – 2)
m = 2/1
m = 2
y – y1 = m(x – x1)
y – (4) = 2(x – (2))
y – 4 = 2x – 4
y = 2x

Question 16.
(-2, 3), (2, 7)
Answer:
Slope(m) = (7 – (3))/(2 – (-2))
= (7 – 3)/(2 + 2)
m = 4/4
m = 1
y – y1 = m(x – x1)
y – (3) = 1(x – (-2))
y – 3 = x + 2
y = x + 5

Question 17.
(4, 1), (8, 2)
Answer:
Slope(m) = (2 – (1))/(8 – (4))
= (2 – 1)/(8 – 4)
m = 1/4
y – y1 = m(x – x1)
y – (1) = 1/4(x – (4))
y – 1 = 1/4 x – 1
y = 1/4 x

Question 18.
(-9, 5), (-3, 3)
Answer:
Slope(m) = (3 – (5))/(-3 – (-9))
= (3 – 5)/(-3 + 9)
m = -2/6
m = -1/3
y – y1 = m(x – x1)
y – (3) = -1/3(x + 3)
y – 3 = -1/3 x – 1
y = -1/3 x + 2

Question 19.
(1, 2), (-2, -1)
Answer:
Slope(m) = (2 – (1))/(8 – (4))
= (-1 – 2)/(-2 – 1)
m = -3/-3
m = 1
y – y1 = m(x – x1)
y – (2) = 1(x – (1))
y – 2 = x – 1
y = x + 1

Question 20.
MODELING REAL LIFE
At 0° C, the volume of a gas is 22 liters. For each degree the temperature T (in degrees Celsius) increases, the volume V (in liters) of the gas increases by \(\frac{2}{25}\). Write an equation that represents the volume of the gas in terms of the temperature.
Answer:
The equation modeling the situation has the form:
V = mT + b
m = 2/25
22 = 2/25(0) + b
b = 22
V = 2/25 T + 22

WRITING AN EQUATION Write an equation of the line that passes through the given points in any form. Explain your choice of form.
Question 21.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 9
Answer:
m = (y2 – y1)/(x2 – x1)
= (2.5 – 1.5)/(0 – (-1))
= 1/1
= 1
Because the line crosses the y – axis at (0, 2.5), the y – intercept is 2.5
y = mx + b
y = (1)x + 2.5
y = x + 2.5

Question 22.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 10
Answer:
m = (y2 – y1)/(x2 – x1)
= (3.5 – 1.5)/(2 – (1))
= 2/1
= 2
y – y1 = m(x – x1)
y – (1.5) = 2(x – (1))
y – 1.5 = 2x – 2
y = 2x – 0.5

Question 23.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 11
Answer:
m = (y2 – y1)/(x2 – x1)
= (-1.5 – 4.5)/(1 – (-1))
= -6/2
= -3
y – y1 = m(x – x1)
y – (-1.5) = -3(x – (1))
y + 1.5 = -3x + 3
y = -3x + 1.5

Question 24.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 12
Answer:
m = (y2 – y1)/(x2 – x1)
= (-0.5 – 3.5)/(1 – (-1))
= -4/2
= -2
y – y1 = m(x – x1)
y – (-0.5) = -2(x – (1))
y + 0.5 = -2x – 2
y = -2x – 2.5

Question 25.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 13
Answer:
m = (y2 – y1)/(x2 – x1)
= (1 – (-1))/(0 – (-3))
= (1 + 1)/(0 + 3)
= 2/3
Because y = 1 when x = 0, the y – intercept is 1.
y = mx + b
y = 2/3 x + 1

Question 26.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 14
Answer:
m = (y2 – y1)/(x2 – x1)
= (4 – 6)/(-3 – (-7))
= -2/4
= -1/2
y – y1 = m(x – x1)
y – (2) = -1/2(x – (1))
y – 2 = -1/2x + 1/2
y = -1/2 x + 5/2

Question 27.
REASONING
Write an equation of the line that passes through the point (8, 2) and is parallel to the graph of the equation y = 4x – 3.
Answer:
y = 4x – 3
Comparing the given equation with y = mx + b, we get
m = 4
y – y1 = m(x – x1)
y – 2 = 4(x – 8)
y – 2 = 4x – 32
y = 4x – 32 + 2
y = 4x – 30

Question 28.
MODELING REAL LIFE
The table shows the amount y (in fluid ounces) of carpet cleaner in a tank after x minutes of cleaning.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 15
a. Write an equation that represents the amount of cleaner x in the tank after minutes.
b. How much cleaner is in the tank when the cleaning begins?
c. After how many minutes is the tank empty? Justify your answer.
Answer:

Question 29.
DIG DEEPER!
According to Dolbear’s law, you can predict the temperature T (in degrees Fahrenheit) by counting the number x of chirps made by a snowy tree cricket in 1 minute.When the temperature is 50°F, a cricket chirps 40 times in 1 minute. For each rise in temperature of 0.25°F, the cricket makes an additional chirp each minute.
a. You count 100 chirps in 1 minute. What is the temperature?
b. The temperature is 96°F.How many chirps do you expect the cricket to make? Justify your answer.
Answer:

Question 30.
PROBLEM SOLVING
The Leaning Tower of Pisa in Italy was built between 1173 and 1350.
a. Write an equation that represents the yellow line.
b. The tower is 56 meters tall. How far from the center is the top of the tower? Justify your answer.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 16
Answer:

Graphing and Writing Linear Equations Connecting Concepts

Using the Problem-Solving Plan
Question 1.
Every item in a retail store is on sale for 40% off. Write and graph an equation that represents the sale price of an item that has an original price of x dollars.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cc 1
Understand the problem.
You know the percent discount of items in a retail store.You are asked to write and graph an equation that represents the sale price of an item that has an original price of x dollars.
Make a plan.
Selling an item for 40% off is the same as selling an item for 60% of its original price. Use this information to write and graph an equation that represents the situation.
Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer:
40% = 0.40 and to find a percent of a number you multiply the number by the percent in decimal form.
So, the equation is d = 0.4p
BIM 8th Grade Answers img_111

Question 2.
Two supplementary angles have angle measures of x° and y°. Write and graph an equation that represents the relationship between the measures of the angles.
Answer:

Question 3.
A mechanic charges a diagnostic fee plus an hourly rate. The table shows the numbers of hours worked and the total costs for three customers.A fourth customer pays $285. Find the number of hours that the mechanic worked for the fourth customer.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cc 2
Answer:

Performance Task

Anatomy of a Hurricane
At the beginning of this chapter, you watched a STEAM Video called “Hurricane!” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cc 3

Graphing and Writing Linear Equations Chapter Review

Review Vocabulary
Write the definition and give an example of each vocabulary term.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations cr 1

Graphic Organizers
You can use a Definition and Example Chart to organize information about a concept. Here is an example of a Definition and Example Chart for the vocabulary term linear equation.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations cr 2

Choose and complete a graphic organizer to help you study the concept.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations cr 3
1. slope
2. slope of parallel lines
3. proportional relationship
4. slope-intercept form
5. standard form
6. point-slope form

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 1

4.1 Graphing Linear Equations (pp. 141–146)
Learning Target: Graph linear equations.Graph the linear equation.

Question 1.
y = \(\frac{3}{5}\)x
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_112

Question 2.
y = -2
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_113

Question 3.
y = 9 – x
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_114

Question 4.
y = -0.25x + 4
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_115

Question 5.
y = \(\frac{2}{3}\)x + 2
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_116

Question 6.
x = -5
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_117

Question 7.
The equation y = 0.53x + 3 represents the cost y (in dollars) of riding in a taxi x miles.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 2
a. Use a graph to estimate how much it costs to ride 5.25 miles in a taxi.
b. Use the equation to find exactly how much it costs to ride 5.25 miles in a taxi.
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_118
y = 0.5x + 3
y = 0.5(5.25) + 3
y ≈ 5.6

Question 8.
The equation y = 9.5x represents the earnings y (in dollars) of an aquarium gift shop employee that works x hours.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 8
a. Graph the linear equation.
b. How much does the employee earn for working 40 hours?
Answer:
Bigideas Math Answers 8th Grade chapter 4 img_119
Determine y for x = 40:
y = 9.5x
y = 9.5(40) = 380

Question 9.
Is y = x2 a linear equation? Explain your reasoning.
Answer:
y = x2
The graph of the given equation passes through the origin, but is not linear, therefore it is not a linear equation.
So, the answer is no.

Question 10.
The sum S of the exterior angle measures of a polygon with n sides is S = 360°.
a. Plot four points (n, S) that satisfy the equation. Is the equation a linear equation? Explain your reasoning.
b. Does the value n = 2 make sense in the context of the problem? Explain your reasoning.
Answer:
8th Grade Big Ideas Math Answer Key Chapter 4 img_120
The value n = 2 does not make sense in the context of the problem because a polygon has at least 3 sides.

4.2 Slope of a Line (pp. 147–154)
Learning Target: Find and interpret the slope of a line.

Describe the slope of the line. Then find the slope of the line.
Question 11.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 11
Answer:
(x1, y1) = (3, 1)
(x2, y2) = (-3, -3)
m = (y2 – y1)/(x2 – x1)
m = (-3 – 1)/(-3 – 3)
m = -4/-6
m = 2/3

Question 12.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 12
Answer:
(x1, y1) = (0, 4)
(x2, y2) = (2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – 4)/(2 – 0)
m = -6/2
m = -3
The slope is negative

Find the slope of the line through the given points.
Question 13.
(-5, 4), (8, 4)
Answer:
(x1, y1) = (-5, 4)
(x2, y2) = (8, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 4)/(8 – (-5))
m = 0/13
m = 0

Question 14.
(-3, 5), (-3, 1)
Answer:
(x1, y1) = (-3, 5)
(x2, y2) = (-3, 1)
m = (y2 – y1)/(x2 – x1)
m = (1 – 5)/(-3 + 3)
m = -4/0
m = undefined

The points in the table lie on a line. Find the slope of the line.
Question 15.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 15
Answer:
(x1, y1) = (0, -1)
(x2, y2) = (1, 0)
m = (y2 – y1)/(x2 – x1)
m = (0 – (-1))/(1 – 0)
m = 1/1
m = 1

Question 16.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 16
Answer:
(x1, y1) = (-2, 3)
(x2, y2) = (0, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 3)/(0 – (-2))
m = 1/2

Question 17.
How do you know when two lines are parallel? Use an example to justify your answer.
Answer:
Two lines are parallel when their slopes are the same. In order for the two lines not to coincide, we must add the condition that their y – intercepts.
Example 1:
d1: y = 3x – 6
d2: 3x – y = 6
The lines d1 and d2 have the same slope and the same y – intercept, therefore they coincide.

Question 18.
Draw a line through the point (-1, 2) that is parallel to the graph of the line in Exercise 11.
Answer:
y = 2/3 x – 1
A (-1, 2)
y = 2/3 x + b
y = 2/3 (-1) + b
b = 8/3
The equation of d1 is:
y = 2/3 x + 8/3
Determine the x intercept of d1:
0 = 2/3 x + 8/3
0 = 2x + 8
2x = -8
x = -8/2
x = -4
Bigideas math answers grade 8 ch 4 img_121

4.3 Graphing Proportional Relationships (pp. 155–160)
Learning Target: Graph proportional relationships.

Tell whether x and y are in a proportional relationship. Explain your reasoning. If so, write an equation that represents the relationship.
Question 19.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 19
Answer:
x and y are not in a proportional relationship because the line does not pass through the origin.

Question 20.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 20
Answer:
x and y are in a proportional relationship because the line does passes through the origin.
Determine the slope k using two points from the graph
k = (10 – 0)/(2 – 0)
k = 10/2
k = 5x

Question 21.
The cost y (in dollars) to provide food for guests at a dinner party is proportional to the number x of guests attending the party. It costs $30 to provide food for 4 guests.
a. Write an equation that represents the situation.
b. Interpret the slope of the graph of the equation.
c. How much does it cost to provide food for 10 guests? Justify your answer.
Answer:
y = kx
30 = 4k
k = 30/4
k = 7.5
y = 7.5x
b. The slope 7.5 represents the unit cost for a guest.
y = 7.5 × 10
y = 75
c. Determine y for x = 10
So it costs $75 to provide food for 10 guests.

Question 22.
The distance y (in miles) you run after weeks is represented by the equation y =8x. Graph the equation and interpret the slope.
Answer:
y = 8x
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_122

Question 23.
You research that hair grows 15 centimeters per year on average. The table shows your friend’s hair growth.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 23
a. Does your friend’s hair grow faster than average? Explain.

Answer:
The rate of growth on average is
15/12 = 1.25 cm/month
The slope/rate of growth for your friend is
(6 – 3)/(4 – 2) = 3/2 = 1.5 cm/month
As 1.5 > 1.25, your friends hair grows faster than average.

b. In the same coordinate plane, graph the average hair growth and the hair growth of your friend. Compare and interpret the steepness of each of the graphs.
Answer:
The equation for the average growth is
y = 1.25x
The equation for the friends growth is
y = 1.5 x
Big Ideas Math 8th Grade Answer Key for Chapter 4 img_123

4.4 Graphing Linear Equations in Slope-Intercept Form (pp. 161–166)
Learning Target: Graph linear equations in slope-intercept form.

Find the slope and the -intercept of the graph of the linear equation.
Question 24.
y = -4x + 1
Answer:
y = mx + b
slope = -4 and y – intercept = 1

Question 25.
y = \(\frac{2}{3}\)x – 12
Answer:
y = mx + b
slope = \(\frac{2}{3}\) and y – intercept = -12

Question 26.
y – 7 = 0.5x
Answer:
Given the equation
y – 7 = 0.5x
y = 0.5x + 7
slope = 0.5 and y – intercept = 7

Graph the linear equation. Identify the -intercept.
Question 27.
y = 2x – 6
Answer:
Given the equation
y = 2x – 6
Comparing the above equation with slope – intercept equation
slope = 2, y – intercept = -6
Slope = rise/run = 2/1
Plot the point that is 1 unit right and 2 units up from (0, -6) = (1, -4)
BIM Grade 8 Answers Chapter 4 img_124
The line crosses the x – axis at (3, 0)
So, the x – intercept is 3.
BIM Grade 8 Answers Chapter 4 img_125

Question 28.
y = -4x + 8
Answer:
y = -4x + 8
slope = -4 and y – intercept = 8
So plot (0, 8)
Slope = rise/run = -4/1
plot the point that is 1 unit right and 4 units down from (0, 8) = (1, 4)
BIM Grade 8 Answers Chapter 4 img_126
The line crosses the x- axis at (2, 0)
So the x – intercept is 2.
BIM Grade 8 Answers Chapter 4 img_127

Question 29.
y = -x – 8
Answer:
Given the equation
y = -x – 8
comparing the above equation with sloope – intercept equation.
Slope = -1 and y – intercept = -8
Slope = rise/run = -1/1
Plot the point that is 1 unit right and 1 unit down from (0, -8) = (1, -9)
BIM Grade 8 Answers Chapter 4 img_128
The line crosses the x-axis at (-8, 0)
So, the intercept is -8.
BIM Grade 8 Answers Chapter 4 img_129BIM Grade 8 Answers Chapter 4 img_129

Question 30.
The cost y (in dollars) of one person buying admission to a fair and going on x rides is y = x + 12.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 30
a. Graph the equation.
b. Interpret the y-intercept and the slope.
Answer:
y = x + 12
Comparing the above equation with slope – intercept equation.
Slope = 1 and y – intercept = 12
So plot (0, 20)
Slope = rise/run = 1/1
Plot the point that is 1 unit right and 1 unit up from (0, 12) = (1, 13)
BIM Grade 8 Answers Chapter 4 img_130
The y – intercept is 12 so the initial cost of admission is $12.
The slope is 1 so for each ride the cost of the person increases $1 per ride.

Question 31.
Graph the linear equation with slope -5 and y-intercept 0.
Answer:
y – intercept = 0. So plot (0, 0)
Plot the point that is 1 unit right and 5 unit down from (0, 0) = (1, -5)
BIM Grade 8 Answers Chapter 4 img_131

4.5 Graphing Linear Equations in Standard Form (pp. 167–172)
Learning Target: Graph linear equations in standard form.

Write the linear equation in slope-intercept form.
Question 32.
4x + 2y = -12
Answer:
4x + 2y = -12
2y = -12 – 4x
y = -6 – 2x
y = -2x – 6

Question 33.
x – y = \(\frac{1}{4}\)
Answer:
Given the equation
x – y = \(\frac{1}{4}\)
y = x – \(\frac{1}{4}\)

Graph the linear equation.
Question 34.
\(\frac{1}{4}\)x + y = 3
Answer:
\(\frac{1}{4}\)x + y = 3
y = 3 – \(\frac{1}{4}\)x
y = –\(\frac{1}{4}\)x + 3
Slope = –\(\frac{1}{4}\) and y – intercept = 3
So plot (0, 3)
Slope = rise/run = –\(\frac{1}{4}\)
Plot the point that is 4 units right and 1 unit down from (0, 3) = (4, 2)
Big Ideas Math Answers 8th Grade Chapter 4 img_132

Question 35.
-4x + 2y = 8
Answer:
-4x + 2y = 8
2y = 8 + 4x
y = 2x + 4
Slope = 2 and y – intercept = 4
So plot (0, 4)
Slope = rise/run = 2/1
Plot the point that is 1 unit right and 2 units up from (0, 4) = (1, 6)
Big Ideas Math Answers 8th Grade Chapter 4 img_133

Question 36.
x + 5y = 10
Answer:
x + 5y = 10
5y = -x + 10
y = -1/5 x + 2
Slope = -1/5 and y – intercept = 2
So, plot (0, 2)
Slope = rise/run = -1/5
Plot the point that is 5 unit right and 1 unit down from (0, 2) = (5, 1)
Big Ideas Math Answers 8th Grade Chapter 4 img_134

Question 37.
–\(\frac{1}{2}\)x + \(\frac{1}{8}\)y = \(\frac{3}{4}\)
Answer:
–\(\frac{1}{2}\)x + \(\frac{1}{8}\)y = \(\frac{3}{4}\)
\(\frac{1}{8}\)y = \(\frac{3}{4}\)  + \(\frac{1}{2}\)x
y = 4x + 6
Slope = 4 and y – intercept = 6
So plot (0, 6)
Slope = rise/run = 4/1
Plot the point that is 1 unit right and 4 units up from (0, 6) = (1, 10)
Big Ideas Math Answers 8th Grade Chapter 4 img_135

Question 38.
A dog kennel charges $30 per night to board your dog and $6 for each hour of playtime. The amount of money you spend is given by 30x + 6y = 180, where x is the number of nights and y is the number of hours of playtime. Graph the equation and interpret the intercepts.
Answer:
Given,
A dog kennel charges $30 per night to board your dog and $6 for each hour of playtime.
The amount of money you spend is given by 30x + 6y = 180,
where x is the number of nights and y is the number of hours of playtime.
30x + 6y = 180
6y = -30x + 180
y = -5x + 30
BIM Answers Grade 8 Chapter 4 img_136
The x – intercept is 6, which means that the dog can stay for 6 nights when there is no playtime.
The y – intercept is 30, which means the dog can play for 30 hours when he does not spend any night at the kennel.

4.6 Writing Equations in Slope-Intercept Form (pp. 173–178)
Learning Target: Write equations of lines in slope-intercept form.

Write an equation in slope-intercept form of the line that passes through the given points.
Question 39.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 39
Answer:
m = (y2 – y1)/(x2 – x1)
m = (1 – (-2))/(3 – 0)
m = (1 + 2)/(3 – 0)
m = 3/3
m = 1
We have to find the y – intercept because the line crosses the y – axis at (0, -2)
y = mx + b
y = x – 2

Question 40.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 40
Answer:
m = (y2 – y1)/(x2 – x1)
m = (4 – 2)/(0 – 4)
m = 2/-4
m = -1/2
We have to find the y – intercept because the line crosses the y – axis at (0, 4)
y = mx + b
y =-1/2 x + 4

Question 41.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 41
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-2 – 1)/(2 – 0)
m = -3/2
We have to find the y – intercept because y = 1 when x = 0, the y – intercept is 1.
y = mx + b
y = -3/2 x + 1

Question 42.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 42
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-3))/(1 – 0)
m = 2/1
m = 2
We have to find the y – intercept because y = -3 when x = 0, the y – intercept is -3.
y = mx + b
y = 2x + (-3)
y = 2x – 3

Question 43.
Write an equation of the line that passes through (0, 8) and (6, 8).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (8 – 8)/(6 – 0)
m = 0/6
m = 0
We have to find the y – intercept because y = 8 when x = 0, the y – intercept is 8.
y = mx + b
y = (0) x + 8
y = 8

Question 44.
Write an equation of the line that passes through (0, -5) and (-5, -5).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-5 – (-5))/(-5 – 0)
m = 0/-5
m = 0
We have to find the y – intercept because y = -5 when x = 0, the y – intercept is -5
y = mx + b
y = (0) x + (-5)
y = -5

Question 45.
A construction crew is extending a highway sound barrier that is 13 miles long. The crew builds \(\frac{1}{2}\) of a mile per week. Write an equation in slope -intercept form that represents the length y (in miles) of the barrier after x weeks.
Answer:
Given,
A construction crew is extending a highway sound barrier that is 13 miles long.
The crew builds \(\frac{1}{2}\) of a mile per week.
y = mx + b
m = \(\frac{1}{2}\)
b = 13
y = \(\frac{1}{2}\)x + 13

4.7 Writing Equations in Point-Slope Form (pp. 179–184)
Learning Target: Write equations of lines in point-slope form.

Write an equation in point-slope form of the line that passes through the given point and has the given slope.
Question 46.
(4, 4); m = 3
Answer:
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (4) = 3(x – 4)
y – 4 = 3(x – 4)

Question 47.
(2, -8); m = –\(\frac{2}{3}\)
Answer:
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (-8) = –\(\frac{2}{3}\)(x – 2)
y + 8 = –\(\frac{2}{3}\)(x – 2)

Write an equation in slope-intercept form of the line that passes through the given points.
Question 48.
(-4, 2), (6, -3)
Answer:
m = (y2 – y1)/(x2 – x1)
m = (-3 – 2)/(6 – (-4))
m = -5/10
m = -1/2
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – 2 = –\(\frac{1}{2}\)(x – (-4))
y – 2 = –\(\frac{1}{2}\)(x + 4)
y = –\(\frac{1}{2}\)x

Question 49.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 49
Answer:
m = (y2 – y1)/(x2 – x1)
m = (5 – 1)/(3 – 2)
m = 4/1
m = 4
y – y1 = m(x – x1)
Substitute m value, x and y value in the equation
y – (-3) = 4(x – 1)
y + 3 = 4x – 4
y = 4x – 7

Question 50.
The table shows your elevation y (in feet) on a ski slope after x minutes.
Big Ideas Math Answers Grade 8 Chapter 4 Graphing and Writing Linear Equations 4.7 cr 50
a. Write an equation that represents your elevation after x minutes.

Answer:
m = (600 – 800)/(2 – 1)
m = -200
800 = -200(1) + b
800 = -200 + b
800 + 200 = b
b = 1000 feet

b. What is your starting elevation?

Answer:
The starting elevation is the y – intercept
b = 1000 feet

c. After how many minutes do you reach the bottom of the ski slope? Justify your answer.
Answer:
0 = -200x + 1000
0 – 1000 = -200x
-1000 = -200x
200x = 1000
x = 5 minutes

Question 51.
A company offers cable television at$29.95 per month plus a one-time installation fee. The total cost for the first six months of service is $214.70. a. Write an equation in point-slope form that represents the total cost you pay for cable television after x months.
b. How much is the installation fee? Justify your answer.
Answer:
y – y1 = m(x – x1)
m = 29.95
y – 214.70 = 29.95(x – 6)
y – 214.70 + 214.70 = 29.95x – 179.97 + 2147.70
y = 29.95x + 35
b = 35

Question 52.
When might it be better to represent an equation in point-slope form rather than slope-intercept form? Use an example to justify your answer.
Answer:
When we are given the slope and a point that is the y – intercept, then the easiest way is to use the slope – intercept form y = mx + b
Example:
m = 2
(0, 5)
y = 2x + 5
m = 2
(1, 3)
y – 3 = 2(x – 1)
Easier when given the slope and a point that is not the y – intercept.

Graphing and Writing Linear Equations Practice Test

Find the slope and the -intercept of the graph of the linear equation.
Question 1.
y = 6x – 5
Answer:
y = 6x – 5
Slope = 6 and y – intercept = -5

Question 2.
y – 1 = 3x + 8.4
Answer:
Given the equation
y – 1 = 3x + 8.4
y = 3x + 8.4 + 1
y = 3x + 9.4
Slope = 3 and y – intercept = 9.4

Question 3.
–\(\frac{1}{2}\)x + 2y = 7
Answer:
Given the equation
–\(\frac{1}{2}\)x + 2y = 7
y = \(\frac{1}{4}\)x + \(\frac{7}{2}\)
Slope = \(\frac{1}{4}\) and y – intercept = \(\frac{7}{2}\)

Graph the linear equation.
Question 4.
y = –\(\frac{1}{2}\)x – 5
Answer:
Given the equation
y = –\(\frac{1}{2}\)x – 5
Slope = –\(\frac{1}{2}\) and y – intercept = -5
So plot (0, -5)
Plot the point that is 2 units right and 1 unit down from (0, -5) = (2, -6)
Draw a line through the two points.
BIM Grade 8 Answers Chapter 4 img_108

Question 5.
-3x + 6y = 12
Answer:
Given the equation
-3x + 6y = 12
6y = 3x + 12
y = \(\frac{1}{2}\)x + 2
Slope = \(\frac{1}{2}\), y – intercept = 2
Slope = rise/run = \(\frac{1}{2}\)
Plot the point that is 2 units right and 1 unit up from (0, 2) = (2, 3)
BIM Grade 8 Answers Chapter 4 img_109

Question 6.
y = \(\frac{2}{3}\)x
Answer:
Given the equation
y = \(\frac{2}{3}\)x
Slope = \(\frac{2}{3}\), y – intercept = 0
Slope = rise/run = \(\frac{2}{3}\)
Plot the point that is 3 units right and 2 unit up from (0, 0) = (3, 2)
Big Ideas Math Answers Grade 8 Ch 4 img_109

Question 7.
Which lines are parallel? Explain.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation pt 7
Answer:
Red line:
(x1, y1) = (-4, 1)
(x2, y2) = (2, 4)
m = (y2 – y1)/(x2 – x1)
m = (4 – 1)/(2 – (-4))
m = 3/6
m = 1/2
Blue line:
(x1, y1) = (-4, -1)
(x2, y2) = (2, 0.5)
m = (y2 – y1)/(x2 – x1)
m = (0.5 – (-1))/(2 – (-4))
m = 1.5/6
m = 1/4
Green Line:
(x1, y1) = (-2, -4)
(x2, y2) = (2, -2)
m = (y2 – y1)/(x2 – x1)
m = (-2 – (-4))/(2 – (-2))
m = 2/4
m = 1/2
Red lines and Green lines are parallel because both have same slope = 1/2

Question 8.
The points in the table lie on a line. Find the slope of the line.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation pt 8
Answer:
(x1, y1) = (-1, -4)
(x2, y2) = (0, -1)
m = (y2 – y1)/(x2 – x1)
m = (-1 – (-4))/(0 – (-1))
m = 3/1
m = 3

Write an equation in slope-intercept form of the line that passes through the given points.
Question 9.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation pt 9
Answer:
(x1, y1) = (-4, 1)
(x2, y2) = (2, 4)
m = (y2 – y1)/(x2 – x1)
m = (-5 – (-1))/(3 – 0)
m = -4/3
Because the line crosses the y – axis at (0, -1), the y – intercept is -1.
y = mx + b
y = -4/3x – 1

Question 10.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation pt 10
Answer:
m = (y2 – y1)/(x2 – x1)
m = (2 – 2)/(0 – (-2))
m = 0/2
m = 0
Because y = 2 when x =0, the y – intercept is 2.
y = mx + b
y = 2

Question 11.
Write an equation in point-slope form of the line that passes through (-4, 1) and (4, 3).
Answer:
m = (y2 – y1)/(x2 – x1)
m = (3 – 1)/(4 – (-4))
m = 2/8
m = 1/4
y – y1 = m(x – x1)
y – 1 = 1/4(x – (-4))
y – 1 = 1/4(x + 4)

Question 12.
The number y of new vocabulary words that you learn after x weeks is represented by the equation y = 15x.
a. Graph the equation and interpret the slope.
b. How many new vocabulary words do you learn after 5 weeks?
c. How many more vocabulary words do you learn after 6 weeks than after 4 weeks?
Answer:
a. 8th Grade Big Ideas Math Answer Key Chapter 4 img_109
b. y = 15 . 5
y = 75 words
c. 15 . 6 – 15 . 4 = 90 – 60 = 30 words

Question 13.
You used $90 worth of paint for a school float. The amount of money you spend is given by 18x + 15y = 90, where x is the number of gallons of blue paint and y is the number of gallons of white paint. Graph the equation and interpret the intercepts.
Answer:
Given,
18x + 15y = 90
15y = -18x + 90
y = -6/5 x + 6
8th Grade Big Ideas Math Answer Key Chapter 4 img_110
The x – intercept is 5 and shows that 5 gallons of blue paint might be bought when no gallon of the white pants is bought.
The y – intercept is 6 and shows that 6 gallons of white paint might be bought when no gallon of blue is bought.

Graphing and Writing Linear Equations Cumulative Practice

Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 1
Question 1.
Which equation matches the line shown in the graph?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 2
A. y =2x – 2
B. y = 2x + 1
C. y = x – 2
D. y = x + 1
Answer:
m = (-2 – 0)/(0 – 1)
m = 2
y = 2x – 2
Thus the correct answer is option A.

Question 2.
Which point lies on the graph of 6x – 5y = 14?
F. (-4, -1)
G. (-2, 4)
H. (-1, -4)
I. (4, -2)
Answer:
6x – 5y = 14
F. 6(-4) – 5(-1) = 14
-24 + 5 = 14
-19 ≠ 14
G. 6(-2) – 5(4) = 14
-12 – 20 = 14
-32 ≠ 14
H. 6(-1) – 5(-4) = 14
-6 + 20 = 14
14 = 14
Thus the correct answer is option H.

Question 3.
You reflect the triangle in the x-axis. What are the coordinates of the image?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 3
A. X'(4, 1), Y'(2, 3), Z'(-2, 1)
B. X'(4, -1), Y'(2, -3), Z'(-2, -1)
C. X'(-4, -1), Y(-2, -3), Z'(2, -1)
D. X'(1, 4), Y'(3, 2), Z'(1, -2)
Answer:
BIM 8th Grade Solution Key ch 4 img_108
Thus the correct answer is option C.

Question 4.
Which of the following is the equation of a line parallel to the line shown in the graph?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 4
Answer:
m = (-4 – 2)/(6 – 4)
m = -6/2
m = -3
Two lines parallel if they have the same slope.
From the given equations, the one having the slope -3 is y = -3x + 5
Thus the correct answer is option H.

Question 5.
What is the value of x?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 5
Answer:
122 = 47 + x
47 + x = 122
x = 122 – 47
x = 75

Question 6.
An emergency plumber charges $49.00 plus $70.00 per hour of the repair. A bill to repair your sink is $241.50. This can be modeled by 70.00 h + 49.00 = 241.50, where h represents the number of hours for the repair. How many hours did it take to repair your sink?
A. 2.75 hours
B. 3.45 hours
C. 4.15 hours
D. 13,475 hours
Answer:
70.00 h + 49.00 = 241.50
70h = 241.50 – 49
70h = 192.5
h = 2.75 hours
Thus the correct answer is option A.

Question 7.
It costs $40 to rent a car for one day. In addition, the rental agency charges you for each mile driven, as shown in the graph.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 7
Part A Determine the slope of the line joining the points on the graph.
Part B Explain what the slope represents.
Answer:
m = (50 – 40)/(100 – 0)
m = 10/100
m = 0.1

Question 8.
What value of makes the equation true?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 8
7 + 2x = 4x – 5
Answer:
7 + 2x = 4x – 5
2x – 4x = -5 – 7
-2x = -12
x = 6

Question 9.
Trapezoid KLMN is graphed in the coordinate plane shown.
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 9
Rotate Trapezoid KLMN 90° clockwise about the origin. What are the coordinates of point M’, the image of point M after the rotation?
F. (-3, -2)
G. (-2, -3)
H. (-2, 3)
I. (3, 2)
Answer: M'(-3, -2)
Thus the correct answer is option F.

Question 10.
Solve the formula K = 3M – 7.
A. M = K + 7
B. M = \(\frac{K+7}{3}\)
C. M = \(\frac{K}{3}\) + 7
D. M = \(\frac{K-7}{3}\)
Answer:
K = 3M – 7
K + 7 = 3M
M = \(\frac{K+7}{3}\)
Thus the correct answer is option B.

Question 11.
What is the distance across the canyon?
Big Ideas Math Solutions Grade 8 Chapter 4 Exponents and Scientific Notation cp 11
F. 3.6 ft
G. 12 ft
H. 40 ft
I. 250 ft
Answer:
100/30 = d/12
3d = 12 × 10
3d = 120
d = 40 feet
Thus the correct answer is option H.

Conclusion:

All the solutions in the above article are beneficial for all the students of middle school students. All the solutions are prepared by the math professionals. The solutions are given clearly with step by step explanations. If you have any doubts regarding the chapter we are always ready to clarify your doubts. All you have to do is to post the comments in the below comment box.

Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles

Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles

Get Big Ideas Math Answer Key for Grade 7 Chapter 9 Geometric Shapes and Angles Pdf here. We are providing free download links to all problems in the upcoming sections. You can understand the concept easily as we are explaining the concept with so many real-time examples. Candidates can improve their problem-solving skills and analytical thinking with the help of Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles.

BIM Grade 7 Chapter 9 Geometric Shapes and Angles Answer Key pdf. Download all the pdf links from here for free of cost and kickstart your exam preparation. Follow the various concepts and solve as many problems as before going to the exam. We are providing the preparatory material for the candidates to score better marks in the exam. Grade 7 Big Ideas Math Answers is providing amazing tips and tricks in the next sections. Check it out!!

Big Ideas Math Book 7th Grade Answer Key 9 Geometric Shapes and Angles

If it is some theory, then you can easily learn it but if it is problematic, then you must have perfection and grip on the subject. Therefore, we are providing a lot of practice material with solutions. First, read the question and try to solve the problems without looking at the solutions. Then refer to the solution and know if it is correct or not. Also, check if it the easiest process or not. With the help of professionals and math experts, we gave the solution to the problems in the easiest methods.

You need not be the expert in all the topics but you must check the easy topics like Geometric Shapes and Angles and prepare perfectly. These topics act as a scoring factor because they are easy to understand and solve. If you have any further doubts, go through Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles, you will get a clear idea about every detail.

Performance Task

Lesson: 1 Circle and Circumference

Lesson: 2 Areas of Circles

Lesson: 3 Perimeters and Areas of Composite Figures

Lesson: 4 Constructing Polygons

Lesson: 5 Finding Unknown Angle Measures

Chapter 9 – Geometric Shapes and Angles

Geometric Shapes and Angles STEAM Video/Performance Task

STEAM Video

Track and Field
Different lanes on a race track have different lengths. How can competitors run in different lanes and have the same finish line?

Watch the STEAM Video “Track and Field.” Then answer the following questions.
1. A track consists of a rectangle and two semicircles. The dimensions of the rectangle formed by the innermost lane are shown. What is the distance around each semicircle on the 400-meter, innermost lane?
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 1
2. How does the width of the rectangle, 63.7 meters, compare to the distance around each semicircle? Explain.

Answer:
1. The distance around each semicircle on the 400-meter, innermost lane = 488 m
2. The distance around each semicircle = 90π + 320

Explanation:
1. The inside perimeter of the track = 400 m
the total length of the two straight portions = 90 + 90 = 180
therefore the length of the remaining portion = 400-180 = 220 m
circumference of the two remaining semi-circular portions = πr + πr = 2πr
2πr = 220
2 x 3.14 x r = 220
r = 35 m
Area of the track = 2 x 90 x 14 +3.14 x (49) x (49) – (35) x (35)
area of the track = 6216 square meter
length of the outer running track = 488 m
2. The perimeter of the track is the two circumferences of the circumferences.
The diameters of the circle and the width of the rectangle = 90 m
90 π + 320
Performance Task.
Finding the Area and Perimeter of a Track
After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given the dimensions of a race track.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 2
You will be asked to solve various perimeter and area problems about the track. Given a race track, what measures do you need to find the outer perimeter?

Answer:
The outer perimeter = 11,2610 sq m

Explanation:
perimeter of the semicircle = (π + 2 ) r
p = (3.14 + 2)36.5
p= (3.16) 36.5
p = 11,2610 sq m

Geometric Shapes and Angles Getting Ready for Chapter 9

Chapter Exploration
Work with a partner.
Question 1.
Perform the steps for each of the figures.

  • Measure the perimeter of the larger polygon to the nearest millimeter.
  • Measure the diameter of the circle to the nearest millimeter.
  • Measure the perimeter of the smaller polygon to the nearest millimeter.
  • Calculate the value of the ratio of the two perimeters to the diameter.
  • Take the average of the ratios. This average is the approximation of π(the Greek letter ).
    Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 3

Question 2.
Based on the table, what can you conclude about the value of π? Explain your reasoning.

Answer:
The value of π = 3.14

Explanation:
We can consider 3 values for the π.
they are (22/7) or 3.14
so i am considering the 3.14

Question 3.
The Greek mathematician Archimedes used the above procedure to approximate the value of π. He used polygons with 96 sides. Do you think his approximation was more or less accurate than yours? Explain your reasoning.
Answer:
The greek mathematician used polygons with the side of polygons as 12,14,48, and finally 96 sides.

Explanation:
The greek mathematician used polygons with the side of polygons as 12,14,48, and finally 96 sides.
yes the accuration is more than i think.

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
diameter of a circle
semi circle
adjacent angles
circumference
composite figure
vertical angles

Answer:
The diameter of the circle = the diameter is the length of the line through the center that touches two points on the edge of the circle.
semi circle =  semicircle is a one-dimensional locus of points that forms half of the circle.
adjacent angles = adjacent angles are two angles that have a common vertex and a common side but do not overlap.
circumference = the circumference is the perimeter of the circle. the circumference would be the arc length of the circle.
composite figure = a figure that consists of two or more geometric shapes.
vertical angles = a pair of non-adjacent angles form when two lines intersect.

Explanation:
The diameter of the circle = the diameter is the length of the line through the center that touches two points on the edge of the circle.
semi circle =  semicircle is a one-dimensional locus of points that forms half of the circle.
adjacent angles = adjacent angles are two angles that have a common vertex and a common side but do not overlap.
circumference = the circumference is the perimeter of the circle. the circumference would be the arc length of the circle.
composite figure = a figure that consists of two or more geometric shapes.
vertical angles = a pair of non-adjacent angles form when two lines intersect.

Lesson 9.1 Circles and Circumference

EXPLORATION 1

Using a Compass to Draw a Circle
Work with a partner. Set a compass to 2 inches and draw a circle.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 1
a. Draw a line from one side of the circle to the other that passes through the center. What is the length of the line? This is called the diameter of the circle.
b. Estimate the distance around the circle. This is called the circumference of the circle. Explain how you found your answer.

Answer:
a. the length of the line = 4 inches
b.  The circumference of the circle = 12.56 inch

Explanation:
a. In the question they said that 2 inches
the length of the line = 4 in
b. the circumference of the circle = 2π r
circle = 2 x 3.14 x 2
circle = 12.56 in

EXPLORATION 2

Exploring Diameter and Circumference
Work with a partner.
a. Roll a cylindrical object on a flat surface to find the circumference of the circular base.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 2
b. Measure the diameter of the circular base. Which is greater, the diameter or the circumference? how many times greater?
c. Compare your answers in part(b) with the rest of the class. What do you notice?
d. Without measuring, how can you find the circumference of a circle with a given diameter? Use your method to estimate the circumference of the circle in Exploration 1.

Answer:
a. The circumference of the circle = 2πr
b. The circumference of the circle is 3.14 times greater than the diameter of the circle.
c. The circumference of the circle is greater than the diameter of the circle.
d. The diameter of the circle = 2r and the circumference of the circle = 2πr

Explanation:
a. The circumference of the circle = 2πr
b. The circumference of the circle is 3.14 times greater than the diameter of the circle.
c. The circumference of the circle is greater than the diameter of the circle.
d. The diameter of the circle = 2r and the circumference of the circle = 2πr

Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 3

Try It

Question 1.
The diameter of a circle is 16 centimeters. Find the radius.

Answer:
radius = 8 cm

Explanation:
The diameter of the circle = 2r
16 = 2r
r = 8 cm

Question 2.
The radius of a circle is 9 yards. Find the diameter.

Answer:
The diameter = 18 yds

Explanation:
The diameter of the circle = 2r
diameter = 2 x 9
r = 18 yds

Find the circumference of the object. Use 3.14 or \(\frac{22}{7}\) for π.
Question 3.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 4

Answer:
circumference = 12.56 cm

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 2 where r = 2cm given
circle = 6.28 x 2
circle = 12.56 cm

Question 4.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 5

Answer:
circumference =43.96 square feet

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 7 where r = 7 ft given
circle = 6.28 x 7
circle = 43.96 square feet

Question 5.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 6

Answer:
circumference =28.26 square in

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 4.5 where r = 4.5 is given
circle = 6.28 x 4.5
circle =28.26 in

Find the perimeter of the semicircular region.
Question 6.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 7

Answer:
perimeter of the semicircle =  5.14 ft

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 1 diameter = 2 given r= 1
perimeter = 5.14 feet

Question 7.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 8

Answer:
perimeter of the semicircle =  17.99 cm

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 3.5 diameter = 7 given r= 3.5
perimeter = 17.99 cm

Question 8.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 9

Answer:
the perimeter of the semicircle =  33.14 in

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 15 given r= 15
perimeter = 33.14 in

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
WRITING
Are there circles for which the value of the ratio of circumference to diameter is not equal to π? Explain.

Answer:
circumference to diameter is equal to π

Explanation:
d. The diameter of the circle = 2r and the circumference of the circle = 2πr
circumference to diameter is equal to π

Question 10.
FINDING A PERIMETER
Find the perimeter of a semicircular region with a straight side that is 8 yards long.

Answer:
perimeter = 11.14 yd

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 4 given r= 4
perimeter = 11.14 yd
Question 11.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 10

Answer:
What is π times the radius?
What is π times the diameter?

Explanation:
the radius of the circle = (c/2 π )
the diameter of the circle =  2r
Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 12.
The wheels of a monster truck are 66 inches tall. Find the distance the monster truck travels when the tires make one 360-degree rotation.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 11

Answer:
The distance = 207.35 inches

Explanation:
The wheel is in the shape of a circle.
diameter = 66 given
radius = (66/2)
radius = 33
The circumference = 2πr
c = 2 x 3.14 x 33
c = 6.28 x 33
c = 207.24 in

Question 13.
DIG DEEPER!
The radius of a dog’s collar should be at least 0.5 inch larger than the radius of the dog’s neck. A dog collar adjusts to a circumference of 10 to 14 inches. Should the collar be worn by a dog with a neck circumference of 12.5 inches? Explain.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 12

Answer:
No, the collar should not be worn by this dog.

Explanation:
Given that the collar should be at least 0.5 inches.
dog collar adjusts to a circumference of 10 to 14 inches.

Question 14.
You resize a picture so that the radius of the midday Sun appears four times larger. How much larger does the circumference of the Sun appear? Explain.

Answer:
4 times larger

Explanation:
they said that if they resize it for 4 times.
therefore the sun appears 4 times larger.

Circles and Circumference Homework & Practice 9.1

Review & Refresh

Two jars each contain 1000 numbered tiles. The double box-and-whisker plot represents a random sample of 10 numbers from each jar.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 13
Question 1.
Compare the samples using measures of center and variation.

Answer:
a. Jar A = median 3, starting 2.
b. Jar B = median  6, starting 2

Explanation:
In the above-given figure, the jar A is starting from 2
jar A contains median = 3
the jar B is starting from 2
jar B contains median = 6

Question 2.
Can you determine which jar contains greater numbers? Explain.

Answer:
Jar B

Explanation:
jar B contains the numbers from 4 to 9
Question 3.
Find the percent of change from 24 to 18.
A. 25% decrease
B. 25% increase
C. 75% increase
D. 75% decrease

Answer:
option A is correct

Explanation:
if the percent of jar changes from 24 to 18
the decrease in the percent = 25

Concepts, Skills, & Problem Solving
EXPLORING DIAMETER AND CIRCUMFERENCE Estimate the circumference of the circular base of the object. (See Exploration 2, p. 361.)
Question 4.
tube of lip balm with radius 0.5 mm

Answer:
c = 3.14 mm

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 0.5 where r = 0.5 mm given
circle = 6.28 x 0.5
circle =3.14 mm

Question 5.
D battery with radius 0.65 in.

Answer:
c = 4.082 in

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 0.65 where r = 0.65 ingiven
circle = 6.28 x 0.65
circle =4.082 in

FINDING A RADIUS Find the radius of the button.
Question 6.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 14

Answer:
radius =2.5 cm

Explanation:
radius = (5/2)
radius = 2.5 cm

Question 7.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 15

Answer:
radius =14 mm

Explanation:
radius = (28/2)
radius = 14 mm

Question 8.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 16

Answer:
radius =1.75 in

Explanation:
radius = (3.5/2)
radius = 1.75 in

FINDING A DIAMETER Find the diameter of the object.
Question 9.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 17

Answer:
diameter = 4 in

Explanation:
diameter of the circle = 2r
where r = 2 given
d = 4 in

Question 10.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 18

Answer:
diameter = 0.64 ft

Explanation:
diameter of the circle = 2r
where r = 0.8 given
d = 0.64 ft

Question 11.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 19

Answer:
diameter = 1.2 cm

Explanation:
diameter of the circle = 2r
where r = 0.6 given
d = 1.2 cm

FINDING A CIRCUMFERENCE Find the circumference of the object. Use 3.14 or \(\frac{22}{7}\) for π.
Question 12.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 20

Answer:
c = 43.96 in

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 7 where r = 7 ingiven
circle = 6.28 x 7
circle =43.96 in

Question 13.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 21

Answer:
c = 18.84 cm

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 7 where r = 3 cmgiven
circle = 6.28 x 3
circle =18.84 cm

Question 14.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 22

Answer:
c = 6.28 mm

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 1where r = 1mgiven
circle = 6.28 x 1
circle =6.28 m

FINDING THE PERIMETER OF A SEMICIRCULAR REGION Find the perimeter of the window.
Question 15.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 23

Answer:
perimeter = 7.71 ft

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 1.5 given d =3 ,r = (d/2)
perimeter = 7.71 ft

Question 16.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 24

Answer:
perimeter = 64.8 cm

Explanation:
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 20 given  ,r = 20 cm
perimeter = 64.8 cm

ESTIMATING A RADIUS Estimate the radius of the object.
Question 17.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 25

Answer:
Radius = 1.417 mm

Explanation:
radius of the circle = (c/2π )
r = (8.9/6.28)
r = 1.417 mm

Question 18.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 26

Answer:
Radius = 19.426 in

Explanation:
radius of the circle = (c/2π )
r = (122/6.28)
r = 19.426 in

Question 19.
MODELING REAL LIFE
A circular sinkhole has a circumference of 75.36 meters. A week later, it has a circumference of 150.42 meters.
a. Estimate the diameter of the sinkhole each week.
b. How many times greater is the diameter of the sinkhole a week later?

Answer:
a. The diameter of the sinkhole each week = 4 in
b. 2 times greater is the diameter of the sinkhole a week later

Explanation:
a. The diameter of the sinkhole each week = 75.36 m
b. 2 times greater is the diameter of the sinkhole a week later
75.36 x 75.36 = 150.42 m
Question 20.
REASONING
Consider the circles A, B, C, and D.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 27
a. Without calculating, which circle has the greatest circumference? Explain.
b. Without calculating, which circle has the least circumference? Explain.

Answer:
a. option D has the greatest circumference.
b. option C has the least circumference.

Explanation:
D. circumference of the circle =2πr
circle = 2 x 3.14 x 50where r =50 ingiven
circle = 6.28 x 50
circle =314 in
Explanation:
C. circumference of the circle =2πr
circle = 2 x 3.14 x 1where r = 1given
circle = 6.28 x1
circle = 6.28
Explanation:
A. circumference of the circle =2πr
circle = 2 x 3.14 x 4 where r = 4given
circle = 6.28 x 4
circle = 25.12
Explanation:
A. circumference of the circle =2πr
circle = 2 x 3.14 x 10 where r = 10given
circle = 6.28 x 10
circle = 62.8

FINDING CIRCUMFERENCES Find the circumferences of both circles.
Question 21.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 28

Answer:
circumference of inside circle  =31.4 square cm
circumference of outside circle = 62.8  square cm

Explanation:
circumference of the inside circle =2πr
circle = 2 x 3.14 x 5 where r = 5 cm given
circle = 6.28 x 5
circle = 31.4 square cm
circumference of the outside circle =2πr
circle = 2 x 3.14 x 2 where r = 2 cm given
circle = 6.28 x 2
circle = 62.8 square cm

Question 22.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 29

Answer:
circumference of inside circle  =28.26 ft
circumference of outside circle = 31.4 square cm

Explanation:
circumference of the inside circle =2πr
circle = 2 x 3.14 x 4.5 where r = 4.5 feet given
circle = 6.28 x 4.5
circle = 28.26 ft
circumference of the outsideside circle =2πr
circle = 2 x 3.14 x 2.5 where r = 2.5 ft given
circle = 6.28 x 2.5
circle = 15.7 square ft

Question 23.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 30

Answer:
circumference of inside circle  =69.08  m
circumference of outside circle = 138.16 m

Explanation:
circumference of the inside circle =2πr
circle = 2 x 3.14 x 5.5 where r = 5.5 feet given
circle = 6.28 x 5.5
circle = 69.08 m
circumference of the outsideside circle =2πr
circle = 2 x 3.14 x 22 where r = 22given
circle = 6.28 x 22
circle = 138.16 m

Question 24.
MODELING REAL LIFE
A satellite is in an approximately circular orbit 36,000 kilometers from Earth’s surface. The radius of Earth is about 6400 kilometers. What is the circumference of the satellite’s orbit?
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 31

Answer:
c = 40,192 km

Explanation:
circumference of the  satellite orbit =2πr
circle = 2 x 3.14 x 6400where r = 6400kmgiven
circle = 6.28 x 6400
circle =40,192km

Question 25.
STRUCTURE
The ratio of circumference to diameter is the same for every circle. Is the ratio of circumference to radius the same for every circle? Explain.

Answer:
The ratio of circumference to radius is  same for every circle.

Explanation:
c/r = 2πr/r
where r get canceled in both numerator and denominator.
c/r = 2π
radius = (c/2π)
the radius is same for every circle.

Question 26.
PROBLEM SOLVING
A wire is bent to form four semicircles. How long is the wire? Justify your answer.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 32

Answer:
The wire is 128 cm long

Explanation:
Given that the four semicircles are 32 cm
32 + 32 + 32 + 32 = 64

Question 27.
CRITICAL THINKING
Explain how to draw a circle with a circumference of π2 inches. Then draw the circle.

Answer:

Explanation:
circumference of circle = 2πr
c = π2

Question 28.
DIG DEEPER!
“Lines” of latitude on Earth are actually circles. The Tropic of Cancer is the northernmost line of latitude at which the Sun appears directly overhead at noon. The Tropic of Cancer has a radius of 5854 kilometers.
To qualify for an around-the-world speed record, a pilot must cover a distance no less than the circumference of the Tropic of Cancer, cross all meridians, and land on the same air field where the flight began.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 33
a. What is the minimum distance that a pilot must fly to qualify for an around-the-world speed record?
b. RESEARCH Estimate the time it will take for a pilot to qualify for the speed record. Explain your reasoning.

Answer:
a. The minimum distance that a pilot must fly to qualify for an around the world-speed record = 18.3376 km
b. The pilot will take for the speed record = 18.3376 km

Explanation:
a. The minimum distance that a pilot must fly to qualify for an around the world-speed record = 18.3376 km
b. The pilot will take for the speed record = 18.3376 km
Question 29.
PROBLEM SOLVING
Bicycles in the late 1800s looked very different than they do today.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 34
a. How many rotations does each tire make after traveling 600 feet? Round your answers to the nearest whole number.
b. Would you rather ride a bicycle made with two large wheels or two small wheels? Explain.

Answer:
a. The rotations each tire make after traveling 600 feet = 188.4 in in
b. two large wheels = 376.8 in
two small wheels = 113.04 in

Explanation:
the rotations each tire make after travelling = 2 x 3.14 x 30  = 188.4 in
b. two large wheels = 188.4 x 2 = 376.8 in
for two small wheels = 113.04 in

Question 30.
LOGIC
The length of the minute hand is 150% of the length of the hour hand.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.1 35
a. What distance will the tip of the minute hand move in 45 minutes? Justify your answer.
b. In 1 hour, how much farther does the tip of the minute hand move than the tip of the hour hand? Explain how you found your answer.

Answer:
The distance will the tip of minute hand move in 45 minutes = 140 %
b. the tip of the minute hand moves 60 times faster than hour hand.

Explanation:
The distance will the tip of minute hand move in 45 minutes = 140 %
b. the tip of the minute hand moves 60 times faster than hour hand.

Lesson 9.2 Areas of Circles

EXPLORATION 1

Estimating the Area of a Circle
Work with a partner. Each grid contains a circle with a diameter of 4 centimeters. Use each grid to estimate the area of the circle. Which estimate should be closest to the actual area? Explain.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 1

Answer:
Area of 1st circle = 200.96 cm
Area of 2nd circle =803.84 cm
Area of 3rd circle =3215.36 cm

Explanation:
area of 1st circle = πr x r
area = 3.14 x 8 x 8
a = 200.96 cm
area of 2nd circle = πr x r
area= 3.14 x 16 x 16
a = 803.84 cm
area of 3rd circle = πr x r
area= 3.14 x 32 x 32
a = 3215.36
EXPLORATION 2

Writing a Formula for the Area of a Circle
Work with a partner. A student draws a circle with radius and divides the circle into 24 equal sections. The student cuts out each section and arranges the sections to form a shape that resembles a parallelogram.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 2
a. Use the diagram to write a formula for the area of a circle in terms of the radius r. Explain your reasoning.Describe the relationship between the radius and the area of a circle.
b. Use the formula to check your estimates in Exploration 1.

Answer:
a. the area of the circle = 1808.64
b. the area of the circle in terms of radius r = 0.0084 cm

Explanation:
The area  of circle = πr x r
a = 3.14 x 24 x 24
a = 1808.64 cm
The radius of the circle = (c/2 π)
circumference = 2πr
c = 2 x 3.14 x 24
c = 150.72 cm
area = (150.72/6.28)
area = 0.0084 cm

Try It
Question 1.
Find the area of a circle with a radius of 6 feet. Use 3.14 for π.

Answer:
The area  of circle = 113.04 sq ft

Explanation:
The area  of circle = πr x r
a = 3.14 x 6 x 6
a = 113.04 square feet
Question 2.
Find the area of a circle with a diameter of 28 meters.Use \(\frac{22}{7}\) for π.

Answer:
The area  of circle = 175.84 sq meters

Explanation:
The area  of circle = πr x r
a = 3.14 x 14 x 14 where d = 28 so r = 14
a = 175.84 square meters

Find the area of the semicircle.
Question 3.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 3

Answer:
Area of semicircle =62.07 sq cm

Explamation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +121/2)
area =(121.14/2)
area = 62.07 sq cm

Question 4.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 4

Answer:
Area of semicircle =9.57 sqm

Explamation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +16/2)
area =(19.14/2)
area = 9.57 sq m

Question 5.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 5

Answer:
Area of semicircle =4.695 sq yd

Explanation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +6.25/2)
area =(9.39/2)
area = 4.695 sq yd

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
ESTIMATING AN AREA
The grid contains a circle with a diameter of 2 centimeters. Use the grid to estimate the area of the circle. How can you change the grid to improve your estimate? Explain.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 6

Answer:
The area  of circle = 50.24 sq centi meters

Explanation:
The area  of circle = πr x r
a = 3.14 x 4 x 4 where d = 8 so r = 4
a = 50.24 square centimeters

Question 7.
WRITING
Explain the relationship between the circumference and area of a circle.

Answer:
The area  of circle = πr x r
circumference of circle = 2πr

Explanation:
The circumference of the circle is 2 times greater than the area of the circle.
The area  of circle = πr x r
circumference of circle = 2πr

Question 8.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 7

Answer:
What is area of a circle with a radius of 100 cm?
What is the area of a circle with a radius  of 500 mm?

Explanation:
The area  of circle = πr x r
area = 3.14 x 100 x 100
area = 31400 sq cm
The area  of circle = πr x r
area = 3.14 x 500 x 500
area = 785000 sq mm
Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
A local event planner wants to cover a circular region with mud for an obstacle course. The region has a circumference of about 157 feet. The cost to cover 1 square foot with mud is $1.50. Approximate the cost to cover the region with mud.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 8

Answer:
Cost to cover =

Question 10.
DIG DEEPER!
A manufacturer recommends that you use a frying pan with a radius that is within 1 inch of the radius of your stove top burner. The area of the bottom of your frying pan is 25π square inches. The circumference of your cook top burner is 9π inches. Does your frying pan meet the manufacturer’s recommendation?

Answer:
no the frying pan does not meet the manufacture

Explanation:
Given that frying pan has radius = 1 inch
area of frying pan = 25π square inches
circumference = 9π  inches

Areas of Circles Homework & Practice 9.2

Review & Refresh

Find the circumference of the object. Use 3.14 or \(\frac{22}{7}\) for π.
Question 1.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 9

Answer:
c = 28.26cm

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 4.5where r = 4.5cmgiven
circle = 6.28 x 4.5
circle =28.26 cm

Question 2.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 10

Answer:
c = 21.98 sq in

Explanation:
circumference of the circle =2πr
circle = 2 x 3.14 x 3.5where r = 3.5ingiven
circle = 6.28 x 3.5
circle =21.98 sq in

You spin the spinner shown.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 11
Question 3.
How many possible outcomes are there?

Answer:
3 possible outcomes

Explanation:
There are 3 possible outcomes.
3 numbers are there in the spin.

Question 4.
In how many ways can spinning an odd number occur?

Answer:
2 ways the spinning an odd number occur.

Explanation:
There are 2 possible ways that the odd numbers can occur.

Concepts, Skills, & Problem Solving
ESTIMATING AN AREA Use the grid to estimate the area of the circle. (See Exploration 1, p. 369.)

Question 5.
diameter of 3 centimeters
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 12

Answer:
area of the circle = 6.75 sq cm

Explanation:
The area  of circle = πr x r
area = 3.14 x 1.5 x 1.5
area = 6.75 sq cm

Question 6.
diameter of 1.6 inches
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 13

Answer:
area of the circle = 141.41 sq in

Explanation:
The area  of circle = πr x r
area = 3.14 x 6.4 x 6.4
area = 141.41 sq in

FINDING AN AREA Find the area of the circle. Use 3.14 or \(\frac{22}{7}\) for π.
Question 7.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 14

Answer:
The area  of circle = 254.34 sq milli meters

Explanation:
The area  of circle = πr x r
a = 3.14 x 4 x 4 where d = 8 so r = 4
a = 254.34 square millimeters

Question 8.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 15

Answer:
The area  of circle = 615.44 sq centi meters

Explanation:
The area  of circle = πr x r
a = 3.14 x 14 x 14 where r = 14
a = 615.44 square centimeters

Question 9.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 16

Answer:
The area  of circle = 314 sq inches

Explanation:
The area  of circle = πr x r
a = 3.14 x 10 x 10 where r = 10
a = 314 square inches

Question 10.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 17

Answer:
The area  of circle = 7.065 sq inches

Explanation:
The area  of circle = πr x r
a = 3.14 x 1.5 x 1.5 where r = 1.5
a = 7.065 square inches

Question 11.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 18

Answer:
The area  of circle = 3.14 sq cm

Explanation:
The area  of circle = πr x r
a = 3.14 x 1 x 1 where r = 1
a = 3.14 square cm

Question 12.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 19

Answer:
area  of circle = 1.76625sq ft

Explanation:
The area  of circle = πr x r
a = 3.14 x 0.75 x 0.75 where r = 0.75
a = 1.76625 square ft

Question 13.
YOU BE THE TEACHER
Your friend finds the area of a circle with a diameter of 7 meters. Is your friend correct? Explain.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 20

Answer:
No, my friend is not correct.

Explanation:
The area  of circle = πr x r
a = 3.14 x 3.5 x 3.5 where r = 0.75
a = 38.465 square meters
Question 14.
MODELING REAL LIFE
The diameter of a flour tortilla is 12 inches. What is the total area of two tortillas?

Answer:
The area  of tortilla = 226.08 sq inches

Explanation:
The area  of tortilla = πr x r
a = 3.14 x 6 x 6 where r = 6
a = 113.04 square inches
for 2 tortilla = 226.08 sq inches

Question 15.
MODELING REAL LIFE
The diameter of a coaster is 7 centimeters. What is the total area of five coasters?

Answer:
The  total area  of coaster = 192.325 cm

Explanation:
The area  of tortilla = πr x r
a = 3.14 x 3.5 x 3.5 where r = 3.5
a = 38.465 square cm
for 5 tortilla = 192.325 centimeters

Question 16.
PROBLEM SOLVING
The HillsboroInlet Lighthouse lights up how much more area than the Jupiter Inlet Lighthouse?
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 21

Answer:
The HillsboroInlet Lighthouse lights are 2 times greater than the Jupiter Inlet Lighthouse.

Explanation:
Hillsboro inlet Lighthouse = 3.14 x 28 x 28
area = 2,461.76 sq mi
jupiter inlet Lighthouse = 3.14 x 18 x 18
area = 1,017.36 sq mi

FINDING THE AREA OF A SEMICIRCLE Find the area of the semicircle.
Question 17.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 22

Answer:
Area of semicircle = 628 sq cm

Explanation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +400/2)
area =(403.14/2)
area = 628 sq cm

Question 18.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 23
Answer:
Area of semicircle =201.57 sq cm

Explanation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +400/2)
area =(403.14/2)
area = 201.57 sq cm

Question 19.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 24

Answer:
Area of semicircle =1.57 sq ft

Explanation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +1/2)
area =(3.14/2)
area = 1.57 sq ft

Question 20.
MODELING REAL LIFE
The plate for a microscope has a circumference of 100π millimeters. What is the area of the plate?

Answer:
Area of the plate = 200π mm

Explanation:
Area of the plate = π x r x r
area = 3.14 x 200 x 200

Question 21.
MODELING REAL LIFE
A dog is leashed to the corner of a house. How much running area does the dog have? Explain how you found your answer.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 25

Answer:
Area of the circle = 942 sq ft

Explanation:
Area of the circle = π x r x r
area = 3.14 x 20 x 20
area = 942 sq ft
The running area is 3/4 the area of a circle with a radius of 20 feet.

Question 22.
REASONING
Target A has a circumference of 20 feet. Target B has a diameter of 3 feet. Both targets are the same distance away. Which target is easier to hit? Explain your reasoning.

Answer:
Target B is easier to hit

Explanation:
Target A =2  π x r
A  = 2 x 3.14 x 3.18
A = 19.9704

Target B = 1.5

Question 23.
DIG DEEPER!
A circular oil spill has a radius of 2 miles. After a day, the radius of the oil spill increases by 3 miles. By how many square miles does the area of the oil spill increase?

Answer:
The area of oil spill increases by 65.94 miles.

Explanation:
Given that the circular oil spill has a radius of 2 miles.
The radius of the oil spill increases by 65.94 sq miles.

Question 24.
FINDING AN AREA
Find the area of the circle in square yards.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles 9.2 26

Answer:
Area of the circle = 7.057935 sq yd

Explanation:
Area of the circle = π x r x r
area = 3.14 x 4.5 x 4.5
area = 63.585 sq ft
area = 7.057935 sq yd

Question 25.
REPEATED REASONING
What happens to the circumference and the area of a circle when you double the radius? triple the radius? Justify your answer.

Answer:
If we double the radius ,area  = π x r x r x r x r
If we double the radius, circumference  = 2πr x r x r
If we triple the radius ,area  = π x r x r x r x r x r
If we triple the radius, circumference  = 2πr x r x r x r x r

Explanation:
circumference doubles and area quadruples;
circumference triples and area is 9 times greater;
double the radius: circumference = 2π2r = 4πr
4πr /2πr  = 2 times larger, area =π (2r) x r =4πrx r
4πrx r/ πrx r = 4 times larger.

Question 26.
CRITICAL THINKING
Is the area of a semicircle with a diameter of x greater than, less than, or equal to the area of a circle with a diameter of \(\frac{1}{2}\)x? Explain.

Answer:
The area of a semicircle with a diameter of x is greater than the area of a circle with a diameter of (0.5)

Explanation:
Area of semicircle = (3.14 + (0.5 x 0.5)/2)
area = 1.695
Area of circle = (3.14 x 0.5 x 0.5)
area = 0.785

Lesson 9.3 Perimeters and Areas of Composite Figures

EXPLORATION 1

Submitting a Bid
Work with a partner. You want to bid on a project for the pool shown. The project involves ordering and installing the brown tile that borders the pool, and ordering a custom-made tarp to cover the surface of the pool. In the figure, each grid square represents 1 square foot. You pay $5 per linear foot for the tile.

  • You pay $4 per square foot for the tarp.
  • It takes you about 15 minutes to install each foot of tile.

a. Estimate the total cost for the tile and the tarp.
b. Write a bid for how much you will charge for the project.Include the hourly wage you will receive.Estimate your total profit.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 1

Answer:

Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 2

Try It

Question 1.
Estimate the perimeter and the area of the figure.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 3

Answer:
50.24 sq mm

Explanation:
The above-given figure is about  50.24 sq mm
Question 2.
Find the perimeter and the area of the figure.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 4

Answer:
perimeter of the figure = 3.16 sq in
area of the figure = 3.14 sq in

Explanation:
perimeter of the semicircle = ( π + 2) r
p = (3.14 + 2 ) 1
p = 3.16 in
area of the figure = π x r x r
area = 3.14 x 1 x1
area = 3.14 in

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
ESTIMATING PERIMETER AND AREA
Estimate the perimeter and area of the figure at the right.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 5

Answer:
The perimeter  and area = 30 ft
area =  π  x r x r

Explanation:

The perimeter = ( π + 2) r
area =  π  x r x r

Question 4.
FINDING PERIMETER AND AREA
Identify the shapes that make up the figure at the left. Then find the perimeter and area of the figure.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 6

Answer:
The perimeter = 9.48 sq ft
area =  27.36 sq ft

Explanation:
The perimeter = ( π + 2) r
perimeter = 3.14 +2 x 3
perimeter = 9.48 sq ft
area = 3.14 x 3 x 3
area = 27.36 sq feet

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
A farmer wants to seed and fence a section of land. Fencing costs $27 per yard. Grass seed costs $2 per square foot. How much does it cost to fence and seed the pasture?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 7

Answer:
1 m

Explanation:
Given that farmer has the fencing cost = $ 27
seed cost = $ 2
5.10 $ is used to cost for  grass seed
$ 27 is used to fence = 1 m

Question 6.
DIG DEEPER!
In each room shown, you plan to put down carpet and add a wallpaper border around the ceiling. Which room needs more carpeting? more wallpaper?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 8

Answer:
Room A needs more carpeting.

Explanation:
Room A =  10 x 11
where length = 11 , breadth = 10 given
Room A = 110
Room B = 12 x 8
B = 96

Perimeters and Areas of Composite Figures Homework & Practice 9.3

Review & Refresh

Find the area of the circle. Use 3.14 or \(\frac{22}{7}\) for π.
Question 1.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 9

Answer:
Area of the circle = 50.24 sq mm

Explanation:
Area of the circle = π x r x r
area = 3.14 x 4 x 4
area = 3.14 x 16
area = 50.24 sq mm

Question 2.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 10

Answer:
Area of the circle = 63.585 sq ft

Explanation:
Area of the plate = π x r x r
area = 3.14 x 4.5 x 4.5
area = 3.14 x
area = 63.585 sq ft

Find the missing dimension. Use the scale 1 : 5.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 11

Answer:
3. Height = 30 ft
4. Length = 6 ft
5. Depth = 100 cm
6. Diameter = 2 in

Explanation:
3. house : height = 6 ft , height = 30 ft given that scale = 1 : 5
4. garden hose : length = 6 ft , length = 20 yd
4. fountain : depth = 20 cm, depth = 100 cm
5. bicycle wheel :  = diameter = 2 in  diameter = 2 ft

Concepts, Skills, & Problem Solving

ESTIMATING PERIMETER AND AREA You build a patio with a brick border. (See Exploration 1, p. 375.)
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 12
Question 7.
Estimate the perimeter of the patio.

Answer:
The perimeter of a patio = 24 units

Explanation:
In the above-given figure,
the perimeter of the patio = 24

Question 8.
Estimate the area of the patio.

Answer:
area of the patio = π r

ESTIMATING PERIMETER AND AREA Estimate the perimeter and the area of the shaded figure.
Question 9.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 13

Answer:
Perimeter = 19.5 units
area =13.5 units

Explanation:
given figure is trapezoid
Perimeter = a + b + c + d
area =( (a + b) x h /2)

Question 10.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 14

Answer:
area =(  3 √ 3/2) a square
perimeter = 6 a

Explanation:
given figure is hexagon
area =(  3 √ 3/2) a square
perimeter = 6 a

Question 11.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 15

Answer:
The perimeter = 24.6 units
Area of the plate = 41.1 sq units

Explanation:
given figure is semicircle
The perimeter = ( π + 2) r
Area of the plate = π x  r x r

Question 12.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 16

Answer:
Perimeter = a + b + c + d
area =( (a + b) x h /2)

Explanation:
given figure is trapezoid
Perimeter = a + b + c + d
area =( (a + b) x h /2)

Question 13.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 17

Answer:
Perimeter = 19 units
area = 24  squnits

Explanation:
given figure is pentagon
Perimeter = 5 a
area = (  perimeter x apotherm /2 )

Question 14.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 18

Answer:
Perimeter = a + b + c
area = (  height x breadth /2 )

Explanation:
given figure is triangle
Perimeter = a + b + c
area = (  height x breadth /2 )

FINDING PERIMETER AND AREA Find the perimeter and the area of the figure.
Question 15.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 19

Answer:
area = 137 sq m
perimeter = 56 m

Explanation:
area of the rectangle = l x w
l = length, w = width
area = 12 x 11
area = 137 sq m
perimeter of the rectangle = 2 ( l + w)
perimeter = 2 (28)
perimeter = 56 m

Question 16.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 20

Answer:
area = 114.07 sq ft
perimeter = 47.4 sq ft

Explanation:
Area of semicircle =( π+r x r/2)
area =( 3.14 +225/2)
area =(228.14/2)
area = 114.07 sq ft
perimeter of the semicircle = (π + 2 ) r
perimeter = (3.14 + 2) 15 given r= 15
perimeter = 3.16 x 15
perimeter = 47.4 sq ft

Question 17.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 21

Answer:
area = 49.5 cm
perimeter = 29 cm

Explanation:
area of the rectangle = l x w
l = length, w = width
area = 7 x 7
area = 49 cm
perimeter of the rectangle = 2 ( l + w)
perimeter = 2 (14)
perimeter = 29 cm

Question 18.
YOU BE THE TEACHER
Your friend finds the perimeter of the figure. Is your friend correct? Explain your reasoning.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 22

Answer:
Yes my friend is correct.

Explnation:
perimeter = length + side +  height + breadth + width + base
p = 4 + 3 + 4 + 5 + 4 + 5
p = 25 in

Question 19.
LOGIC
A running track has six lanes. Explain why the starting points for the six runners are staggered. Draw a diagram as part of your explanation.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 23

Answer:

Explanation:
The starting points for the six runners are staggered because each runner can run the same distance.

Explanation:
The starting points are staggered so that each runner can run the same distance and use the same finish line.
this is necessary because the circumference is different for each lane.
the above-diagram shows this because the diameter is greater n the outer lanes.

Question 20.
PROBLEM SOLVING
You run around the perimeter of the baseball field at a rate of 9 feet per second. How long does it take you to run around the baseball field?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 24

Answer:
It take to run around the baseball field = 1,58,962.5 sq feet

Explanation:
The area of the circle = π x r x r
area = 3.14 x 225 x 225
area = 1,58,962.5 sq feet

Question 21.
STRUCTURE
The figure at the right is made up of a square and a rectangle. Find the area of the shaded region.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 25

Answer:
The area of the shaded region =24 sq m

Ex planation:
Area of triangle = ( b x h )/2
area =( 8 x 7)/ 2
area = 48/2
area = 24 sq m
Question 22.
DIG DEEPER!
Your friend makes a two-dimensional model of a dividing cell as shown. The total area of the dividing cell is 350 square inches. What is the area of the shaded region?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles 9.3 26

Answer:
The area of the shaded region = 1.89 sq in

Explanation :
area of semicircle = (π + r x r/2)
area = (3.14 + 64/2)
area = ( 3.78 / 2)
area = 1.89 sq in
Question 23.
CRITICAL THINKING
How can you add a figure to a composite figure without increasing its perimeter? Can this be done for all figures? Draw a diagram to support your answer.

Answer:

Explanation:
The perimeter does not increases.

Lesson 9.4 Constructing Polygons

EXPLORATION 1

Using Technology to Draw Polygons
Work with a partner.
a. Use geometry software to draw each polygon with the given side lengths or angle measures, if possible. Complete the table.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 1
b. Without constructing, how can you tell whether it is possible to draw a triangle given three angle measures? three side lengths? Explain your reasoning.
c. Without constructing, how can you tell whether it is possible to draw a quadrilateral given four angle measures? four side lengths? Explain your reasoning.

Answer:
b. Yes it is possible to draw a triangle with the given three angles measures, three side lengths.
c. yes it is possible to draw a quadrilateral with the given 4 angle measures, four side lengths.

Explanation:
1. given that sides = 4 cm , 6 cm , 7cm

2. given that sides = 2 cm , 3 cm , 3 cm, 5 cm

Try It

Draw a triangle with the given angle measures, if possible.
Question 1.
45°, 45°, 90°

Answer:

Explanation:
The above triangle is an equilateral triangle.
it forms with the given angles 45°, 45°, 90°.

Question 2.
100°, 55°, 25°

Answer:

Explanation:
The above triangle is scalene  triangle.
it forms with the given angles 100°, 55°, 25°.

Question 3.
60°, 60°, 80°

Answer:

Explanation:
The above triangle is an equilateral triangle.
it forms with the given angles60°, 60°, 80°

Question 4.
Draw a triangle with side lengths of 1 inch and 2 inches that meet at a 60° angle.

Answer:

Explanation:
The above triangle is a scalene triangle.
it forms with the given angles 60° , 1 inch and 2 inch.

Draw a triangle with the given side lengths, if possible.
Question 5.
2 cm, 2 cm, 5 cm

Answer:

Explanation:
given the sides of a triangle 2cm , 2 cm , 5 cm

Question 6.
4 cm, 3 cm, 3 cm

Answer:

Explanation:
given that 2 sides are same and one side is different.

Question 7.
1 cm, 4 cm, 5 cm

Answer:

Draw a quadrilateral with the given angle measures, if possible.
Question 8.
100°, 90°, 65°, 105°

Answer:

Explanation:
The quadrilateral formed with the given angles 100°, 90°, 65°, 105°.

Question 9.
100°, 40°, 20°, 20°

Answer:

Explanation:
The quadrilateral formed with the given angles 100°, 40°, 20°, 20°.

Question 9.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

DRAWING POLYGONS Draw a polygon with the given side lengths or angle measures, if possible.
Question 10.
25 mm, 36 mm, 38 mm

Answer:

Explanation:
The polygon formed with the given sides is a triangle.

Question 11.
10°, 15°, 155°

Answer:

Explanation:
The polygon formed with the given sides is a triangle.

Question 12.
20°, 45°, 50°, 65°

Answer:

Explanation:
The polygon formed with the given sides is a  hexagon.

Question 9.
100°, 40°, 20°, 20°

Answer:

Explanation:
The polygon formed with the given sides is a  hexagon.
Question 9.
100°, 40°, 20°, 20°
Answer:

Question 9.
100°, 40°, 20°, 20°
Answer:

Explanation:
The polygon formed with the given sides is a  quadrilateral.

Question 13.
50°, 90°, 110°, 110°

Answer:

Question 14.
USING SIDE LENGTH
Can you construct one, many, or triangle(s) with side lengths of 3 inches, 4 inches, and 8 inches? Explain.
Answer:
We can construct only one triangle

Explanation:
Given the side lengths of 3 inches, 4 inches, and 8 inches.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 15.
A triangular pen has fence lengths of 6 feet, 8 feet, and 10 feet. Create a scale drawing of the pen.
Answer:

Question 16.
The front of a cabin is the shape of a triangle. The angles of the triangle are 40°, 70°, and 70°. Can you determine the height of the cabin? If not, what information do you need?
Answer:

Question 17.
DIG DEEPER!
Two rooftops have triangular patios. One patio has side lengths of 9 meters,10 meters, and 11 meters.e other has side lengths of 6 meters,10 meters, and 15 meters. Which patio has a greater area? Explain.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 2
Answer:
The patio which has a side length of 6 meters, 10 meters, and 15 meters.

Explanation:
The patio has a greater side length.

Constructing Polygons Homework & Practice 9.4

Review & Refresh

Find the perimeter and area of the figure.
Question 1.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 3
Answer:
area = 12 in
perimeter = 14 in

Explanation:
area of the rectangle = l x w
l = length, w = width
area = 4 x 3
area = 12 in
perimeter of the rectangle = 2 ( l + w)
perimeter = 2 (7)
perimeter = 14 in

Question 2.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 4
Answer:

perimeter of the figure = 9.48 sq cm
area of the figure = 28.26 sq cm

Explanation:
perimeter of the semicircle = ( π + 2) r
p = (3.14 + 2 ) 3
p = 9.48 cm
area of the figure = π x r x r
area = 3.14 x 3 x3
area = 28.26 sq cm

Use a tree diagram to find the sample space and the total number of possible outcomes of the indicated event.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 5
Question 3.
choosing a toothbrush
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 6
Answer:
Extra soft, soft, Medium

Explanation:
In the above given figure the strength of the toothbrush = extra soft , soft , meedium

Question 4.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 7
Answer:
The size of the toy hop is small, medium , large.

Explanation:
given that the colour of the toy hoop is blue , green , orange, pink, purple , yellow.

Concepts, Skills, & Problem Solving

USING TECHNOLOGY TO DRAW POLYGONS Use geometry software to draw the polygon with the given side lengths or angle measures, if possible. (See Exploration 1, p. 381.)
Question 5.
30°, 65°, 85°
Answer:

Question 6.
2 in., 3 in., 5 in.
Answer:

Question 7.
80°, 90°, 100°, 110°
Answer:
Not possible.

Question 8.
2 cm, 2 cm, 5 cm, 5 cm
Answer:

CONSTRUCTING TRIANGLES USING ANGLE MEASURES Draw a triangle with the given angle measures, if possible.
Question 9.
40°, 50°, 90°
Answer:

Question 10.
20°, 40°, 120°
Answer:

Question 11.
38°, 42°, 110°
Answer:

Question 12.
54°, 60°, 66°
Answer:

Question 13.
YOU BE THE TEACHER
Your friend determines whether he can draw a triangle with angle measures of 10°, 40°, and 130°. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 8
Answer:
Yes .

Explanation:
yes we cannot draw the triangle with the angle measures of 10, 40, 130

CONSTRUCTING TRIANGLES USING ANGLES AND SIDES Draw a triangle with the given description.
Question 14.
side lengths of 1 inch and 2 inches meet at a 50° angle
Answer:
yes.

Explanation:
we can draw a triangle with 1 inch 2 inch that meets at  50 degrees.

Question 15.
side lengths of 7 centimeters and 9 centimeters meet at a 120° angle
Answer:
yes.

Explanation:
we can draw a triangle with 7 cm 9 cm that meets at  120 degrees.

Question 16.
a 95° angle connects to a 15° angle by a side of length 2 inches
Answer:
no.

Explanation:
we cannot draw a triangle with 2 inches 15 degrees that meets at  120 degrees.

Question 17.
a 70° angle connects to a 70° angle by a side of length 4 centimeters
Answer:
yes.

Explanation:
we can draw an angle with 4 cm  70 degrees that meets at  120 degrees.

CONSTRUCTING TRIANGLES USING SIDE LENGTHS Draw a triangle with the given side lengths, if possible.
Question 18.
4 in., 5 in., 10 in.
Answer:

Question 19.
10 mm, 30 mm, 50 mm
Answer:

Question 20.
5 cm, 5 cm, 8 cm
Answer:

Question 21.
8 mm, 12 mm, 13 mm
Answer:

Question 22.
MODELING REAL LIFE
Can you construct a triangular case using two pieces of wood that are 12 inches long and one piece of wood that is 25 inches long? Explain.
Answer:
Yes we can construct a triangle .

Explanation:
We can costruct the triangle by using two pieces of wood that are 12 inches long and the one piece of wood is 25 inches.

Question 23.
MODELING REAL LIFE
Can you construct a warning triangle using three pieces of plastic that are each 6 inches long? Explain.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 9
Answer:
Yes.

Explanation:
we can construct the three pieces of plastic by using 3 6 inches long.

Question 24.
LOGIC
You are constructing a triangle. You draw the first angle, as shown. Your friend says that you must be constructing an acute triangle. Is your friend correct? Explain your reasoning.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 10
Answer:
Yes my friend is correct.

Explanation:
it is a acute angle triangle.

USING ANGLES AND SIDES Determine whether you can construct one, many, or no triangle(s) with the given description. Explain your reasoning.
Question 25.
a triangle with one angle measure of 60and one side length of 4 centimeters
Answer:

Explanation:
we cannot construct one trinangle with the help of given sidelengths.

Question 26.
a scalene triangle with side lengths of 3 centimeters and 7 centimeters
Answer:

Question 27.
an isosceles triangle with two side lengths of 4 inches that meet at an 80° angle
Answer:

Question 28.
a triangle with one angle measure of 60°, one angle measure of 70°, and a side length of 10 centimeters between the two angles
Answer:

Question 29.
a triangle with one angle measure of 20°, one angle measure of 35°, and a side of length 3 inches that is between the two angles
Answer:

Question 29.
REASONING
A triangle is shown.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 11
a. Construct a triangle with side lengths twice those of the triangle shown. Does the new triangle have the same angle measures?
b. How can you change the side lengths of the triangle so that the measure of ∠A increases?
Answer:
a. Yes the new triangle have the same angle.
b. angle A increases .

Explanation:
Given that the triangle with side lengths twice those of the triangle shown.
If we can change the side lengths of triangle .

CONSTRUCTING QUADRILATERALS Draw a quadrilateral with the given angle measures, if possible.
Question 31.
60°, 60°, 120°, 120°
Answer:

Question 32.
50°, 60°, 110°, 150°
Answer:

Question 33.
20°, 30°, 150°, 160°
Answer:

Question 34.
10°, 10°, 10°, 150°
Answer:

Explanation:
Given angles are 10 degrees, 10 degrees, 10 degrees, 10 degrees.

CONSTRUCTING SPECIAL QUADRILATERALS Construct a quadrilateral with the given description.
Question 35.
a rectangle with side lengths of 1 inch and 2 inches
Answer:

Question 36.
a kite with side lengths of 4 centimeters and 7 centimeters
Answer:

Question 37.
a trapezoid with base angles of 40°
Answer:
Answer

Question 38.
a rhombus with side lengths of 10 millimeters
Answer:

Question 39.
REASONING
A quadrilateral has side lengths of 6 units, 2 units, and 3 units as shown. How many quadrilaterals can be formed given a fourth side with a fixed length? Explain.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 12
Answer:
2 quadrilaterals can be formed.

Explanation:
Given that the quadrilateral has side lengths of 6 units, 2 units, and 3 units.
so 2 quadrilaterals can be formed.

Question 40.
REASONING
What types of quadrilaterals can you form using four side lengths of 7 units? Use drawings to support your conclusion.
Answer:

Question 41.
MODELING REAL LIFE
A triangular section of a farm is enclosed by fences that are 2 meters, 6 meters, and 7 meters long. Estimate the area of the section.
Answer:
Area of the section = 12 sq meters.

Question 42.
MODELING REAL LIFE
A chemical spill expert sets up a triangular caution zone using cones. Cones A and B are 14 meters apart. Cones B and C are 22 meters apart. Cones A and C are 34 meters apart. Estimate the area of the caution zone.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 13
Answer:
Area of the area of the caution Zone = 308 sq meters.

Explanation:
Area of the triangle = l x b
area = 22 x 14
area = 308 sq meters.

Question 43.
MODELING REAL LIFE
A search region is in the shape of an equilateral triangle. The measure of one side of the region is 20 miles. Make a scale drawing of the search region. Estimate the area of the search region.
Answer:

Explanation:
Given that the equilateral triangle .

Question 44.
REASONING
A triangle has fixed side lengths of 2 and 14.
a. How many triangles can you construct? Use the figure below to explain your reasoning.
Big Ideas Math Solutions Grade 7 Chapter 9 Geometric Shapes and Angles 9.4 14
b. Is the unknown side length of the triangle also fixed? Explain.
Answer:
We can construct 14 triangles.
b. No the side length of the triangle cannot fixed.

Explanation:
a. We can construct 14 triangles.
b. No the side length of the triangl cannot fixed.

Lesson 9.5 Finding Unknown Angle Measures

EXPLORATION 1

Using Rules About Angles
Work with a partner. The diagram shows pairs of angles and vertical angles. Vertical angles cannot be adjacent.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 1
a. Which pair(s) of angles are adjacent angles? Explain.
b. Which pair(s) of angles are vertical angles? Explain.
c. Without using a protractor, find the values of x, y, and z. Explain your reasoning.
d. Make a conjecture about the measures of any two vertical angles.
e. Test your conjecture in part(d) using the diagram below. Explain why your conjecture is or is not true.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 2
Answer:
A. ∠ACD, ∠AEB
b. ∠ACD, ∠AEB
c. 125
d. ∠ACD, ∠AEB
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 3

Try It

Question 1.
Name a pair of (a) adjacent angles, (b) complementary angles, (c) supplementary angles, and (d) vertical angles in the figure.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 4
Answer:
a. ∠JKL, ∠JKQ, . ∠MNJ,. ∠PJN
b. ∠JKQ
c. ∠JNK, ∠ JPL. ∠JMQ
D. ∠JMQ, ∠JPL.

Explanation:
The above angles are adjacent, vertical, supplementary,complementary.

Classify the pair of angles. Then find the value of x.
Question 2.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 5
Answer:
x = 95 ˚

Explanation:
x = (180 – 85)
x = 95 ˚

Question 3.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 6
Answer:
x =  180 ˚

Explanation:
x = 180 ˚

Question 4.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 7
Answer:
x = 30 ˚

Explanation:
(2x – 3) = 60
2x = (60/3)
2x = 20
x = 10

Find the measure of the indicated angle in the diagram.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 8.1
Question 5.
∠NJM
Answer:
12 x ˚

Question 6.
∠KJP
Answer:
16 x ˚

Question 7.
∠KJM
Answer:
6x ˚

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.
NAMING ANGLES
Name a pair of (a) adjacent angles, (b) complementary angles, (c) supplementary angles, and (d) vertical angles in the figure at the left.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 8
Answer:
a. ∠ABC
b. ∠ABD
c. ∠ABE
d. ∠ABE

Explanation:
The above angles are adjacent, vertical, supplementary,complementary.

FINDING ANGLE MEASURES Find the value of x.
Question 9.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 9
Answer:
x = 60˚

Explanation:
4x = x
4x – x = 180
3x = 180
x = 60˚

Question 10.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 10
Answer:
x = 12.5˚

Explanation:
2x  – 10= 2x + 40
4x  = 50
x = 12.5˚

Question 11.
WHICH ONE DOESN’T BELONG?
Which pair of angles does not belong with the other three? Explain your reasoning.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 11
Answer:
∠FBA, ∠EBD does not belong with the other three.

Explanation:
the 3 angles are different measures,

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 12.
What is the angle between any two windmill blades in the windmill at the left? Justify your answer.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 12

Answer:
The angle between any two wind mills blades in the windmill at the left = 60 °

Explanation:
60 + 60 + 60 = 180

Question 13.
A hockey puck strikes a wall at an angle of 30°. The puck then travels away from the wall at the same angle. Find the value of y. Explain your reasoning.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 13
Answer:
y = 150 °

Explanation:
In the above figure said that hockey puck strikes a wall at an angle of 30 °.
so 180 – 30 = 150

Question 14.
DIG DEEPER!
The laptop screen turns off when the angle between the keyboard and the screen is less than 20°. How many more degrees can the laptop screen close before the screen turns off?
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 14
Answer:
The laptop screen close before the screen turns off = 60 degrees.

Explanation:
(z + 40) = (z – 20)
z – z = (-  20 -40)
z = -60

Finding Unknown Angle Measures Homework & Practice 9.5

Review & Refresh

Draw a triangle with the given side lengths, if possible.
Question 1.
1 in., 3 in., 4 in.
Answer:

Explanation:
In the above question , they said to draw 1 in, 3 in, 4 in.

Question 2.
4 cm, 4 cm, 7 cm
Answer:

Solve the inequality. Graph the solution.
Question 3.
– 8y ≤ 40
Answer:

Explanation:
– 8y ≤ 40
y = (40/8)
y = 5

Question 4.
1.1z > – 3.3
Answer:

Explanation:
z = 3.3

Question 5.
\(\frac{1}{3}\)x ≥ 2.5
Answer:

Concepts, Skills, & Problem Solving

USING RULES ABOUT ANGLES The diagram shows pairs of adjacent vertical angles and angles. B(See Exploration 1, p. 389.)
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 15
Question 6.
Which pair(s) of angles are adjacent angles? Explain.
Answer:
angle AEC, angle ABD.

Explanation:
In the above given figure angle AEC, angle ABD are adjacent.

Question 7.
Which pair(s) of angles are vertical angles? Explain.
Answer:
angle ABC, angle ADE

Explanation:
In the above given figure angle AEC, angle ABD are adjacent.

NAMING ANGLES Use the figure shown.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 16
Question 8.
Name a pair of adjacent angles.
Answer:
∠ADC, ∠AEF, ∠ABC

Question 9.
Name a pair of complementary angles.
Answer:
∠ADE, ∠ABD

Question 10.
Name a pair of supplementary angles.
Answer:
∠ABE, ∠ACF

Question 11.
Name a pair of vertical angles.
Answer:
∠AEF, ∠ABC

Question 12.
YOU BE THE TEACHER
Your friend names a pair of angles with the same measure. Is your friend correct? Explain your reasoning.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 17
Answer:
yes my friend is correct

Explanation:
The angles both have the same measure.

ADJACENT AND VERTICAL ANGLES Tell whether the angles are adjacent, vertical, or neither. Explain.
Question 13.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 18
Answer:
vertical.

Explanation:
The given angles are vertical.

Question 14.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 19
Answer:
Adjacent.

Explanation:
The given angles are adjacent.

Question 15.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 20

Answer:
Adjacent, vertical

Explanation:
The given angles are adjacent, vertical.

COMPLEMENTARY AND SUPPLEMENTARY ANGLES Tell whether the angles are complementary supplementary, or neither. Explain.
Question 16.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 21
Answer:
The angles are neither complementary nor supplementary.

Explanation:
complementary = 90 degree
supplementary = 180 degree

Question 17.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 22
Answer:
The angles are complementary.

Explanation:
complementary = 90 degree
supplementary = 180 degree

Question 18.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 23
Answer:
The angles are complementary.

Explanation:
complementary = 90 degree
supplementary = 180 degree

Question 19.
YOU BE THE TEACHER
Your friend names a pair of supplementary angles. Is your friend correct? Explain.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 24
Answer:
yes my friend is correct.

Explanation:
angle LMN and angle PMQ are supplementary angles.

USING PAIRS OF ANGLES Classify the pair of angles. Then find the value of x.
Question 20.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 25
Answer:
Acute angle.
x = 145

Explanation:
x = (180 – 35)
x = 35

Question 21.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 26
Answer:
verticle angle.
x = 52

Explanation:
x = (180 – 128)
x = 52

Question 22.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 27
Answer:
obtuse angle.
x = 63

Explanation:
x = (180 – 117)
x = 63

Question 23.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 28
Answer:
intersection angles
x = 25

Explanation:
(4x – 25) = 75
4x = 75 + 25
4x = 100
x = (100/4)
x = 25

Question 24.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 29
Answer:
x = 15

Explanation:
2x = 30
x = (30/2)
x = 15
4x = 60
x = (60/4)
x = 15

Question 25.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 30
Answer:
x = 3.33

Explanation:
(x + 20 ) = 7 x
20 = 7x – x
20 = 6x
x = (20/6)
x = 3.33

Question 26.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 31
Answer:
x = 15

Explanation:
3x = 45
x = (45/3)
x = 15

Question 27.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 32
Answer:
x = 20

Explanation:
(x – 20 ) = x
20 = x – x
x = 20

Question 28.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 33
Answer:
x = 25

Explanation:
(3x + 25) = 2x
3x – 2x = 25
x = 25

Question 29.
MODELING REAL LIFE
What is the measure of each angle formed by the intersection? Explain.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 34
Answer:
angle 2 = 50°
angle 3 = 40°
angle 1 = 40°

Explanation:
In the above figure the angle 4 is given.

Question 30.
MODELING REAL LIFE
A tributary joins a river at an angle x. Find the value of x. Explain.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 35
Answer:
x = 21

Explanation:
(2x + 21 ) = x
2x – x = 21
x = 21

Question 31.
MODELING REAL LIFE
The iron cross is a skiing trick in which the tips of the skis are crossed while the skier is airborne. Find the value of x in the iron cross shown.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 36
Answer:
The value of x in the iron cross = 43

Explanation:
(2x + 41) = 127
2x = 127 – 41
2x = 86
x = 43

FINDING ANGLE MEASURES Find all angle measures in the diagram.
Question 32.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 37
Answer:
x = 90˚

Question 33.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 38
Answer:
23.33

Explanation:
(3x + 5) = 75
3x = 75 – 5
3x = 70
x = (70/3)
x = 23.33

Question 34.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 39
Answer:
x = 68
x = 67

Explanation:
(2x + 4) = 140
2x = (140 – 4)
2x = 136
x = (136/2)
x = 68
(2x + 6) = 140
2x = (140 – 6)
2x = 134
x = (134/2)
x = 67

OPEN-ENDED Draw a pair of adjacent angles with the given description.
Question 35.
Both angles are acute.
Answer:

Question 36.
One angle is acute, and one is obtuse.
Answer:

Question 37.
The sum of the angle measures is 135°.
Answer:

REASONING Copy and complete each sentence with always, sometimes, or never.
Question 38.
If x and y are complementary angles, then both x and y are________ acute.
Answer:
Right acute.

Explanation:
if x and y are complimentary then the x and y are right acute.

Question 39.
If x and y are supplementary angles, then is x ________ acute.
Answer:
left acute.

Explanation:
if x and y are complimentary then the x and y are right acute.

Question 40.
If x is a right angle, then is x ________ acute.
Answer:
Right acute.

Explanation:
if x and y are complimentary then the x and y are right acute.

Question 41.
If x and y are complementary angles, then x and y are ________ adjacent.
Answer:
Right adjacent.

Explanation:
if x and y are complimentary then the x and y are right adjacent.

Question 42.
If x and y are supplementary angles, then x and y are _______ vertical.
Answer:
left vertical.

Explanation:
if x and y are supplementary then the x and y are left vertical.

Question 43.
REASONING
Draw a figure in which ∠1 and ∠2 are acute vertical angles, ∠3 is a right angle adjacent to ∠2, and the sum of the measure of ∠1 and the measure of ∠4 is 180°.
Answer:

Question 44.
STRUCTURE
Describe the relationship between the two angles represented by the graph shown at the right.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 40
Answer:
90°

Explanation:
The relationship between the two angles represented by the graph =90°

Question 45.
STRUCTURE
Consider the figure shown at the left. Use a ruler to extend both rays into lines. What do you notice about the three new angles that are formed?
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 41
Answer:
The 3 angles that are formed = 30°, 60°, 90°

Explanation:
The given angles are right angles.

Question 46.
OPEN-ENDED
Give an example of an angle that can be a supplementary angle but cannot be a complementary angle to another angle. Explain.
Answer:
Acute angle

Question 47.
MODELING REAL LIFE
The vanishing point of the picture is represented by point B.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 42
a. The measure of ∠ABD is 6.2 times greater than the measure of ∠CBD. Find the measure of ∠CBD.
b. ∠FBE and ∠EBD are congruent. Find the measure of ∠FBE.
Answer:
a. The measure of  ∠CBD = 30°
b. The measure of ∠FBE = 60°

Explanation:
Given that the measure of ∠ABD is 6.2 times greater than the measure of ∠CBD  = 30°
∠FBE and ∠EBD are congruent so ∠FBE = 60°

Question 48.
CRITICAL THINKING
The measures of two complementary angles have a ratio of 3 : 2. What is the measure of the larger angle?
Answer:
The measure of the larger angle = 3

Explanation:
given that, the measures of two complementary angles have a ratio = 3 : 2

Question 49.
REASONING
Two angles are vertical angles. What are their measures if they are also complementary angles? supplementary angles?
Answer:
when two angles are vertical.
complementary angles = Two angles are called complementary when their measures add to 90°
supplementary angles = two angles are called supplementary when their measures add to 180°

Question 50.
PROBLEM SOLVING
Find the values of x and y.
Big Ideas Math Answer Key Grade 7 Chapter 9 Geometric Shapes and Angles 9.5 43
Answer:
x = 2.857
y = 2
x = 4

Explanation:
7  x = 20
x = (20/7)
x = 2.857
2y = 20
y = (20/10)
y  = 2
5x = 20
x = (20/5)
x = 4

Geometric Shapes and Angles Connecting Concepts

Using the Problem-Solving Plan
Question 1.
A dart is equally likely to hit any point on the board shown. Find the theoretical probability that a dart hitting the board scores 100 points.
Understand the problem.
You are given the dimensions of a circular dartboard. You are asked to find the theoretical probability of hitting the center circle.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cc 1
Make a plan.
Find the area of the center circle and the area of the entire dart board. To find the theoretical probability of scoring 100 points, divide the area of the center circle by the area of the entire dart board.
Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer:
Area of center =31,400 sq in
area of entire dart board = 1,962.5 sq in

Explanation:
Area of center circle = π r ²
a = 3.14 x 100 x 100
a = 31,400  sq in
area of entire dart board =  π r ²
a = 3.14 x 25 x 25
a = 3.14 x 625
a = 1,962.5 sq in

Question 2.
A scale drawing of a window is shown. Find the perimeter and the area of the actual window. Justify your answer.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cc 2
Answer:
Area of semicircle = 1.695sq ft
perimeter of semicircle = 1.58 sq ft

Explanation:
area of the semicircle = ( π + r x r /2)
s . c = (3.14 + 0.5 x 0.5 /2)
s. c = (3.14 + 0.25 /2)
s . c = (3.39 /2
s. c = 1.695 sq ft
perimeter of the semicircle = (π + 2 ) x r
p= 3.14 + 2 x 0.5
p = 3.16 x 0.5
p = 1.58 sq ft

Question 3.
∠CAD makes up 20% of a pair of supplementary angles. Find the measure of ∠DAE. Justify your answer.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cc 3
Answer:
∠DAE = 30 %

Explanation:
Given that ∠CAD = 20%
so
∠DAE = 30 %

Performance Task

Finding the Area and Perimeter of a Track
At the beginning of the this chapter, you watched a STEAM video called “Track and Field”. You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cc 4

Geometric Shapes and Angles Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 1

Graphic Organizers

You can use a Four Square to organize information about a concept. Each of the four squares can be a category, such as definition, vocabulary, example, non-example, words, algebra, table, numbers, visual, graph, or equation. Here is an example of a Four Square for circumference.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 2

Choose and complete a graphic organizer to help you study each topic.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 3
1. area of a circle
2. semicircle
3. composite figure
4. constructing triangles
5. constructing quadrilaterals
6. complementary angles
7. supplementary angles
8. vertical angles

Answer:
1. area of a circle = π r ²
2. semicircle = ( π  +  r ²/2)
3. composite figure = The figure that consists of two or more geometric shapes.
4. constructing triangles = A triangle is a 3 – sided polygon made up of three sides having 3 angles.
5. constructing quadrilateral = quadrilateral can be categorized by the lengths of its sides and the size of its angles.
6. complementary angles = Two angles are called complimentary when their measures add to 90°
7. supplementary angles = two angles are called supplementary when their measures add to 180°
8. vertical angles = The angles opposite each other when two lines cross.

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 4

9.1 Circles and Circumference (pp. 361–368)
Learning Target: Find the circumference of a circle.

Question 1.
What is the radius of a circular lid with a diameter of 5 centimeters?
Answer:
radius = 50 mm

Explanation:
radius  = ( d / 2)
radius  = (5/ 2) cm
r = 2.5  cm

Question 2.
The radius of a circle is 25 millimeters. Find the diameter.
Answer:
Diameter = 50 mm

Explanation:
diameter = 2 x radius
diameter = 2 x 25  mm
d = 50 mm

Find the circumference of the object. Use 3.14 or \(\frac{22}{7}\) for π.
Question 3.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 5
Answer:
circumference of the object = 37.68 sq mm

Explanation:
D. circumference of the circle =2πr
circle = 2 x 3.14 x 6 where r =6 given
circle = 6.28 x  6
circle =37.68 sq mm

Question 4.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 6
Answer:
circumference of the object = 4.71 sq ft

Explanation:
D. circumference of the circle =2πr
circle = 2 x 3.14 x 0.75 where r =0.75 given
circle = 6.28 x  0.75
circle =4.71 sq ft

Question 5.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 7
Answer:
circumference of the object = 4.71 sq cm

Explanation:
D. circumference of the circle =2πr
circle = 2 x 3.14 x 3 .5 where r =3.5 given
circle = 6.28 x  3.5
circle =21.98  sq cm

Question 6.
You are placing non-slip tape along the perimeter of the bottom of a semicircle-shaped doormat. How much tape will you save applying the tape to the perimeter of the inside semicircle of the doormat? Justify your answer.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 8
Answer:
the tape saved = 47 .4 sq in
Explanation:
perimeter of the semicircle = ( π + 2 ) r
p = ( 3.14 + 2) 15
p = (3 .16 ) 15
p = 47.4 sq in

Question 7.
You need to carry a circular cake through a 32-inch wide doorway without tilting it. The circumference of the cake is 100 inches. Will the cake fit through the doorway? Explain.
Answer:
yes the cake fit through the doorway.
Explanation:
radius of the circle = (c/2π)
r= (100/6.28)
r = 15.923 sq in

Question 8.
Estimate the radius of the Big Ben clock face in London.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 9
Answer:
Radius of the Big Ben clock = 7.0063 m

Explanation:
radius of the circle = (c/2π)
r= (44/6.28)
r = 7.0063 m

Question 9.
Describe and solve a real-life problem that involves finding the circumference of a circle.
Answer:
The circumference of a circle = 2 π r

Explanation:
circle = 2 π r
where r = radius , π = 3.14

9.2 Areas of Circles (pp. 369-374)
Learning Target: Find the area of a circle.

Find the area of the circle. Use 3.14 or \(\frac{22}{7}\) for π.
Question 10.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 10
Answer:
The area of the circle = 50.24 sq in

Explanation:
Area of the circle = π x r x r
area = 3.14 x 4 x 4
area = 3.14 x 16
area = 50.24 sq in

Question 11.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 11
Answer:
The area of the circle = 379.94 sq cm

Explanation:
Area of the circle = π x r x r
area = 3.14 x 11 x 11
area = 3.14 x 121
area = 379.94 sq cm

Question 12.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 12
Answer:
The area of the circle = 1384.74 sq mm

Explanation:
Area of the circle = π x r x r
area = 3.14 x 21 x 21
area = 3.14 x 441
area = 1384.74 sq mm

Question 13.
A desktop is shaped like a semicircle with a diameter of 28 inches. What is the area of the desktop?
Answer:
The area of the desktop = 615.44 sq in

Explanation:
Area of the desktop = π x r x r
area = 3.14 x 14 x 14
area = 3.14 x 196
area = 615.44 sq in

Question 14.
An ecologist is studying an algal bloom that has formed on the entire surface of a circular pond. What is the area of the surface of the pond covered by the algal bloom?
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 14
Answer:
The area of the surface of the pond covered by the algol bloom = 615.44 ft

Explanation:
Area of the pond = π x r x r
area = 3.14 x 14 x 14
area = 3.14 x 196
area = 615.44 sq ft

Question 15.
A knitted pot holder is shaped like a circle. Its radius is 3.5 inches. What is its area?
Answer:
The area of the pot holder = 38.465 sq in

Explanation:
Area of the pot holder = π x r x r
area = 3.14 x 3.5 x 3.5
area = 3.14 x 12.25
area = 38.465 sq in

9.3 Perimeters and Areas of Composite Figures (pp. 375–380)
Learning Target: Find perimeters and areas of composite figures.

Find the perimeter and the area of the figure.
Question 16.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 16
Answer:
Area of semicircle = 1.695 sq in
perimeter of semicircle = 15.8 sq in

Explanation:
area of the semicircle = ( π + r x r /2)
s . c = (3.14 + 5 x 5 /2)
s. c = (3.14 + 25/2)
s . c = (3.39 /2 )
s. c = 1.695 sq in
perimeter of the semicircle = (π + 2 ) x r
p= 3.14 + 2 x 5
p = 3.16 x 5
p = 15.8 sq in

Question 17.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 17
Answer:
Area of semicircle = 6.07 sq ft
perimeter of semicircle = 9.48 sqft

Explanation:
area of the semicircle = ( π + r x r /2)
s . c = (3.14 + 3 x 3 /2)
s. c = (3.14 + 9/2)
s . c = (12.14 /2 )
s. c = 6.07 sq ft
perimeter of the semicircle = (π + 2 ) x r
p= 3.14 + 2 x 3
p = 3.16 x 3
p = 9.48 sq ft

Question 18.
GARDEN
You want to fence part of a yard to make a vegetable garden. How many feet of fencing do you need to surround the garden?
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 18
Answer:
The fencing need to surround the garden = 32 sq feet

Explanation:
area  of the rectangle = l + b
area = 18 + 14
area = 32 sq feet

9.4 Constructing Polygons (pp. 381-388)

Learning Target: Construct a polygon with given measures.

Draw a triangle with the given description, if possible.
Question 19.
a triangle with angle measures of 15°, 75°, and 90°
Answer:

Explanation:
Given triangle with angle measures.

Question 20.
a triangle with a 3-inch side and a 4-inch side that meet at a 30° angle
Answer:

Question 21.
a triangle with side lengths of 5 centimeters, 8 centimeters, and 2 centimeters
Answer:

Draw a quadrilateral with the given angle measures, if possible.
Question 22.
110°, 80°, 70°, 100°
Answer:

Question 23.
105°, 15°, 20°, 40°
Answer:

9.5 Finding Unknown Angle Measures (pp. 389–396)
Learning Target: Use facts about angle relationships to find unknown angle measures.

Use the figure shown.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 24
Question 24.
Name a pair of adjacent angles.
Answer:
x , y , v , w .

Explanation:
In the above-given figure, the adjacent angles are x, y, v, w.

Question 25.
Name a pair of complementary angles.
Answer:
u and z

Explanation:
complementary angles = u , z

Question 26.
Name a pair of supplementary angles.
Answer:
x , y , v , z

Explanation:
supplementary are x , y , v , z
Question 27.
Name a pair of vertical angles.
Answer:
x , y , v, w

Explanation:
pair of vertical angles are x , y , v , w

Classify the pair of angles. Then find the value of x.
Question 28.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 28
Answer:
x = 111 degrees.

Explanation :
x = 56
x =180  – 69
x = 111 degree

Question 29.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 29
Answer:
x = 81 degrees.

Explanation :
x + 3  = 84
x =84   – 3
x = 81  degree

Question 30.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 30
Answer:
x = 3.33degrees.

Explanation :
(4x + 10) = x
10 = x – 4 x
3 x = 10
x = 3.33  degree

Question 31.
Describe two ways to find the measure of ∠2.
Big Ideas Math Answers 7th Grade Chapter 9 Geometric Shapes and Angles cr 31
Answer:
angle 2 = 65

Explanation:
x = 180 – 115
x = 65
2 = 65

Question 32.
Using the diagram from Exercises 24–27, find all the angle measures when ∠XUY = 40°.
Answer:

Geometric Shapes and Angles Practice Test

Question 1.
Find the radius of a circle with a diameter of 17 inches.
Answer:
radius of a circle = 8.5 in

Explanation:
radius of a circle = (d / 2)
radius =( 17 / 2)
radius = 8.5 in
Find (a) the circumference and (b) the area of the circle. Use 3.14 or \(\frac{22}{7}\) for π.
Question 2.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 2
Answer:
Area of the circle = 3.14 m
circumference of the circle = 6.28 m

Explanation:
Area of the circle = π x r x r
area = 3.14 x 1 x 1
area = 3.14 m
circumference of the circle = 2 x π x r
c = 2 x 3.14 x 1
c = 6.28 m

Question 3.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 3
Answer:
Area of the circle = 3846.5 sq in
circumference of the circle = 219. 8 sq in

Explanation:
Area of the circle = π x r x r
area = 3.14 x 35 x 35
area = 3.14 x 1,225 sq in
area = 3846.5 sq in
circumference of the circle = 2 x π x r
c = 2 x 3.14 x 35
c = 6.28 x 35
c = 219.8 sq in

Find (a) the perimeter and (b) the area of the figure. Use 3.14 or \(\frac{22}{7}\) for π.
Question 4.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 4
Answer:
Area of semicircle = 2.695 sq ft
perimeter of semicircle = 4. 74 sq ft

Explanation:
area of the semicircle = ( π + r x r /2)
s . c = (3.14 + 1.5 x 1.5 /2)
s. c = (3.14 + 2.25 /2)
s . c = (5.39 /2 )
s. c = 2.695 sq ft
perimeter of the semicircle = (π + 2 ) x r
p= 3.14 + 2 x 1.5
p = 3.16 x 1.5
p = 4. 74 sq ft

Question 5.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 5
Answer:
Area of semicircle = 9.57 sq ft
perimeter of semicircle = 12.64 sq ft

Explanation:
area of the semicircle = ( π + r x r /2)
s . c = (3.14 + 4 x 4 /2)
s. c = (3.14 + 16 /2)
s . c = (19.14 /2 )
s. c = 9.57  sq ft
perimeter of the semicircle = (π + 2 ) x r
p= 3.14 + 2 x 4
p = 3.16 x 4
p = 12.64 sq ft

Draw a figure with the given description, if possible.
Question 6.
a triangle with sides of length 5 inches and 6 inches that meet at a 50° angle.
Answer:

Question 7.
a triangle with side lengths of 3 inches, 4 inches, and 5 inches
Answer:

Question 8.
a quadrilateral with angle measures of 90°, 110°, 40°, and 120°
Answer:

Classify each pair of angles. Then find the value of x.
Question 9.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 9
Answer:
x = 9 degrees.

Explanation:
(8x + 2) = 74
8x = 74 – 2
8x = 72
x = (72/8)
x = 9

Question 10.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 10
Answer:
x = 50 degrees.

Explanation:
(x + 6) = 56
x = 56 – 6
x = 50
Question 11.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 11
Answer:
x = 67 degrees.

Explanation:
x = 180 – 113
x = 67 degrees.

Question 12.
A museum plans to rope off the perimeter of the 60 ftL-shaped exhibit. How much rope does it need?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 12
Answer:
Area of the museum = 2,826 sq ft

Explanation:
Area of the museum  = π x r x r
area = 3.14 x 30 x 30
area = 3.14 x 900
area =  2,826 sq ft

Geometric Shapes and Angles Cumulative Practice

Question 13.
Draw a pair of adjacent angles that are neither complementary nor supplementary.
Answer:

Question 14.
The circumference of a circle is 36.2 centimeters. What is the length of the diameter of the circle?
Answer:
Diameter of the circle = 11.52866 cm

Explanation:
Diameter of the circle = 2 x r
radius of the circle = (c / 2 π )
circumference = 36.2 cm
radius = (36.2 / 6.28)
radius = 5.7643
daimeter = 2 x r
diameter = 5.7643 x 2
diameter = 11.52866 cm

Question 15.
The circular rug is placed on a square floor. The rug touches all four walls. How much of the floor space is not covered by the rug?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles pt 15
Answer:
Area of the circle = 176.625sq ft

Explanation:
Area of the circle  = π x r x r
area = 3.14 x 7.5 x 7.5
area = 3.14 x 56.25
area =  176.625 sq ft

Geometric Shapes and Angles Cumulative Practice

Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 1
Question 1.
To make 6 servings of soup, you need 5 cup of chicken broth. You want to know how many servings you can make with 2 quarts of chicken broth. Which proportion should you use?
A. \(\frac{6}{5}=\frac{2}{x}\)
B. \(\frac{6}{5}=\frac{x}{2}\)
C. \(\frac{6}{5}=\frac{x}{8}\)
D. \(\frac{5}{6}=\frac{x}{8}\)
Answer:
option B is correct.

Explanation:
Given that in the question to make 6 servings of soup you need 5 cup of chicken broth.

Question 2.
What is the value of x?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 2
Answer:
x = 42 degrees.

Explanation:
(2x + 1) = 85
2x = 85 – 1
2x = 84
x = (84/2)
x = 42

Question 3.
Your mathematics teacher described an inequality in words.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 3
Which inequality matches your mathematics teacher’s description?
F. 7n – 5 < 42 G. (7 – 5)n > 42
H. 5 – 7n > 42
I. 7n – 5 > 42
Answer:
option G is correct.

Explanation:
5 is less than the product of 7 and an unknown number is greater than 42.
(7 – 5)n > 42

Question 4.
What is the approximate area of the circle below? (Use \(\frac{22}{7}\) for π).
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 4
A. 132 cm2
B. 264 cm2
C. 5544 cm2
D. 22,176 cm2
Answer:
Area of the circle = 63.585 sq ft

Explanation:
Area of the circle  = π x r x r
area = 3.14 x 42 x 42
area = 3.14 x 1,764
area =  5,538.96 cm

Question 5.
You have a 50% chance of selecting a blue marble from Bag A and a 20% chance of selecting a blue marble from Bag B. Use the provided simulation to answer the question, “What is the estimated probability of selecting a blue marble from both bags?”
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 5
F. 12%
G. 16%
H. 24%
I. 88%
Answer:
option F is correct.

Explanation:
The digits 1 and 2 in the ones place represent selecting a blue marble from bag B.
The digits 1 through 5  in the tens place represent selecting a blue marble from bag A.

Question 6.
Which proportion represents the problem?
“What number is 12% of 125?”
A. \(\frac{n}{125}=\frac{12}{100}\)
B. \(\frac{12}{125}=\frac{n}{100}\)
C. \(\frac{125}{n}=\frac{12}{100}\)
D. \(\frac{12}{n}=\frac{125}{100}\)
Answer:
option B is correct.

Explanation:
(12/125 ) x 100
(12/5) x 4
Question 7.
What is the approximate perimeter of the figure below? (Use 3.14 for π)
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 7
Answer:
The perimeter of the semicircle = 18. 84

Explnation:
perimeter = ( π + 2 x r)
perimeter = (6.28 x 3 )
perimeter = 18 . 84

Question 8.
A savings account earns 2.5% simple interest per year. The principal is $850. What is the balance after 3 years?
F. $63.75
G. $871.25
H. $913.75
J. $7225
Answer:

Question 9.
Two ponds each contain about 400 fish. The double box-and-whisker plot represents the weights of a random sample of 12 fish from each pond. Which statement about the measures of center and variation is true?
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 9
A. The variation in the samples is about the same, but the sample from Pond A has a greater median.
B. The variation in the samples is about the same, but the sample from Pond B has a greater median.
C. The measures of center and variation are about the same for both samples.
D. Neither the measures of center nor variation are the same for the samples.
Answer:
option D is correct.

Explanation:
Neither the measures of center nor variation are same for the samples.

Question 10.
A lawn sprinkler sprays water onto part of a circular region, as shown below.
Big Ideas Math Answers Grade 7 Chapter 9 Geometric Shapes and Angles cp 11
Part A What is the area, in square feet, of the region that the sprinkler sprays with water? Explain your reasoning. (Use 3.14 for π.)
Part B What is the perimeter, in feet, of the region that the sprinkler sprays with water? Explain your reasoning. (Use 3.14 for π.)
Answer:
part A The region that sprinkler sprays with water = 1,256 ft
part B The region that sprinkler sprays with water = 125 .6 ft

Explanation:
area of the circle = π x r x r
area = 3.14 x 20 x 20
area = 1256 ft
perimeter of the circle =  2 x π x r
perimeter = 2 x 3.14 x 20
perimeter = 125. 6 ft

Question 11.
What is the least value of x for which x – 12 ≥ – 8 is true?
F. – 20
G. – 4
H. 4
I. 5
Answer:
option F is correct.

Explanation:
x – 12 ≥ – 8
x = -20

Final Words:

Access Big Ideas Math Book 7th Grade Answer Key 9 Geometric Shapes and Angles from the direct links presented above. Hit the direct links and prepare yourself for the exam. With the help of the problems, you can test yourself and your capability of solving the problems. Cumulative practice, Chapter review, the Practice test will help you throughout your preparation.

Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data

Big Ideas Math Answers Grade 2 Chapter 13

Do you feel maths is a difficult subject and worried about the test? If yes, then don’t worry we are here to help you out to overcome the difficulties in maths. Get free access to download Big Ideas Math Answers 2nd Grade 13th Chapter Represent and Interpret Data pdf from this page. Students can find various tricks to solve the questions. Learn how to represent data in different ways with the help of BIM Book Grade 2 Chapter 13 Represent and Interpret Data Answers.

Big Ideas Math Book Grade 2 Answer Key Chapter 13 Represent and Interpret Data

It is necessary for the students to go through the topics included in this chapter before starting their preparation. The list of lessons in Chapter 13 Represent and Interpret Data are Sort and Organize Data, Real and Interpret Picture Graphs, Make Picture Graphs, Real and Interpret Bar Graphs, Make Bar Graphs, Make Line Plots, and Measure Objects and Make Line Plots.

You can understand the concepts quickly and easily by referring to Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data. The quick links are attached at the end for the reference of students.

Vocabulary

Lesson: 1 Sort and Organize Data

Lesson: 2 Real and Interpret Picture Graphs

Lesson: 3 Make Picture Graphs

Lesson: 4 Real and Interpret Bar Graphs

Lesson: 5 Make Bar Graphs

Lesson: 6 Make Line Plots

Lesson: 7 Measure Objects and Make Line Plots

Chapter- 13: Represent and Interpret Data

Represent and Interpret Data Vocabulary

Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v1
Organize It
Use the review words to complete the graphic organizer.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v2

Answer:
The number of students who choose math = 6.
The number of students who choose science = 5

Explanation:
In the above-given figure,
the number of students who choose math = 6
the number of students who choose science = 5
the more number of students choose maths than science.

.Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13-1

Organize It
Use your vocabulary cards to identify the word.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v3

Answer:
1. The pencil lengths of each student = 4, 5, 6, and 7.

Explanation:
The number of pencil lengths of all students = 4, 5, 6, and 7.
the max length of the pencil = 5
In the above-given figure is a line plot.
given that the pencil lengths in inches = 4, 5, 6, and 7.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13-2

Answer:
2. The more number of students whose favorite hobby is singing = 7.

Explanation:
In the above-given figure,
the total number of students = 8
the number of students whose favorite hobby dancing = 6.
the number of students whose favorite hobby running = 5.
the number of students whose favorite hobby singing = 7.
The more number of students whose favorite hobby is singing = 7.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13-3

Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v4.1
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v4
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v5.1
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data v5

Lesson 13.1 Sort and Organize Data

Explore and Grow

Look at your color tiles. Complete the tally chart.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 1
Answer:
The tile colors are blue, green, red, and yellow.

Explanation:
In the above-given figure,
given the tile, colors are blue, green, red, and yellow.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-1

Write and answer a question about your tally chart.

Answer:
The more number of students they choose yellow = 5.

Explanation:
In the above-given figure,
the number of students who choose blue = 2.
the number of students who choose green = 3.
the number of students who choose red = 4.
the number of students who choose yellow = 5.
the more number of students who choose yellow = 5.

Show and Grow

Question 1.
Use the data to complete the tally chart.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 2
How many students chose math? ________
Which subject is the most favorite? _________

Answer:
The students who choose math = 7.
The most favorite subject = math.

Explanation:
The students who choose math = 7.
The students who choose science = 4.
The students who choose social studies = 2.
The students who choose language arts = 3.

Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-2

Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-3

Apply and Grow: Practice

Question 2.
Use the data to complete the tally chart.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 3
Which animal is the least favorite? ________
How many students did not choose fox? How do you know?

Answer:
The least favorite animal = owl
the students did not choose fox = 12.

Explanation:
In the above-given figure,
the least favorite animal = owl.
the students did not choose fox = 12.
the total number of students = 18
18 – 6 = 12
12 students did not choose fox.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-4

Did more students choose fox or owl?
_______
How many more? ______ more

Did fewer students choose reindeer or polar bear?
________
How many fewer? ______ fewer

Answer:
The more students choose fox = 7.
the more students = 7
the fewer students choose a polar bear.
fewer = 3

Explanation:
The students who choose fox = 7.
the students who choose more = 7
the students who choose fewer = polar bear
polar bear = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-4

Question 3.
Reasoning
Which sentences are correct?
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 4
You survey 30 students.
11 students chose whale.
4 more students chose seal than penguin.
20 students did not choose seal.

Answer:
4 more students choose seal than penguin is correct.

Explanation:
In the above-given figure,
the penguin = 14
whale = 6
seal = 10

Think and Grow: Modeling Real Life

Newton wants to survey25 students. How many more students does he need to survey?
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 5
Addition equation:

_______ students

Answer:
The more students does he need to survey = 6 students.

Explanation:
In the above-given figure,
the total number of students = 19.
newton wants 25 students.
19 + 6 = 25
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-5

Show and Grow

Question 4.
Descartes wants to survey 20 students. How many more students does he need to survey?
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 6
_______ students

Answer:
The more students do he need to survey = 2 students.

Explanation:
In the above-given figure,
the total number of students = 18.
Descartes wants 20 students.
18 + 2 = 20.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-6

How many more students need to choose sneakers so that the numbers of students who choose sneakers and sandals are equal?
______ students

Answer:
The students need to choose sneakers so that the number of students who choose sneakers and sandals is equal = 2 students.

Explanation:
In the above-given figure,
the sandals = 6
sneakers = 4
6 – 4 = 2
2 students.

Sort and Organize Data Homework & Practice 13.1

Question 1.
Use the data to complete the tally chart.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 7
Which activity is the most favorite? _______
Did more students choose catch or sidewalk chalk? _______
How many more?
______ more

Answer:
The activity the most favorite = tag.
The more students choose catch than sidewalk chalk.
0ne more.

Explanation:
The most favorite activity = tag.
Most students choose catch than the sidewalk.
sidewalk = 2
catch = 4
bubbles are more.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-7

Question 2.
Modeling Real Life
Newton wants to survey 15 friends. How many more friends does he need to survey?
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 8

Answer:
The more friends he needs to survey = 4.

Explanation:
Given that newton wants 15 friends to survey.
but in the figure, he has only 11 friends.
11 + 4 = 15
the more friends he wants to survey = 4
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-8

Question 3.
Writing
In Exercise 2, what question did Newton ask?
__________________
__________________

Answer:
Newton wants to survey 15 friends.

Explanation:
In the above-given question, he wants to survey more friends.
the more friends = 4
in the figure given that 11 friends.
11 + 4 = 15.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-9

Question 4.
Modeling Real Life
You want to survey 30 students. How many more students do you need to survey?
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.1 9
______ students
How many more students need to choose coloring so that the numbers of students who choose coloring and puppets are equal?
______ students

Answer:
The more students you need to survey = 10 students.

Explanation:
In the above-given figure,
The total number of students = 20
The more number of students you need to survey = 10.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-10
Review & Refresh

Question 5.
153 − 10 = ______

Answer:
153 – 10 = 143

Explanation:
153 – 10 = 143
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-12

Question 6.
978 − 10 = ______

Answer:
978 – 10 = 968

Explanation:
978 – 10 = 968
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-13

Question 7.
642 − 100 = ______

Answer:
642 – 100 = 542

Explanation:
642 – 100 = 542
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-14

Question 8.
1,000 −100 = ______

Answer:
1000 – 100 = 900

Explanation:
1000 – 100 = 900
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.1-15

Lesson 13.2 Real and Interpret Picture Graphs

Explore and Grow

How are the tally chart and the picture graph the same? How are they different?
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 1
__________________
___________________

Answer:
Yes, the tally chart and picture graph represents the same.

Explanation:
The favorite bird = 2.
cat = 4
dog = 7.
So in the tally chart and the picture graph represents the same.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-1

Show and Grow

Question 1.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 2
How many students chose butterfly? _______
Which insect is the most favorite? ______
Which insect is the least favorite? _______

Answer:
The students who choose butterfly = 7
The most favorite insect= 8
The least favorite insect = 2

Explanation:
In the above-given figure,
The insects bumblebee = 2
ladybug = 8
grasshopper = 6
butterfly = 7
The students who choose butterfly = 7
The most favorite insect= 8
The least favorite insect = 2
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-2

Apply and Grow: Practice

Question 2.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 3
How many students chose sporting event? _______
Which school trip is the least favorite? _______
How many more students chose science center than the park? _______

Answer:
The students who choose sporting event = 8
The least favorite school trip = park
The more students who choose science center than the park = 3

Explanation:
In the above-given figure,
the science center = 5
park = 2
sporting event = 8
museum = 3
The students who choose sporting event = 8
The least favorite school trip = park
The more students who choose science center than the park = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-3

Question 3.

Number Sense
Use the numbers to complete the sentences.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 4
Newton has ______ siblings.
Your friend has _______ siblings.
Your friend has ______ more siblings than your cousin.

Answer:
Newton has 2 siblings.
your friend has 4 siblings.
your friend has 3 more siblings than your cousin.

Explanation:
In the above-given figure,
your cousin has 1 sibling.
your friend has 4 siblings.
newton has 2 siblings.
Descartes has 3 siblings.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-4

Think and Grow: Modeling Real Life

Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 5
Do more students like crabs and sea turtles or octopuses and jellyfish?
Addition equations:
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 6
More students like ______ and _______.

Answer:
The more students like sea turtle = 9
More students like octopus and sea turtle.

Explanation:
In the above-given figure,
the students like crab = 6
the students who like octopus = 8
the students who like jellyfish = 3
the students who like sea turtle = 9
The more students like sea turtle = 9
More students like octopus and sea turtle.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-5

Show and Grow

Question 4.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 7
Do you see more cars and motorcycles or vans and trucks?
You see more _______ and ______.

Answer:
We see more cars.

Explanation:
In the above-given figure,
the cars = 9
vans = 8
truck  4
motorcycles = 2
we see more cars than other vehicles.

Real and Interpret Picture Graphs Homework & Practice 13.2

Question 1.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 8
Which fruit do exactly 7 students like most? _______
How many students like the fruit with the fewest votes? _______
How many more students chose banana than pear? _____

Answer:
The fruits exactly 7 students like most = banana.
the students who like the fewest = apple.
the more students choose banana than pears = 3

Explanation:
In the above-given figure,
The students who like apple = 4
the students who like pears = 5
The students who like grapes = 7
the students who like banana = 8
The fruits exactly 7 students like most = banana.
the students who like the fewest = apple.
the more students choose banana than pears = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-6

Question 2.
Writing
Use the picture graph.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 9
Write two true statements about the picture graph.
____________________
____________________

Answer:
The students who choose bus = 6
The students who choose walk= 4
The students who choose car = 2
The students who choose subway = 5

Explanation:
In the above- given figure,
the students who choose bus = 6
The students who choose walk= 4
The students who choose car = 2
The students who choose subway = 5
the more students choose the bus.
fewer students choose cars.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-7

Question 3.
Modeling Real Life
Use the picture graph.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 10
Do more students like to play video games and read or play outside and watch TV?
More students like to ______ and ______.

Answer:
The more students like to play video games.

Explanation:
In the above-given figure,
the students who like to play outside = 6
the students who like to play video games = 9
the students who like to watch tv= 7
the students who like to read = 3
the more students like to play videogames and watching tv
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.2-8
Review & Refresh

Question 4.
What is the best estimate of the length of a keyboard?
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.2 11
Answer: 18 inches

Lesson 13.3 Make Picture Graphs

Explore and Grow

Look at your color tiles. Complete the tally chart and the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 1

Answer:
The number of students who like blue = 2
The number of students who like green = 3
The number of students who like red = 4
The number of students who like yellow = 5
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-1

Show and Grow

Question 1.
Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 2
Which fruit is the most favorite? ______
Which fruit is the least favorite? ________
How many more students chose apple than banana? _________

Answer:
The favorite fruit = apple.
the least favorite = strawberry
The students choose apple than banana = 1

Explanation:
In the above-given figure,
The students who choose orange = 4
The students who choose strawberry = 2
The students who choose apple = 6
The students who choose banana = 5
The favorite fruit = apple.
the least favorite = strawberry
The students choose apple than banana = 1
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-2

Apply and Grow: Practice

Question 2.
Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 3
How many students chose the most favorite bird? ________
How many students chose flamingo or penguin? _______

Answer:
The students who choose penguin = 7
the students who choose flamingo or penguin = 6 or 7

Explanation:
In the above-given figure,
the students who choose penguin = 7
the students who choose flamingo = 5
the students who choose own = 3
the students who choose parrot = 2
The students who choose penguin = 7
the students who choose flamingo or penguin = 6 or 7

Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-3

Question 3.
Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 4
Newton gives 4 coins to Descartes. How many coins does Newton have now? How do you know?
____________________
____________________

Answer:
Newton has zero coins.

Explanation:
In the above-given figure,
the newton has 4 coins.
Descartes has 6 coins.
I have 6 coins.
Given that newton gives 4 coins to Descartes.
so newton has 0 coins.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-4

Think and Grow: Modeling Real Life

You ask 20 students to name their eye colors. 9 have brown eyes. 3 more students have brown eyes than blue eyes. The rest have green eyes. Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 5

Answer:
The students who have green eyes = 8

Explanation:
In the above-given question, given that
9 have brown eyes.
3 students have blue eyes.
the students who have green = 8
20 – 8 = 12
9 + 3 = 12.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-5

Show and Grow

Question 4.
You ask 15 students which community helper is their favorite. 6 choose police officer. 2 fewer students choose doctor than police officer. The rest choose firefighter. Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 6
How did you find how many students chose firefighter?

Answer:
The students who choose firefighter = 5

Explanation:
The students who choose police officer = 6
The students who choose doctor = 2
The students who choose fire fighter= 5
8 + 5 = 15
15 – 10 = 5
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-6

Make Picture Graphs Homework & Practice 13.3

Question 1.
Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 7
Who has the least medals? _______
How many more medals do you have than your friend? _________
How many medals do Newton and Descartes have in all? ________

Answer:
Your friend has the least medals.
the medals I have more than my friend = 2
Newton and Descartes have all the medals = 11

Explanation:
In the above-given figure,
I have 4 medals.
my friend has 2 medals.
Newton has 5 medals.
descartes have 6 medals.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-10

Question 2.
Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 8

2 more students chose tomato soup. How many students chose tomato now? How do you know?

Answer:
The students who choose tomato soup = 8

Explanation:
In the above-given figure,
The students who choose alphabet = 7
The students who choose vegetable = 3
the students who choose tomato = 6
given that 2 more students choose tomato soup.
6 + 2 = 8
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-11

Question 3.
Modeling Real Life
You ask 20 students about their favorite way to exercise. 4 like to walk. 6 like to swim. The rest like to bike. Complete the picture graph.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 9
How did you find how many students like to bike?

Answer:
The students who like to walk = 4.
The students who like to swim = 6.
Rest of the students who like bike = 10

Explanation:
In the above-given figure,
The students who like to walk = 4.
The students who like to swim = 6.
Rest of the students who like bike = 10
20 – 4 + 6
20 – 10
10
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-12

Review & Refresh

Question 4.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 10

Answer:
415 – 273 = 142

Explanation:
415 – 273 = 142
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-7

Question 5.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 11

Answer:
583 – 127 = 456

Explanation:
583 – 127 = 456
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-8

Question 6.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.3 12

Answer:
892 – 105 = 787

Explanation:
892 – 105 = 787
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.3-9

Lesson 13.4 Real and Interpret Bar Graphs

Explore and Grow

How are the tally chart and the bar graph the same? How are they different?
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 1
Based on the given values they are the same.
If they give different values they are different.

Answer:

The number of students who choose baseball = 6
The number of students who choose basketball = 5
The number of students who choose soccer = 7

Explanation:
In the above-given figure,
The number of students who choose baseball = 6
The number of students who choose basketball = 5
The number of students who choose soccer = 7
The students whose favorite spot = soccer.

Show and Grow

Question 1.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 2
How many students chose computer? _______
Which activity is the least favorite? ________

Answer:
The students who choose computer = 8.
The least favorite activity = reading.

Explanation:
In the above-given figure,
The students who choose painting = 4
the students who choose computer = 8
the students who choose reading = 3
the students who choose games = 6
The students who choose computer = 8.
The least favorite activity = reading
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-1

Apply and Grow: Practice

Question 2.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 3.1
How many students chose breakfast? _______
Which meal is the most favorite? ________
How many students chose the meal that is least favorite? ________
How many more students chose snack than lunch? _______

Answer:
The students who choose breakfast = 7.
the most favorite meal = Dinner.
The students chose the meal that is least favorite = 4
The students choose snack than lunch = 2

Explanation:
The students who choose breakfast = 7.
The students who choose lunch = 4.
The students who choose dinner = 9.
The students who choose snack = 6.
The students who choose breakfast = 7.
the most favorite meal = Dinner.
The students chose the meal that is least favorite = 4
The students choose snack than lunch = 2
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-2

Question 3.
DIG DEEPER!
Order the meals in Exercise 2 from the least favorite to the most favorite.
______, ______, ________, ________

Answer:
Lunch, snack, breakfast, and dinner.

Explanation:
In the above-given figure
given that order the meals from least favorite to the most favorite.
the least favorite = lunch.
the most favorite = dinner.
lunch, snack, breakfast, dinner.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-3

Think and Grow: Modeling Real Life

Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 3
A student chooses a food that has 1 more vote than eggs and toast combined. Which food does the student choose?
The student chooses ________.

Answer:
The students who choose 1 more than eggs = toast.

Explanation:
In the above-given figure,
The students who choose cereal = 9
The students who choose toast = 5
The students who choose pancakes = 8
The students who choose eggs = 3
The students who choose 1 more than eggs = toast.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-4

Show and Grow

Question 4.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 4
A student chooses an activity that has the same number of votes as crafts and hiking combined. Which activity does the student choose?
The student chooses ________

Answer:
The student who chooses an activity that has the same number of votes as crafts and hiking combined = Archery.

Explanation:
In the above-given figure,
the student who chooses archery = 6
the student who chooses crafts = 4
the student who chooses Hiking = 2
the student who chooses swimming = 7
The student who chooses an activity that has the same number of votes as crafts and hiking combined = Archery.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-5

Real and Interpret Bar Graphs Homework & Practice 13.4

Question 1.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 5
How many students chose police officer? _________
How many more students chose teacher than nurse? _________

Answer:
The students who choose police officer = 7.
The more students choose teacher than nurse = 1

Explanation:
In the above-given figure,
The students who choose teacher job = 5
The students who choose police officer job = 7
The students who choose sports player job = 8
The students who choose nurse job = 4
The students who choose police officer = 7.
The more students choose teacher than nurse = 1
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-6

Question 2.
Writing
How can you use a bar graph to find how many students were surveyed?
__________________
__________________

Answer:
The number of students surveyed = 24 students.

Explanation;
In the above-given graph
The students who choose teacher job = 5
The students who choose police officer job = 7
The students who choose sports player job = 8
The students who choose nurse job = 4
The total number of students were surveyed = 24 students.

Question 3.
Modeling Real Life
Use the bar graph.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 6
A student chooses a writing tool that has 3 fewer votes than crayon and marker combined. Which writing tool does the student choose?
The student chooses ______.

Answer:
The student chooses a writing tool that has 3 fewer votes than crayon and marker combined = pen.

Explanation:
In the above-given graph,
the student who chooses pencil = 4
the student who chooses pen = 6
the student who chooses crayon = 2
the student who chooses marker = 5
The student chooses a writing tool that has 3 fewer votes than crayon and marker combined = pen.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-7

Review & Refresh

Find the missing number.
Question 4.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 7
Answer:
576 + 153  = 729

Explanation:
729 – 576 = 153
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-8

Question 5.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 8
Answer:
431 + 389 = 820

Explanation:
820 – 431 = 389
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-9

Question 6.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 13.4 9

Answer:
128 + 521 = 649

Explanation:
649 – 128 = 521
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.4-10

Lesson 13.5 Make Bar Graphs

Explore and Grow

Look at your Instrument Cards. Complete the tally chart and the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 1
Answer:
The students who choose drum = 2
The students who choose trumpet = 3
The students who choose tuba = 4

Explanation:
In the above-given figure,
The students who choose drum = 2
The students who choose trumpet = 3
The students who choose tuba = 4
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-1

Show and Grow

Question 1.
Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 2
Which book type is the least favorite? _________
How many more students chose fiction than history? _______

Answer:
The least favorite book = history
The more students choose fiction than history = 3

Explanation:
In the above-given figure,
The students who choose history = 3
The students who choose fiction = 7
The students who choose science = 5
The students who choose poetry = 6
The least favorite book = history
The more students choose fiction than history = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-2

Apply and Grow: Practice

Question 2.
Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 3
How many raisins are there? _________
How many more almonds are there than dried fruit? _______

Answer:
The raisins are = 6
The more almonds are there than dried fruit = 3

Explanation:
In the above-given figure,
The number of peanuts = 9
the number of raisins = 6
The number of almonds = 7
the number of dried fruit = 4
The raisins are 6
the more almonds are there than dried fruits = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-3

Question 3.
Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 4
Which drink was chosen the most? _______
How many more people chose iced tea than water? _______

Answer:
The number of people who choose the drink most = Lemonade
The people who choose iced tea over water = 3

Explanation:
The number of people who choose water = 5
The number of people who chose fruit punch = 2
The number of people who choose lemonade = 9
The number of people who choose  iced tea 8
The number of people who choose the drink most = Lemonade
The people who choose iced tea over water = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-4

Think and Grow: Modeling Real Life

You classify 30 animals as fish, mammals, or reptiles.11 are fish. 7 are reptiles. The rest are mammals. Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 5

Answer:
The mammals = 12.

Explanation:
In the above-given figure,
Given that the fishes are 11.
reptiles = 7
mammals = 12
11 + 7 = 18
30 – 18 = 12
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-5

Show and Grow

Question 4.
You ask 29 students if they want to collect seashells, fossils, or stickers. 8 said fossils. 12 said stickers. The rest said seashells. Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 6

DIG DEEPER!
You ask 3 more students which object they want to collect. 1 said fossils. 2 said seashells. How many fewer students chose fossils than seashells now?
_______ students

Answer:
The fewer students choose fossils than seashells now = 2.

Explanation:
In the above-given figure,
The number of students who choose seashells = 9
The number of students who choose fossils = 8
The number of students who choose stickers = 12
given that 29 students.
11 + 9 = 20
29 – 20 = 9
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-6
Make Bar Graphs Homework & Practice 13.5

Question 1.
Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 7
How many more votes did Spot receive than Sparkle? ______

Answer:
The more votes did spot receive than sparkle = 7

Explanation:
In the above-given figure,
The number of votes for spot = 9
The number of votes for sparkle = 2
The number of votes for flip = 7
The number of votes for star = 6
The more votes did spot receive than sparkle = 7
the spot receives 7 more votes than sparkle.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-7

Question 2.
Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 8
How many total animals are in the pet store? _______

Answer:
The total animals in the pet store = 20.

Explanation:
In the above-given figure,
Given that fish = 7
hamster = 5
snake = 4
rabbit = 4
total animals in the pet = 20 animals.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-8

Question 3.
Modeling Real Life
You, your friend, and your cousin hand out a total of 25 flyers. You hand out 12. Your friend hands out 9. Your cousin hands out the rest. Complete the bar graph.
Big Ideas Math Answer Key Grade 2 Chapter 13 Represent and Interpret Data 13.5 9
DIG DEEPER!
You hand out 1 more flyer. Your friend hands out4 more. How many fewer flyers does your cousin hand out than your friend now?
_______ flyers

Answer:
The many fewer flyers do your cousin hand out than your friend now = 9

Explanation:
In the above-given figure,
The flyers handed out by me = 12.
The flyers handed out by my friend = 9.
The flyers handed out by my cousin = 4.
given that flyers handed out by me = 12
totally there are 25 flyers.
12 + 9 = 25
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-9
Review & Refresh

Question 4.
861 − 410 = ______

Answer:
861 – 410 = 451

Explanation:
861 – 410 = 451
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-10

Question 5.
624 − 320 = ______

Answer:
624 – 320 = 304

Explanation:
624 – 320 = 304
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.5-11

Lesson 13.6 Make Line Plots

Explore and Grow

How are the thumb lengths shown on the number line?
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 1
Answer:
The thumb lengths for thumb 1, thumb 4, thumb 7 = 4 cms.
thumb 2 , thumb 6 = 5 cms.
thumb 3 = 3 cm
thumb 5 = 6cm

Explanation:
In the above-given figure,
given the thumb lengths.
The thumb lengths for thumb 1, thumb 4, thumb 7 = 4 cms.
thumb 2 , thumb 6 = 5 cms.
thumb 3 = 3 cm
thumb 5 = 6cm

Show and Grow

Question 1.
Complete the line plot.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 2
How many long jumps are 43 inches long? _____
What is the most common long jump length? ______ inches

Answer:
One long-jump is 43 inches long = child 3.
the most common long jump length = child 1 and child 4.

Explanation:
In the above-given figure,
given that the long jump lengths.
child 1 and child 4 = 41
child 2 = 44
child 5 = 45
One long-jump is 43 inches long = child 3.
the most common long jump length = child 1 and child 4.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-1

Apply and Grow: Practice

Question 2.
Complete the line plot.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 3
What is the most common puppy length? ______ inches
How many fewer puppies are 11 inches long than 10 inches long? ______
How many puppies are 8 or 9 inches long? ______
How many puppies are shorter than 11 inches? How do you know?
_____________________
____________________

Answer:
The most common puppy length = puppy 2, 3, 5, 6, and 7.
The fewer puppies are 11 inches longer than 10 inches long = puppy 1.
The puppies which are 8 or 9 inches long = 3 puppies.
The puppies which are shorter than 11 inches = 6 puppies.

Explanation:
In the above-given figure,
given the puppy lengths.
the puppy lengths with puppy 1 = 11 inches.
puppy 2 = 9 inches.
puppy 3 = 10 inches.
puppy 4 = 12 inches.
puppy 5 = 9 inches.
puppy 6 = 10 inches.
puppy 7 = 10 inches.
puppy 8 = 8 inches.
The most common puppy length = puppy 2, 3, 5, 6, and 7.
The fewer puppies are 11 inches longer than 10 inches long = puppy 1.
The puppies which are 8 or 9 inches long = 3 puppies.
The puppies which are shorter than 11 inches = 6 puppies.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-2

Think and Grow: Modeling Real Life

9 people measure the length of a guitar. The line plot shows the measured lengths. How long do you think the guitar is? Explain.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 4.2
______ inches

Answer:
The length of the guitar = 35 inches.

Explanation:
In the above-given figure,
the number of inches in guitar lengths are:
the guitar length with 34 inches.
the guitar length with 35 inches.
the guitar length with 36 inches.

Show and Grow

Question 3.
8 people measure the length of a playground. The line plot shows the measured lengths.How long do you think the playground is? Explain.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 5
______ meters
DIG DEEPER!
Why are the measurements different?

Answer:
The length o the playground = 51 meters.

Explanation:
In the above-given figure,
the number of meters of playgrounds is:
given that 8 people measure the length of the playground.
6 people choose 50 meters.
1 people choose 49 meters.
1 people choose 48 meters.

Make Line Plots Homework & Practice 13.6

Question 1.
Complete the line plot.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 6
How many leaves are 13 centimeters long? _______
What is the most common leaf length? _______ centimeters
How many more leaves are 14 centimeters long than 12 centimeters long? ______
How many leaves are 14 or 15 centimeters? ________

Answer:
No leaves are 13 centimeters long.
the most common leaf lengths are leaf 2, leaf 3, and leaf 6.
the more leaves are 14 centimeters long than 12 centimeters long = leaf 1 and leaf 4.
The leaves are 14 or 15 centimeters = 5 leaves.

Explanation:
In the above-given figure,
given that, the leaf lengths.
the leaf 1 = 14 centimeters.
the leaf 2 = 15 centimeters
the leaf 3 = 15 centimeters
the leaf 4 = 14 centimeters
the leaf 5 = 12 centimeters
the leaf 6 = 15 centimeters
No leaves are 13 centimeters long.
the most common leaf lengths are leaf 2, leaf 3, and leaf 6.
the more leaves are 14 centimeters long than 12 centimeters long = leaf 1 and leaf 4.
The leaves are 14 or 15 centimeters = 5 leaves.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-3
Question 2.
Complete the line plot.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 7
How many sharks are longer than 14 feet? How do you know?
____________

Answer:
The sharks which are longer than 14 feet = 3 sharks.

Explanation:
In the above-given figure,
given that the shark lengths in feets.
shark 1 = 14
shark 2 = 15
shark 3 = 16
shark 4 = 14
shark 5 = 12
shark 6 = 16
The sharks which are longer than 14 feet = 3 sharks.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-4

Question 3.
Modeling Real Life
8 people measure the length of a bus. The line plot shows the measured lengths. How long do you think the school bus is? Explain.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 8
______ meters

Answer:
The length of the school bus = 105 meters.

Explanation:
In the above-given figure,
the lengths of the school bus is given.
given that 8 people measure the length of a bus.
the lone plot shows the measured lengths.
2 people measure the 14 meters.
1 people measure the 12 inches.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-5

Review & Refresh

Question 4.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 9
Answer:
127.

Explanation:
36 + 4 + 22 + 65
40 + 22 + 65
62 + 65
127
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-5

Question 5.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 10
Answer:
208.

Explanation:
95 + 68 + 45
163 + 45
208
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-6

Question 6.
Big Ideas Math Answers 2nd Grade Chapter 13 Represent and Interpret Data 13.6 11
Answer:
226.

Explanation:
76 + 50 + 18 + 82
126 + 18 + 82
144 + 82
226
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-7

Lesson 13.7 Measure Objects and Make Line Plots

Explore and Grow

Use a ruler to measure the caterpillars on Lengths of Caterpillars. Use the lengths to complete the chart and the line plot.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 1

Answer:
The long length of the caterpillar = 9.

Explanation:
We have to assume the lengths of the cater pillars.
cater pillar 1 = 6 cms.
cater pillar 2 = 7 cms.
cater pillar 3 = 9 cms.
cater pillar 4 = 9 cms.
cater pillar 5 = 7 cms.
cater pillar 6 = 10 cms.
cater pillar 7 = 8 cms.
cater pillar 8 = 7 cms.
The long length of the caterpillar = 9
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-8

Show and Grow

Question 1.
Measure the lengths of 4 pencils. Complete the line plot.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 2
What is the length of the longest pencil? _______ inches
How much longer is the longest pencil than the shortest pencil? _______ inches

Answer:
The length of the longest  pencil = 10 inches.
The longer is the longest pencil than the shortest penccil = 5 inches.

Explanation:
In the above-given figure,
given that the length of the 4 pencils.
the length of the pencil 1 = 5 inches.
pencil 2 = 6 inches.
pencil 3 = 8 inches.
pencil 4 = 10 inches.
The length of the longest  pencil = 10 inches.
The longer is the longest pencil than the shortest penccil = 5 inches.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-9

Apply and Grow: Practice

Question 2.
Measure the lengths of 8 shoes. Complete the line plot.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 3
What is the length of the longest shoe? ______ inches
What is the length of the shortest shoe? _______ inches
How much shorter is the shortest shoe than the longest shoe? _______ inches
You measure 5 more shoes and they are each6 inches long. How does the line plot change?
_________________________
_________________________

Answer:
The length of the longest shoe = 10 inches.
The length of the shortest shoe = 6 inches.
The shorter is the shortest shoe than the longest shoe = 5 inches.
The line plot changes.

Explanation:
In the above-given figure,
we have to assume  the examples.
length of the shoe 1 = 6 inches.
length of the shoe 2 = 7 inches.
length of the shoe 3 = 6 inches.
length of the shoe 4 = 7 inches.
length of the shoe 5 = 8 inches.
length of the shoe 6 = 9 inches.
length of the shoe 7 = 10 inches.
length of the shoe 8 = 8 inches.
The length of the longest shoe = 10 inches.
The length of the shortest shoe = 6 inches.
The shorter is the shortest shoe than the longest shoe = 5 inches.
The line plot changes.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-10

Think and Grow: Modeling Real Life

A fire station is building a new garage for emergency vehicles. Complete the sentence. Explain.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 4
The garage should be more than _______ meters long.
____________________

Answer:
The garage should be more than 16 meters long.

Explanation:
In the above-given figure,
Given that the fire station is building a new garage for emergency vehicles.
the garage should be more than 6 meters long and the shorter than 16 meters long.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-11

Show and Grow

Question 3.
You want to put some school supplies in a pencil box. Complete the sentence. Explain.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 5
The pencil box should be more than _______ centimeters long.
______________________
Answer:
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-11

Measure Objects and Make Line Plots Homework & Practice 13.7

Question 1.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 6
Measure the lengths of 5 socks. Complete the line plot.
What is the length of the longest sock? ______ inches
What is the length of the shortest sock? _______ inches
How much longer is the longest sock than the shortest sock? _______ inches

Answer:
The length of the longest sock = 10 inches.
The length of the shortest sock = 7 inches.
The longer is the longest sock than the shortest sock = 3 inches.

Explanation:
In the above-given figure,
we have to assume the:
sock 1 = 7
sock 2 = 8
sock 3 = 8
sock 4 = 9
sock 5 = 10
The length of the longest sock = 10 inches.
The length of the shortest sock = 7 inches.
The longer is the longest sock than the shortest sock = 3 inches.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-12

Question 2.
Measure the lengths of 4 hands. Complete the line plot.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 7
How many hands are longer than 5 inches? How do you know?
__________________________
___________________________

Answer:
The hands that are longer than 5 inches = 3

Explanation:
We have to assume the hands in inches.
given that hand 1 = 7 inches.
hand 2 = 6 inches.
hand 3 = 5 inches.
hand 4 = 6 inches.
The hands that are longer than 5 inches = 3
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.6-13

Question 3.
Modeling Real Life
Newton wants to put his dog bones in a box. Complete the sentence. Explain.
Big Ideas Math Answers Grade 2 Chapter 13 Represent and Interpret Data 13.7 8
The box should be more than ______ inches long.
____________________

Answer:
The bo should be more than 11 inches.

Explanation:
In the above-given figure,
Given that newton wants to put his dog bones in a box.
the box should be more than 11 inches.
the box should be less than 4 inches.
Review & Refresh

Question 4.
10 more than 347 is _______.

Answer:
357

Explanation:
Given that,
10 more than 347 = 357.
347 + 10 = 357

Question 5.
100 less than 926 is _________.

Answer:
826

Explanation:
Given that 100 less than 926 = 826
926 – 100 = 826

Represent and Interpret Data Performance Task

You measure the lengths of 8 writing tools. The tools and their lengths are shown.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 1
Question 1.
Organize the writing tool lengths on the line plot.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 2

Answer:
The maximum length of the pencil = 12cms.

Explanation:
In the above-given figure,
given that
pencil 1 = 9 cm
pencil 2 = 9 cm
pencil 3 = 6 cm
pencil 4 = 9 cm
pencil 5 = 12 cm
pencil 6 = 12 cm
pencil 7 = 12 cm
pencil  8 = 12 cm
the maximum length of the pencil = 12 cms
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-1

Question 2.
Use the line plot to complete the equation. Why is the sum 8?
_____ + ______ + ______ = 8

Answer:
2 + 2 + 4 = 8

Question 3.
Measure your pencil. Add the length of your pencil to the line plot in Exercise 1.

Answer:
My pencil length is = 11 cms.

Explanation:
In the above-given figure,
given that
pencil 1 = 9 cm
pencil 2 = 9 cm
pencil 3 = 6 cm
pencil 4 = 9 cm
pencil 5 = 12 cm
pencil 6 = 12 cm
pencil 7 = 12 cm
pencil  8 = 12 cm
the maximum length of the pencil = 12 cms
Assuming my pencil length = 11 cms
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-2

Question 4.
Compare the lengths of your pencil and one of the writing tools above.

Answer:
The length of my pencil = 11 cms.

Explanation:
In the above-given figure,
given that
pencil 1 = 9 cm
pencil 2 = 9 cm
pencil 3 = 6 cm
pencil 4 = 9 cm
pencil 5 = 12 cm
pencil 6 = 12 cm
pencil 7 = 12 cm
pencil  8 = 12 cm
the maximum length of the pencil = 12 cms
Assuming my pencil length = 11 cms
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-2

Represent and Interpret Data Activity

Spin and Graph
To Play: Spin 10 times. Complete the tally chart. Then complete the bar graph. Answer the Spin and Graph Questions about your graph.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data 3

Represent and Interpret Data Chapter Practice

13.1 Sort and Organize Data

Question 1.
Use the data to complete the tally chart.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 1
How many students chose magic show? _______
Which event is the least favorite? __________

Answer:
The students who choose the magic show = 5 students.
The last favorite = dancing.

Explanation:
In the above-given figure,
The students who choose magic show = 5
The students who choose face painting = 4
The students who choose games = 2
The students who choose dancing = 1
The students who choose the magic show = 5 students.
The last favorite = dancing.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-3

Question 2.
Modeling Real Life
You want to survey30 students. How many more students do you need to ask?
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 2
______ students
How many more students need to have red hair so that the numbers of students with red hair and blonde hair are equal?
______ students

Answer:
The more students need to ask = 3 students.
The more students need to have red hair so that the numbers of students with red hair and blonde hair are equal = 7 students.

Explanation:
In the above-given figure,
the hair color which is in red = 2
the hair color which is in black = 5
the hair color which is brown = 10
the hair color which is in blonde = 9
The more students need to ask = 3 students.
The more students need to have red hair so that the numbers of students with red hair and blonde hair are equal = 7 students.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-4

13.2 Read and Interpret Picture Graphs

Question 3.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 3
Which type of book do exactly 7 students like best? _______
How many more students chose magazine than fiction? ________

Answer:
The type of book does exactly 7 students like best = comics.
The more students choose magazine than fiction = 4.

Explanation:
In the above-given figure,
The students who choose comics = 7
The students who choose fiction = 4
The students who choose nonfiction = 5
The students who choose magazine = 8
The type of book does exactly 7 students like best = comics.
The more students choose magazine than fiction = 4.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-5

13.3 Make Picture Graphs

Question 4.
Complete the picture graph.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 4
How many fewer students chose tiger than cheetah? ______
How many students chose the least favorite cat? _______
How many students chose panther or lion? __________

Answer:
The fewer students choose tiger than cheetah = 4
The students who choose least favorite cat = tiger
the students who choose panther = 5
the students who choose lion = 4

Explanation:
In the above-given figure,
Given that,
The fewer students choose tiger than cheetah = 4
The students who choose least favorite cat = tiger
the students who choose panther = 5
the students who choose lion = 4
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-6

13.4 Read and Interpret Bar Graphs

Question 5.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 5
How much rain fell on Tuesday? ______
Which day did it rain the least? _______
How much more rain fell on Monday than Wednesday? _________

Answer:
The rain fell on Tuesday = 3 inches.
The day it rains the least = thursday.
the more rain fell on Monday than Wednesday = 1 inch.

Explanation:
In the above-given graph,
the rain fell on Monday = 5 inches.
the rain fell on Tuesday = 3 inches.
the rain fell on Wednesday = 4 inches.
the rain fell on Thursday = 2 inches.
The rain fell on Tuesday = 3 inches.
The day it rains the least = Thursday.
the more rain fell on Monday than Wednesday = 1 inch.

Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-7
13.5 Make Bar Graphs

Question 6.
Complete the bar graph.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 6
How many hurricanes were there in 2012 and 2013? _______

Answer:
The hurricanes were there in 2012 and 2013 = 12.

Explanation:
In the above-given figure,
given that the hurricanes in the north Atlantic.
in 2012 = 10
2013 = 2
2014 = 5
2015 = 4
The hurricanes were there in 2012 and 2013 = 12.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-8

13.6 Read and Interpret Line Plots

Question 7.
Complete the line plot.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 7
How many feathers are longer than 13 centimeters? How do you know?
__________________
__________________

Answer:
The feathers which are longer than 13 centimeters = 2 feathers.

Explanation:
In the above-given figure,
The feathers with the lengths in centimeters.
feather 1 = 14 cms.
feather 2 = 13 cms.
feather 3 = 15 cms.
feather 4 = 12 cms.
The feathers which are longer than 13 centimeters = 2 feathers.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-9

13.7 Measure Objects and Make Line Plots

Question 8.
Use an inch ruler to measure the lengths of 5 toys to the nearest inch. Then complete the line plot.
Big Ideas Math Solutions Grade 2 Chapter 13 Represent and Interpret Data chp 8
What is the length of the longest toy? _______ inches

Answer:
The length of the longest toy = 9inches.

Explanation:
In the above-given figure,
we have to assume that
the length of the toy 1 = 6 inches.
the length of the toy 2 = 7 inches.
the length of the toy 3 = 6 inches
the length of the toy 4 = 8 inches.
the length of the toy 5 = 9 inches.
The length of the longest toy = 9inches.
Big-Ideas-Math-Solutions-Grade-2-Chapter-13-Represent and Interpret Data-13.7-10

Conclusion:

I hope that the solutions explained here are helpful for the students while preparing for the exams. So, download Big Ideas Math 2nd Grade 13th Chapter Represent and Interpret Data Answer Key pdf for free of cost and begin preparation. Students can clear their doubts by writing a comment in the below comment section. Also, bookmark our site to get the latest edition solutions for Big Ideas Math Book Grade 2 Chapters.

Big Ideas Math Answers Grade 8 Chapter 6 Data Analysis and Displays

Big Ideas Math Answers Grade 8 Chapter 6

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Big Ideas Math Book 8th Grade Answer Key Chapter 6 Data Analysis and Displays

Test your preparation standards taking the help of these handy resources for Big Ideas Math Book 8th Grade Ch 6 Data Analysis and Displays Solutions and improve on the areas of need. Achieve Learning Targets and succeed in your math journey easily. Download the 8th Grade Big Ideas Math Book Ch 6 Data Analysis and Displays Answers in PDF Format via quick links and prepare anywhere and anytime.

Performance Task

Lesson: 1 Scatter Plots

Lesson: 2 Lines of Fit

Lesson: 3 Two-Way Tables

Lesson: 4 Choosing a Data Display

Chapter: 6 – Data Analysis and Displays 

Data Analysis and Displays STEAM Video/Performance Task

STEAM Video

Fuel Economy
The fuel economy of a vehicle is a measure of the effciency of the vehicle’s engine. What are the benefits of using a car with high fuel economy?
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 1
Watch the STEAM Video “Fuel Economy.” Then answer the following questions.
1. Tory says that the footprint of a vehicle is the area of the rectangle formed by the wheel base and the track width. What is the footprint of a car with a wheel base of 106 inches and a track width of 61 inches?
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 2
2. The graph shows the relationship between the fuel economy and the footprint for four vehicles.
a. What happens to the fuel economy as the footprint increases?
b. Plot the point (50, 40) on the graph. What does this point represent? Does the point fit in with the other points? Explain.

Answer:
1.The footprint of a car = 6,466 sq inches.

Explanation:
In the above-given question,
Tory says that the footprint of a vehicle is the area of the rectangle formed by the wheelbase and the track width.
area of rectangle = length  x width
Given that the footprint of a car = 106 inches.
width with 61 inches.
area = 106 x 61
footprint = 6,466 sq inches.

Answer:
2. a.The fuel economy increases when the footprint increases.

Explanation:
In the above-shown video,
tory says that whenever the footprint increases the fuel economy also increases.
whenever the footprint decreases the fuel economy decreases.

Answer:
2.b.The point (50, 40) represents the outlier.

Explanation:
In the above-given graph,
the point (50, 40) lies in the graph.
it represents the outlier of the graph.
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6-1

Performance Task

Cost vs. Fuel Economy
After completing this chapter, you will be able to use the STEAM concepts you learned to answer the questions in the Video Performance Task. You will be given fuel economies and purchase prices of hybrid and non hybrid car models.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 3
You will be asked to create graphs to compare car models. Why might you want to know the relationship between the fuel economy and the purchase price of a vehicle?

Answer:
The relationship between the fuel economy and the purchase price of a vehicle is proportional.

Explanation:
In the above-given figure,
Given that the city fuel Economy and the purchase price of the cars.
for car A (21.8, 24)
for car B(22.4, 22)
for car C(40.1, 18)
if the fuel economy increases the purchase price also increases.
whenever the economy decreases the purchase price also decreases.

Data Analysis and Displays Getting Ready for Chapter 6

Chapter Exploration
1. Work with a partner. The table shows the number of absences and the final grade for each student in a sample.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 4
a.Write the ordered pairs from the table. Then plot them in a coordinate plane.
b. Describe the relationship between absences and final grade.
c. MODELING A student has been absent6 days. Use the data to predict the student’s final grade. Explain how you found your answer.

Answer:
a. (0, 95), (3, 88), (2, 90), (5, 83), (7, 79), (9, 70), (4, 85), (1, 94), (10, 65), (8, 75).
b. the relationships between the absences and the final grade is decreasing when the absences increases.
c. The student’s final grade is 80.

Explanation:
a. From the above-given figure,
The ordered pairs are:
(0, 95), (3, 88), (2, 90), (5, 83), (7, 79), (9, 70), (4, 85), (1, 94), (10, 65), (8, 75).
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6-2

B. whenever the final grade is decreasing the absences also decrease.
whenever the final grade increases the absence also increases.
c. Given that the student has been absent for 6 days.
The student’s final grade is 80.

2. Work with a partner. Match the data sets with the most appropriate scatter plot. Explain your reasoning.
a. month of birth and birth weight for infants at a day care
b. quiz score and test score of each student in a class
c. age and value of laptop computers
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 5

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
scatter plot
two-way table
line of fit
joint frequency

Answer:
Scatter plot = A scatter plot uses dots to represent values for two different numeric variables. The position of each dot on the horizontal and vertical axis indicates values for an individual data point.
Two-way table = A two-way table is a way to display frequencies or relative frequencies for two categorical variables.
Line of fit = Line of fit refers to a line through a scatter plot of data points that best expresses the relationship between those points.
Joint frequency = Joint frequency is joining one variable from the row and one variable from the column.

Explanation:
Scatter plot = A scatter plot uses dots to represent values for two different numeric variables. The position of each dot on the horizontal and vertical axis indicates values for an individual data point.
Two-way table = A two-way table is a way to display frequencies or relative frequencies for two categorical variables.
Line of fit = Line of fit refers to a line through a scatter plot of data points that best expresses the relationship between those points.
Joint frequency = Joint frequency is joining one variable from the row and one variable from the column.

Lesson 6.1 Scatter Plots

EXPLORATION 1

Work with a partner. The weights and circumferences of several sports balls are shown.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 1
a. Represent the data in the coordinate plane. Explain your method.
b. Is there a relationship between the size and the weight of a sports ball? Explain your reasoning.
c. Is it reasonable to use the graph to predict the weights of the sports balls below? Explain your reasoning.
Kickball : circumference = 26 in.
Bowling ball : circumference = 27 in.
Answer:
a.(21, 30), (5, 9), (1.6, 5.3), (16, 28), (2, 8), (1.4, 7), (7, 12), (10, 26).

Explanation:
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-1

Answer:
b. The weight is measured in inches and size is measured in ounces.

Explanation:
In the above-given figure,
the size and the weight of the balls are given.
size and weight of basketball = (21, 30).
size and weight of baseball = (5, 9).
size and weight of golfball = (1.6, 5.3).
size and weight of soccerball = (16, 28).
size and weight of tennis = (2, 8).
size and weight of racquetball = (1.4, 7).
size and weight of softball = (7, 12).
size and weight of volleyball = (10, 26)

Answer:
c. No, it is not reasonable to use the graph.

Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 2

Try It

Question 1.
Make a scatter plot of the data. Identify any outliers, gaps, or clusters.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 3

Answer:
outliers = (120, 70)
gaps =(10, 62) to (45, 85)
clusters =(80, 95), (90, 97), (80, 91)

Explanation:
outliers =(120, 70)
gaps = (10, 62) to (45, 85)
clusters = (80, 95), (90, 97), (80, 91)

Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-2

Question 2.
Describe the relationship between the data in Example 1.

Answer:
Linear relationship.

Explanation:
In the above-given graph,
the relationship used is a linear relationship.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
SCATTER PLOT
Make a scatter plot of the data. Identify any outliers, gaps, or clusters. Then describe the relationship between the data.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 4
Answer:
outliers = (3,24)
clusters = 22 to 36
gaps = (4, 27), (8, 36)

Explanation:
outliers = (3,24)
clusters = 22 to 36
gaps = (4, 27), (8, 36)

Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-3

Question 4.
WHICH ONE DOESN’T BELONG?
Using the scatter plot, which point does not belong with the other three? Explain your reasoning.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 5

Answer:
The point (3.5, 3) does not belong with the other three.

Explanation:
In the above-given figure
The points (1,8),  (3, 6.5), and (8, 2) lies in the coordinate plane.
the point (3.5, 3) does not belong with the other three.
the point (3.5, 3) is an outlier.
Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
The table shows the high school and college grade point averages (GPAs) of 10 students. What college GPA do you expect for a high school student with a GPA of 2.7?
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 6

Answer:
The college GPA I expect for a high school student with a GPA of 2.7 is 2.45.

Explanation:
In the above-given points,
given that the college GPA for high school students.
college GPA for 2.4 = high school students of 2.6
so I am expecting the 2.45 for 2.7.

Question 6.
The scatter plot shows the ages of 12 people and the numbers of pets each person owns. Identify any outliers, gaps, or clusters. Then describe the relationship between the data.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 7

Answer:
outliers = (40, 6)
clusters = (20, 2) to (70, 1)
gaps = (0, 30), (1, 35), (2, 50) and so on.

Explanation:
Given that,
the person’s age (years) in the x-axis.
a number of pets owned in the y-axis.
outliers = (40, 6)
clusters = (20, 2) to (70, 1)
gaps = (0, 30), (1, 35), (2, 50) and so on.

Scatter Plots Homework & Practice 6.1

Review & Refresh

Solve the system. Check your solution.
Question 1.
y = – 5x + 1
y = – 5x – 2

Answer:
There is no solution for the given equation.

Explanation:
Given that y = – 5x + 1
y = – 5x – 2
so there is no solution for the given equation.

Question 2.
2x + 2y = 9
x = 4.5 – y

Answer:
9 = 9

Explanation:
Given that,
2x + 2y = 9
x = 4.5 – y
2(4.5 – y) + 2y = 9
9 – 2y + 2y = 9
-2y and + 2y get cancelled on both sides.
9 = 9

Question 3.
y = – x
6x + y = 4

Answer:
x = (4/5 , -4/5)

Explanation:
Given that y = -x
6x + y = 4
6x + (-x) = 4
6x – x = 4
5x = 4
x = (4/5)

Question 4.
When graphing a proportional relationship represented by y = mx, which point is not on the graph?
A. (0, 0)
B. (0, m)
C. (1, m)
D. (2, 2m)

Answer:
Point A is not on the graph.

Explanation:
In the above question,
given that the points are:
(0, 0)
(0, m)
(1, m)
(2, 2m)
the point (0, 0) is not in the graph.

Concepts, Skills, &Problem Solving

USING A SCATTER PLOT The table shows the average prices (in dollars) of jeans sold at different stores and the numbers of pairs of jeans sold at each store in one month. (See Exploration 1, p. 237.)

Question 5.
Represent the data in a coordinate plane.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 8

Answer:
The points are (22, 152), (40, 94), (28, 134), (35, 110), and (46, 81)

Explanation:
In the above-given figure,
The points are (22, 152), (40, 94), (28, 134), (35, 110), and (46, 81)
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-4

Question 6.
Is there a relationship between the average price and the number sold? Explain your reasoning.

Answer:
The linear relationship.

Explanation:
In the above-given figure,
the relationship given is linear relationship.

MAKING A SCATTER PLOT Make a scatter plot of the data. Identify any outliers, gaps, or clusters.
Question 7.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 9

Answer:
Outliers = (102, 63)
gaps = x from 40 to 44
clusters = 82 to 89

Explanation:
outliers = (102, 63)
gaps = x from 40 to 44
clusters = 82 to 89
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-5

Question 8.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 10

Answer:
Outliers = (0, 5.5)
gaps = x from 4.5 to 5.5
clusters = 1.5 to 2.5

Explanation:
outliers = (0, 5.5)
gaps = x from 4.5 to 5.5
clusters = 1.5 to 2.5
Big-Ideas-Math-Solutions-Grade-8-Chapter-6-Data Analysis and Displays-6.1-6

IDENTIFYING RELATIONSHIPS Describe the relationship between the data. Identify any outliers, gaps, or clusters.
Question 9.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 11

Answer:
Outliers = (15, 10)
gaps = from x = 15 to x = 25
clusters = 0
Negative linear relationship.

Explanation:
Outliers = (15, 10)
gaps = from x = 15 to x = 25
clusters = 0
There are no clusters.

Question 10.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 12

Answer:
There are no clusters.
gaps = from x = 4 to x = 36
outliers.

Explanation:
In the above-given figure,
there are no clusters.
gaps = from x = 4 to x = 36
no outliers.

Question 11.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 13

Answer:
There is no relationship.
there are no clusters.
no gaps.
no outliers.

Explanation:
In the above-given graph,
there are no clusters.
no gaps.
no clusters.
there is no relationship.

Question 12.
CRITICAL THINKING
The table shows the average price per pound for honey at a store from 2014 to 2017. Describe the relationship between the data.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 14

Answer:
The relationship is a positive linear relationship.

Explanation:
In the above-figure,
given points are:
(2014, $4.65), (2015, $5.90), (2016, $6.50), and (2017, $7.70)
so the above given is a positive linear relationship.

Question 13.
MODELING REAL LIFE
The scatter plot shows the amount of rainfall and the amount of corn produced by a farm over the last 10 years. Describe the relationship between the amount of rainfall and the amount of corn produced.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 15

Answer:
The relationship is a positive linear relationship.

Explanation:
In  the above-given figure,
outliers = (49, 80)
clusters = from x = 190 to 220.

Question 14.
OPEN-ENDED
Describe a set of real-life data that has a negative linear relationship.
Answer:

Question 15.
MODELING REAL LIFE
The scatter plot shows the total earnings (wages and tips) of a food server during one day.
Big Ideas Math Answer Key Grade 8 Chapter 6 Data Analysis and Displays 6.1 16
a. About how many hours must the server work to earn $70?
b. About how much does the server earn for 5 hours of work?
c. Describe the relationship shown by the data.

Answer:
a. 3.5 h
b. 85 $
c. positive linear relationship.

Explanation:
In the above-given graph,
given that,
a. the hours must server work to earn $70 = 3.5 h
b. The server earns for 5 hours of work = $ 85.
c. the relationship is shown by the data = positive linear relationship.

Question 16.
PROBLEM SOLVING
Th