Big Ideas Math Book Geometry Answer Key Chapter 4 Transformations<\/h2>\n
Master the concepts of Big Ideas Math Geometry Chapter 4 Transformations by solving them on a regular basis. You just need to tap on the relevant links and prepare accordingly. Clarify all your doubts by availing the quick resources on Topicwise BIM Geometry Ch 4 Transformations. You can make use of them without paying a single penny and get to know the concepts in a better way.<\/p>\n
Transformations<\/strong><\/p>\n Lesson: 1 Translations<\/strong><\/p>\n Lesson: 2 Reflections<\/strong><\/p>\n Lesson: 3\u00a0Rotations<\/strong><\/p>\n Quiz<\/strong><\/p>\n Lesson: 4 Congruence and Transformations<\/strong><\/p>\n Lesson: 5 Dilations<\/strong><\/p>\n Lesson: 6 Similarity and Transformations<\/strong><\/p>\n Review<\/strong><\/p>\n Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure.<\/p>\n Question 1. Question 2. Question 3. Question 4. Tell whether the two figures are similar. Explain your reasoning.<\/p>\n Question 5. Question 6. Question 7. Monitoring Progress<\/strong><\/p>\n Use dynamic geometry software to draw the polygon with the given vertices. Use the software to find the side lengths and angle measures of the polygon. Round your answers to time nearest hundredth.<\/p>\n Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. Exploration 1<\/strong><\/p>\n Translating a Triangle in a Coordinate Plane<\/p>\n b. Copy the triangle and translate (or slide) it to form a new figure, called an image, \u2206A’B’C’ (read as triangle A prime, B prime. C prime\u201d). c. What is the relationship between the coordinates of the vertices of \u2206ABC and d. What do you observe about the side lengths and angle measures of the two triangles? Exploration 2<\/strong><\/p>\n Translating a Triangle in a Coordinate Plane<\/p>\n b. Use the rule you wrote in part (a) to translate \u2206ABC 4 units left and 3 units down. What are the coordinates of the vertices of the image. \u2206A’B’C’? c. Draw \u2206A’B’C.’ Are its side lengths the same as those of \u2206ABC? Justify your answer. Exploration 3<\/strong><\/p>\n Comparing Angles of Translations<\/p>\n Work with a partner.<\/p>\n a. In Exploration 2, is \u2206ABC a righL triangle? Justify your answer. b. In Exploration 2, is \u2206A’B’C’ a right triangle? Justify your answer. c. Do you think translations always preserve angle measures? Explain your reasoning. Communicate Your Answer<\/p>\n Question 4. Question 5. Monitoring Progress<\/strong><\/p>\n Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. Question 7. Vocabulary and Core Concept Check<\/p>\n Question 1. Question 2. Monitoring Progress and Modeling with Mathematics<\/p>\n In Exercises 3 and 4, name the vector and write its component form.<\/p>\n Question 3. Question 4. In Exercises 5 – 8, the vertices of \u2206DEF are D(2, 5), E(6, 3), and F(4, 0). Translate \u2206DEF using the given vector. Graph \u2206DEF and its image.<\/p>\n Question 5. Question 6. Question 7. Question 8. In Exercises 9 and 10, find the component form of the vector that translates P(- 3, 6) to P’.<\/p>\n Question 9. Question 10. In Exercises 11 and 12, write a rule for the translation of \u2206LMN to \u2206L’M’W’.<\/p>\n Question 11. Question 12. In Exercises 13 – 16, use the translation. Question 13. Question 14. Question 15. Question 16. In Exercises 17 – 20, graph \u2206PQR with vertices P (-2, 3) Q(1, 2), and R(3, – 1) and its image after the translation. Question 18. Question 19. Question 20. P (-2, 3) Q(1, 2), and R(3, – 1) In Exercises 21 and 22. graph \u2206XYZ with vertices X(2, 4), Y(6, 0). and Z(7, 2) and its image after the composition.<\/p>\n Question 21. Question 22. In Exercises 23 and 24, describe the composition of translations.<\/p>\n Question 23. Question 24. Question 25. Question 26. Question 27. Question 28. b. Translate a point using Translation B, followed by Translation A. Write an algebraic rule for the final image of the point after this composition. c. Compare the rules you wrote for parts (a) and (b) Does it matter which translation you do first? Explain your reasoning. MATHEMATICAL CONNECTIONS<\/strong> Question 30. Question 31. Question 32. Question 33. Question 34. b. Compare the areas. Make a conjecture about the areas of a preimage and its image after a translation. Question 35. Question 36. b. Corresponding Angles Converse (Theorem 3.5) Question 37. Question 38. Question 39. Question 40. Question 41. Question 42. Maintaining Mathematical Proficiency<\/p>\n Tell whether the figure can be folded in half so that one side matches the other.<\/p>\n Question 43. Question 44. Question 45. Question 46. Simplify the expression.<\/p>\n Question 47. Question 48. Question 49. Question 50. Exploration<\/strong><\/p>\n Reflecting a Triangle Using a Reflective Device<\/p>\n Work with a partner: a. Use the straightedge to draw a line that does not pass through the triangle. Label it m. b. Place a reflective device on line in. c. Use the reflective device to plot the images of the vertices of \u2206ABc. Label the images of vertices A, B. and C as A’, B’, and C’, respectively. d. Use a straightedge to draw \u2206A’B’C by connecting the vertices. Exploration 2<\/p>\n Reflecting a Triangle in a Coordinate Plane<\/p>\n Work with a partner: Use dynamic geometry software to draw any triangle and label it \u2206ABC. b. What is the relationship between the coordinates of the vertices of \u2206ABC and c. What do you observe about the side lengths and angle measures of the two triangles? d. Reflect \u2206ABC in the x-axis to form \u2206A’B’C’. Then repeal parts (b) and (c). Communicate Your Answer<\/p>\n Question 3. Monitoring progress<\/strong><\/p>\n Graph \u2206ABC from Example 1 and its image after a reflection in the given line. Question 2. Question 3. Question 4. The vertices of \u2206JKL are J(1, 3), K(4, 4), and L(3, 1).<\/p>\n Question 5. Question 6. Question 7. Question 8. Question 9. Question 10. Question 11. Determine the number of lines of symmetry for the figure.<\/p>\n Question 12. Question 13. Question 14. Question 15. Question 16. Vocabulary and Core Concept Check<\/p>\n Question 1. Question 2. Monitoring Progress and Modeling with Mathematics<\/p>\n In Exercises 3-6, determine whether the coordinate plane shows a reflection in the x-axis, y-axis, or neither.<\/p>\n Question 3. Question 4. Question 5. Question 6. In Exercises 7 – 12, graph \u2206JKL and its image after a reflection in the given line. Question 8. Question 9. Question 10. Question 11. Question 12. In Exercises 13-16, graph the polygon and its image after a reflection in the given line.<\/p>\n Question 13. Question 14. Question 15. Question 16. In Exercises 17-20. graph \u2206RST with vertices R(4, 1), s(7, 3), and T(6, 4) and its image after the glide reflection.<\/p>\n Question 17. Question 18. Question 19. Question 20. In Exercises 21 – 24, determine the number of lines of symmetry for the figure.<\/p>\n Question 21. Question 22. Question 23. Question 24. Question 25. Question 26. Question 27. Question 28. In Exercises 29 – 32, find point C on the x-axis so AC + BC is a minimum.<\/p>\n Question 29. Question 30. Question 31. Question 32. Question 33. Question 34. a. Which figure is a reflection of Figure A in the line x = a? Explain. b. Which figure is a reflection of Figure A in the line y = b? Explain. c. Which figure is a reflection of Figure A in the line y = x? Explain. d. Is there a figure that represents a glide reflection? Explain your reasoning. Question 35. Question 36. Question 37. Question 38. Question 39. Maintaining Mathematical Proficiency<\/p>\n Use the diagram to \u00cdind the angle measure.<\/p>\n Question 40. Question 41. Question 42. Question 43. Question 44. Question 45. Question 46. Question 47. Question 48. Question 49. Exploration 1<\/strong><\/p>\n Rotating a Triangle in a Coordinate Plane<\/p>\n Work with a partner:<\/p>\n a. Use dynamic geometry software to draw any triangle and label it \u2206ABC. b. Rotate the triangle 90\u00b0 counterclockwise about the origin to from \u2206A’B’C\u2019. c. What is the relationship between the coordinates of the vertices of \u2206ABC and those of \u2206A’B’C’? d. What do you observe about the side lengths and angle measures of the two triangles? Exploration 2<\/strong><\/p>\n Rotating a Triangle in a Coordinate Plane<\/p>\n Work with a partner: b. Use the rule you wrote in part (a) to rotate \u2206ABC 90\u00b0 counterclockwise about the origin. What are the coordinates of the vertices of the image. \u2206A’B’C’? c. Draw \u2206A’B’C’. Are its side lengths the same as those of \u2206ABC? Justify your answer. Exploration 3<\/strong><\/p>\n Rotating a Triangle in a Coordinate Plane<\/p>\n Work with a partner. a. The point (x, y) is rotated 180\u00b0 counterclockwise about the origin. Write a rule to b. Use the rule you wrote in part (a) to rotate \u2206ABC (front Exploration 2) 180\u00b0 counterclockwise about the origin. What are the coordinates of the vertices of the image, \u2206A’B’C’? Communicate Your Answer<\/p>\n Question 4. Question 5. Monitoring Progress<\/strong><\/p>\n Question 1. Question 2. Question 3. Question 4. Question 5. Question 6. Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.<\/p>\n Question 7. Question 8. Question 9. Vocabulary and Core Concept Check<\/p>\n Question 1. Question 2. What are the coordinates of the vertices of the image after a 270\u00b0 clockwise rotation about the origin? What are the coordinates of the vertices of the image after turning the figure 90\u00b0 to the left about the origin? What are the coordinates of the vertices of the image after a 270\u00b0 counterclockwise rotation about the origin? Monitoring Progress and Modeling with Mathematics<\/p>\n In Exercises 3-6. trace the polygon and point P. Then draw a rotation o the polygon about point P using the given number of degrees.<\/p>\n Question 3. Question 4. Question 5. Question 6. In Exercises 7-10. graph the polygon and its image after a rotation of the given number of degrees about the origin.<\/p>\n Question 7. Question 8. Question 9. Question 10. In Exercises 11-14, graph \\(\\overline{X Y}\\) with endpoints X(-3, 1) and Y(4, – 5) and its image after the composition.<\/p>\n Question 11. Question 12. Question 13. Question 14. In Exercises 15 and 16, graph \u2206LMN with vertices 2 L(1, 6), M(- 2, 4), and N(3, 2) and its image after the composition.<\/p>\n Question 15. Question 16. In Exercises 17-20, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.<\/p>\n Question 17. Question 18. Question 19. Question 20. REPEATED REASONING<\/strong> (A) 30\u00b0\u00a0 \u00a0 \u00a0 \u00a0 (B) 45\u00b0\u00a0 \u00a0 \u00a0 \u00a0(C) 60\u00b0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(D) 72\u00b0 Question 21. Question 22. Question 23. Question 24. ERROR ANALYSIS<\/strong> Question 25. Question 26. Question 27. Question 28. b. You rotate the door 360\u00b0. What does this mean in the Context of the situation? Explain. Question 29. Question 30. Question 31. Question 32. Question 33. Question 34. Question 35. b. Question 36. Question 37. Question 38. b. You pick up Piece 1. How many different ways can it fit in the puzzle? c. Before putting Piece 1 into the puzzle, you connect it to Piece 2. Now how many ways can it fit in the puzzle? Explain. Question 39. Maintaining Mathematical Proficiency<\/p>\n The figures are congruent. Name the corresponding angles and the corresponding sides.<\/p>\n Question 40. Question 41. Graph quadrilateral ABCD with vertices A(- 4, 1), B(- 3, 3), C(0, 1), and D(- 2, 0) and its Question 1. Question 2. Question 3. Graph the polygon with the given vertices and its image after a reflection in the given line.<\/p>\n Question 4. Question 5. Question 6. Question 7. Graph \u2206ABC with vertices A(2, – 1), B(5, 2), and C(8, – 2) and its image after the glide reflection.<\/p>\n Question 8. Question 9. Determine the number of lines of symmetry for the figure.<\/p>\n Question 10. Question 11. Question 12. Question 13. Graph the polygon and its image after a rotation of the given number of degrees about the origin.<\/p>\n Question 14. Question 15. Question 16. Graph \u2206LMN with vertices L(- 3, – 2), M (- 1, 1), and N(2, – 3) and its image after Question 17. Question 18. Question 19. Exploration 1<\/strong><\/p>\n Reflections in Parallel Lines<\/p>\n Work with a partner. Use dynamic geometry software to draw any scalene triangle and label it \u2206ABC.<\/p>\n a. Draw an line b. Draw a line parallel to c. Draw the line through point A that is perpendicular to d. Find the distance between points A and A”. Find the distance between the two parallel lines. What do You notice? e. Hide \u2206A’B’C’. Is there a single transformation that maps \u2206ABC to \u2206A”B”C”? Explain. f. Make conjectures based on your answers in parts (c)-(e). Test our conjectures by changing \u2206ABC and the parallel lines. Exploration 2<\/strong><\/p>\n Reflections in Intersecting Lines<\/p>\n Work with a partner: Use dynamic geometry software to draw any scalene triangle and label it \u2206ABC.<\/p>\n a. Draw an line b. Draw any line c. Find the measure of \u2220EDF. Rotate \u2206ABC counterclockwise about point D using an angle twice the measure of \u2220EDF. d. Make a conjecture about a figure reflected in two intersecting lines. Test your conjecture by changing \u2206ABC and the lines. Communicate your Answer<\/p>\n Question 3. Question 4. Monitoring Progress<\/strong><\/p>\n Question 1. Question 2. Question 3. Use the figure. The distance between line k and line m is 1.6 centimeters.<\/p>\n Question 4. Question 5. Question 6. Question 7. Question 8. Vocabulary and Core Concept Check<\/p>\n Question 1. Question 2. Monitoring Progress and Modeling with Mathematics<\/p>\n In Exercises 3 and 4, identify an congruent figures in the coordinate plane. Explain.<\/p>\n Question 3. Question 4. In Exercises 5 and 6, describe a congruence transformation that maps the blue preimage to the green image.<\/p>\n Question 5. Question 6. In Exercises 7-10. determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning.<\/p>\n Question 7. Question 8. Question 9. Question 10. In Exercises 11-14, k || m, \u2206ABC is reflected in line k, and \u2206A’B’C” is reflected in line in.<\/p>\n Question 11. Question 12. Question 13. Question 14. In Exercises 15 and 16, find the angle of rotation that maps A onto A”.<\/p>\n Question 15. Question 16. Question 17. Question 18. In Exercises 19 – 22, find the measure of the acute or right angle formed by intersecting lines so that C can be mapped to C\u2019 using two reflections.<\/p>\n Question 19.\n
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<\/a>Transformations Maintaining Mathematical Proficiency<\/h3>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-1.png\")
\nAnswer: The red figure is the mirror image of the blue figure so it is a reflection.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-2.png\")
\nAnswer: The blue figure turns to form the red figure so it is a rotation.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-3.png\")
\nAnswer: A dilation produces an image similar to the original figure. So, the blue polygon is a dilation of the red figure.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.png\")
\nAnswer: The blue figure is similar to the red figure and one is a dilation of the other.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-5.png\")
\nAnswer:
\nLength of A\/Length of B = 12\/14 = 6\/7
\nWidth of A\/Width of B = 5\/7
\nThus Rectangle A and Rectangle B are not similar.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-6.png\")
\nAnswer:
\nBase of triangle A\/Base of triangle B = 6\/9 = 2\/3
\nHeight of triangle A\/Height of triangle B = 8\/12 = 2\/3
\nHypotenuse of triangle A\/Hypotenuse of triangle B = 10\/12 = 5\/6
\nTriangle A and Triangle B are similar.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-7.png\")
\nAnswer:
\nLength of A\/Length of B = 6\/3 = 2
\nWidth of A\/Width of B = 10\/5 = 2
\nThe two figures are similar.<\/p>\n<\/a>Transformations Mathematical Practices<\/h3>\n
\nA(0, 2), B(3, – 1), C(4, 3)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_119.png\")
\nPoints:
\nA(0, 2), B(3, – 1), C(4, 3)
\nSegments:
\nAB = 3 units
\nBC = 4 units
\nAC = 4 units<\/p>\n
\nA(- 2, 1), B(- 2, – 1), C(3, 2)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_120.png\")
\npoints:
\nA(- 2, 1), B(- 2, – 1), C(3, 2)
\nSegments:
\nAB = 2 units
\nBC = 5 units
\nAC = 5 units<\/p>\n
\nA(1, 1), B(- 3, 1), C(- 3, – 2), D(1, – 2)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_121-300x260.png\")
\nPoints:
\nA(1, 1), B(- 3, 1), C(- 3, – 2), D(1, – 2)
\nSegments:
\nAB = 2 units
\nBC = 3 units
\nDC = 2 units
\nAD = 3 units
\nAngles:
\n\u2220A = \u2220B = \u2220C = \u2220D = 90\u00b0<\/p>\n
\nA(1, 1) B(- 3, 1), C(- 2, – 2), D(2, – 2)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_122-300x260.png\")
\nPoints:
\nA(1, 1) B(- 3, 1), C(- 2, – 2), D(2, – 2)
\nSegments:
\nAB = 4 units
\nBC = 4 units
\nCD = 4 units
\nDA = 4 units<\/p>\n
\nA(- 3, 0), B(0, 3), C(3, 0), D(0, – 3)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_124-300x260.png\")
\nPoints:
\nA(- 3, 0), B(0, 3), C(3, 0), D(0, – 3)
\nSegments:
\nAB = 3.5 units
\nBC = 3.5 units
\nCD = 3 units
\nAD = 6 units<\/p>\n
\nA(0, 0), B(4, 0), C(1, 1), D(0, 3)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_125-300x260.png\")
\nPoints:
\nA(0, 0), B(4, 0), C(1, 1), D(0, 3)<\/p>\n<\/a>4.1 Translations<\/h3>\n
![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-8.png\")
\nWork with a partner.
\na. Use dynamic geometry software to draw any triangle and label it \u2206ABC.
\nAnswer:<\/p>\n
\nUSING TOOLS STRATEGICALLY<\/strong>
\nTo be proficient in math, you need to use appropriate tools strategically, including dynamic geometry software.
\nAnswer:<\/p>\n
\nthose of \u2206A’B’C’?
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-9.png\")
\nWork with a partner.
\na. The point (x, y) is translated a units horizontally and b units vertically. Write a rule to determine the coordinates of the image of (x, y).
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-10.png\")
\nAnswer: (x + a, y + b)<\/p>\n
\nAnswer:
\nA'(-4, 0), B'(0, 2) and C'(-1, -6)<\/p>\n
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_126-300x260.png\")
<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nHow can you translate a figure in a coordinate plane?
\nAnswer:<\/p>\n
\nEn Exploration 2. translate \u2206A’B’C’ 3 units right and 4 units up. What are the coordinates of the vertices of the image, \u2206A”B”C”? How are these coordinates
\nrelated to the coordinates of the vertices of the original triangle. \u2206ABC?
\nAnswer:<\/p>\n<\/a>Lesson 4.1 Translations<\/h3>\n
\nName the vector and write its component form.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-11.png\")
\nAnswer:<\/p>\n
\nThe vertices of \u2206LMN are L(2, 2), M(5, 3), and N(9, 1). Translate \u2206LMN using the vector (- 2, 6).
\nAnswer:
\nThe vertices of \u2206LMN are L(2, 2), M(5, 3), and N(9, 1)
\nL(2, 2) \u2192 L'(0, 8)
\nM(5, 3) \u2192 M'(3, 9)
\nN(9, 1) \u2192 N'(7, 7)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_127-295x300.png\")
<\/p>\n
\nIn Example 3. write a rule to translate \u2206A’B’C’ back to \u2206ABC.
\nAnswer:<\/p>\n
\nGraph \u2206RST with vertices R(2, 2), S(5, 2), and T(3, 5) and its image alter the translation (x, y) \u2192 (x + 1, y + 2).
\nAnswer:
\nR(2, 2), S(5, 2), and T(3, 5)
\ntranslation (x, y) \u2192 (x + 1, y + 2).
\nx = 2 and y = 2 to find R’
\nR(2, 2) \u2192 (2 + 1, 2 + 2) = R'(3, 4)
\nS(5, 2) \u2192 (5 + 1, 2 + 2) = S'(6, 4)
\nT(3, 5) \u2192 (3 + 1, 5 + 2) = R'(4, 7)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_129.png\")
<\/p>\n
\nGraph \\(\\overline{T U}\\) with endpoints T(1, 2) and U(4, 6) and its image after the composition.
\nTranslation: (x, y) \u2192 (x – 2, y – 3)
\nTranslation: (x, y) \u2192 (x – 4, y + 5)
\nAnswer:<\/p>\n
\nGraph \\(\\overline{V W}\\) with endpoints V(- 6, – 4) and W(- 3, 1) and its image after the composition.
\nTranslation: (x, y) \u2192 (x + 3, y + 1)
\nTranslation: (x, y) \u2192 (x – 6, y – 4)
\nAnswer:<\/p>\n
\nIn Example 6, you move the gray square 2 units right and 3 units up. Then you move the gray square 1 unit left and 1 unit down. Rewrite the composition as a single transformation.
\nAnswer:<\/p>\n<\/a>Exercise 4.1 Translations<\/h3>\n
\nVOCABULARY<\/strong>
\nName the preimage and image of the transformation \u2206ABC – \u2206A’B’C’.
\nAnswer:
\n\u2206ABC is the preimage and \u2206A’B’C’ is the image.<\/p>\n
\nCOMPLETE THE SENTENCE<\/strong>
\nA _______ moves every point of a figure the same distance in the same direction.
\nAnswer: A translation moves every point of a figure the same distance in the same direction.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-12.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-3.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-13.png\")
\nAnswer:
\nA vector is quantity that has both direction and magnitude.
\nThe vertor is ST to more from the initial point S to the terminal point T, you move 2 units left and 4 units down.
\nSo, the component form is (-2, 4)<\/p>\n
\n(6, 0)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-5.png\")
<\/p>\n
\n(5, – 1)
\nAnswer:
\nD(2, 5), E(6, 3) and F(4, 0)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_130-300x238.png\")
\nD'(7, 4)
\nE'(11, 2)
\nF'(9, -1)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_131-300x177.png\")
<\/p>\n
\n(- 3, – 7)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-7.png\")
<\/p>\n
\n(- 2, – 4)
\nAnswer:
\nD(2, 5), E(6, 3) and F(4, 0)
\n
\nD'(0, 1)
\nE'(4, -1)
\nF'(2, -4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_132-300x254.png\")
<\/p>\n
\nP'(0, 1)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-9.png\")
<\/p>\n
\nP'(- 4, 8)
\nAnswer:
\nP(-3, 6) \u2192 P'(-3 + x, 6 + y) = P'(- 4, 8)
\n-3 + x = -4
\nx = -1
\n6 + y = 8
\ny = 8 – 6
\ny = 2
\nThe component form is (x, y) = (-1, 2)<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-14.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-11.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-15.png\")
\nAnswer:
\nTo go from M to M’, you move 3 units right and 1 unit up
\nSo you unmove along vector (3, 1)<\/p>\n
\n(x, y) \u2192 (x – 8,y + 4)<\/p>\n
\nWhat is the image of A(2, 6)?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-13.png\")
<\/p>\n
\nWhat is the image of B(- 1, 5)?
\nAnswer:
\n(x, y) \u2192 (x – 8,y + 4)
\nB(- 1, 5)
\n(-1, 5) \u2192 (-1 – 8,5 + 4) = B'(-9, 9)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Geometry-Chapter-4-Transformations-img_132-300x273.png\")
<\/p>\n
\nWhat is the preimage of C'(- 3, – 10)?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-15.png\")
<\/p>\n
\nWhat is the preimage of D'(4, – 3)?
\nAnswer:
\n(x, y) \u2192 (x – 8,y + 4)
\nD'(4, – 3)
\nx – 8 = 4
\nx = 12
\ny + 4 = -3
\ny = 7
\nD(12, 7)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_133-300x196.png\")
<\/p>\n
\nQuestion 17.
\n(x, y) \u2192 (x + 4, y + 6)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-17.png\")
<\/p>\n
\n(x, y) \u2192 (x + 9, y – 2)
\nAnswer:
\nP (-2, 3) Q(1, 2), and R(3, – 1)
\n(x, y) \u2192 (x + 9, y – 2)
\nP'(7, 1)
\nQ'(10, 0)
\nR'(12, -3)
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_134-300x144.png\")
<\/p>\n
\n(x, y) \u2192 (x – 2, y – 5)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-19.png\")
<\/p>\n
\n(x, y) \u2192 (x – 1, y + 3)
\nAnswer:<\/p>\n
\ntranslation: (x, y) \u2192 (x – 1, y + 3)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answers-Geometry-Chapter-4-Transformations-img_136-300x253.png\")
\n(x, y) \u2192 (x – 1, y + 3)
\nx = -2 and y = -3 from point P(-2, 3)
\n(-2, 3) \u2192 (-2 – 1, 3 + 3) = P'(-3, 6)
\nQ(1, 2) \u2192 (1 – 1, 2 + 3) = Q'(0, 5)
\nR(3, -1) \u2192 (3 – 1, -1 + 3) = R'(2, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answers-Geometry-Chapter-4-Transformations-img_137-300x247.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x + 12, y + 4)
\nTranslation: (x, y) \u2192 (x – 5, y – 9)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-21.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x – 6, y)
\nTranslation: (x, y) \u2192 (x + 2, y + 7)
\nAnswer:
\nX(2, 4), Y(6, 0). and Z(7, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answers-Geometry-Chapter-4-Transformations-img_138-300x198.png\")
\nTranslation: (x, y) \u2192 (x – 6, y)
\nX(2, 4) \u2192 (2 – 6, 4) = X'(-4, 4)
\nY(6, 0) \u2192 (6 – 6, 0) = Y'(0, 0)
\nZ(7, 2) \u2192 (7 – 6, 2) = Z'(1, 2)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Solution-Key-Geometry-Chapter-4-transformations-img_139-300x164.png\")
\ntranslation: (x, y) \u2192 (x + 2, y + 7)
\n(x, y) \u2192 (x + 2, y + 7)
\nX'(-4, 4) \u2192 (-4 + 2, 4 + 7) = X”(-2, 11)
\nY'(0, 0) \u2192 (0 + 2, 0 + 7) = Y”(2, 7)
\nZ'(1, 2) \u2192 (1 + 2, 2 + 7) = Z”(3, 9)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Geometry-Chapter-4-Transformations-img_140-300x243.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-16.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-23.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-17.png\")
\nAnswer:
\nvertex 6 units right and 4 units down.
\n1. translation (x, y) \u2192 (x + 6, y – 4)
\nvertex 6 units left
\n2. translation (x, y) \u2192 (x – 6, y)<\/p>\n
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in graphing the image of quadrilateral EFGH after the translation (x, y) \u2192 (x – 1, y – 2).
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-18.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-25.png\")
<\/p>\n
\nMODELING WITH MATHEMATICS<\/strong>
\nIn chess, the knight (the piece shaped like a horse) moves in an L pattern. The hoard shows two consecutive moves of a black knight during a game. Write a composition of translations for the moves. Then rewrite the composition as a single translation that moves the knight from its original position to its ending position.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-19.png\")
\nAnswer:
\nAssuming the initial position of the knight is (x, y)
\nThe first translation moved the knight 2 units right and 1 unit down.
\nThe first translation is (x, y) \u2192 (x + 2, y – 1)
\nThe second translation moved the knight 1 unit right and 2 unit down.
\nThe first translation is (x, y) \u2192 (x + 1, y – 2)
\nSingle translation rule is the sum of both first and second translations
\n(x, y) \u2192 (x + 2 + 1, y – 1 – 2) = (x + 3, y – 3)<\/p>\n
\nPROBLEM SOLVING<\/strong>
\nYou are studying an amoeba through a microscope. Suppose the amoeba moves on a grid-indexed microscope slide in a straight line from square B3 to square G7.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-20.png\")
\na. Describe the translation.
\nb. The side length of each grid square is 2 millimeters. How far does the amoeba travel?
\nc. The amoeba moves from square B3 to square G7 in 24.5 seconds. What is its speed in millimeters per second?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-27.png\")
<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nTranslation A maps (x, y) to (x + n, y + t). Translation B maps (x, y) to (x + s, y + m).
\na. Translate a point using Translation A, followed by Translation B. Write an algebraic rule for the final image of the point after this composition.
\nAnswer:
\nTranslation A maps (x, y) to (x + n, y + t)
\nTranslation B maps (x, y) to (x + s, y + m)
\nTranslating a point using translation A followed by translation B.
\nFirst: Applying translation A from the initial point (x, y) maps it to the final point (x + n, y + t)
\nApplying translation B from the initial point (x + n, y + t) maps it to the final point (x + n + s, y + t + m)
\nThe single translation rule for the composition
\n(x, y) \u2192 (x + n + s, y + t + m)<\/p>\n
\nAnswer:
\nTranslating a point using translation A followed by translation B.
\nFirst: Applying translation A from the initial point (x, y) maps it to the final point (x + s, y + m)
\nApplying translation B from the initial point (x + s, y + m) maps it to the final point (x + n + s, y + t + m)
\nThe single translation rule for the composition
\n(x, y) \u2192 (x + n + s, y + t + m)<\/p>\n
\nAnswer:
\nTranslating a point using translation A followed by translation B is the same as translation using B followed by A.
\n(x, y) \u2192 (x + n + s, y + t + m)
\nis equivalent to the single rule of the second case
\n(x, y) \u2192 (x + s + n, y + m + t)<\/p>\n
\nIn Exercises 29 and 30, a translation maps the blue figure to the red figure. Find the value of each variable.
\nQuestion 29.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-21.png\")
\nAnswer:
\nThe blue figure is congruent to the red figure.
\nr = 100 because corresponding angles are congruent.
\n162 = 3w
\nw = 54 because corresponding sides are congruent.
\n2t = 10
\nt = 5 because corresponding sides are congruent.
\ns = 8 because corresponding sides are congruent.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-22.png\")
\nAnswer:
\nSince translation is a rigid motion, then the lengths and angles of the image are the same as the preimage.
\na = 180 – 90 – 55
\na = 35
\nb + 6 = 20
\nb = 20 – 6
\nb = 14
\n4c – 6 = 14
\n4c = 14 + 6
\n4c = 20
\nc = 20\/4 = 5
\nc = 5<\/p>\n
\nUSING STRUCTURE<\/strong>
\nQuadrilateral DEFG has vertices D(- 1, 2), E(- 2, 0), F(- 1, – 1), and G( 1, 3). A translation maps quadrilateral DEFG to quadrilateral D’E’F’G’. The image of D is D'(- 2, – 2). What are the coordinates of E’, F’, and G\u2019?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-31.png\")
<\/p>\n
\nHOW DO YOU SEE IT?<\/strong>
\nWhich two figures represent a translation? Describe the translation.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-23.png\")
\nAnswer:
\nTo move from figure 5 to figure 7 you must move 4 units right and 8 units up.<\/p>\n
\nREASONING<\/strong>
\nThe translation (x, y) \u2192 (x + m, y + n) maps \\(\\overline{P Q}\\) to \\(\\overline{P’ Q’}\\). Write a rule for the translation of \\(\\overline{P’ Q’}\\) to \\(\\overline{P Q}\\). Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-33.png\")
<\/p>\n
\nDRAWING CONCLUSIONS<\/strong>
\nThe vertices of a rectangle are Q(2, – 3), R(2, 4), S(5, 4), and T(5, – 3),
\na. Translate rectangle QRST 3 units left and 3 units down to produce rectangle Q’R’S’T’. Find the area of rectangle QRST and the area of rectangle Q’R’S’T’.
\nAnswer:
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_141-269x300.png\")
\nThe length of each rectangle = 7 units
\nWidth of each rectangle = 3 units
\nThe area of QRST = 7 \u00d7 3 = 21 sq. units
\nThe area of Q’R’S’T’ = 7 \u00d7 3 = 21 sq. units<\/p>\n
\nAnswer:
\nThe area of QRST = The area of Q’R’S’T’ = 21 sq. units
\nTranslation creates an image that is congruent to the original figure.<\/p>\n
\nPROVING A THEOREM<\/strong>
\nProve the Composition Theorem (Theorem 4.1).
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-35.png\")
<\/p>\n
\nPROVING A THEOREM<\/strong>
\nUse properties of translations to prove each theorem.
\na. Corresponding Angles Theorem (Theorem 3. 1)
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_144.jpg\")
\n\\(\\overline{A B}\\)|| \\(\\overline{C D}\\)
\nUsing \\(\\overline{E F}\\) a translation vector.
\nTranslating \u2220BEF along \\(\\overline{E F}\\) until F becomes the image of E.
\n\\(\\overline{E B}\\) || \\(\\overline{E’ B’}\\)
\nSo, \\(\\overline{E’ B’}\\) coincides \\(\\overline{F D}\\) as they are parallel and have a common endpoint.
\nSince translation is a rigid motions.
\nThen \u22201 \u2245 \u22202
\nThus, If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.<\/p>\n
\nAnswer:
\nGiven,
\n\u2220AEF \u2245 \u2220EFD
\nAssume \\(\\overline{A B}\\) and \\(\\overline{C D}\\) are not parallel.
\nIf two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.<\/p>\n
\nWRITING<\/strong>
\nExplain how to use translations to draw a rectangular prism.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-37.png\")
<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nThe vector PQ = (4, 1) describes the translation of A(- 1, w) Onto A'(2x + 1, 4) and B(8y – 1, 1) Onto B'(3, 3z). Find the values of w, x, y, and z.
\nAnswer:
\nThe vector PQ = (4, 1) describes the translation of A(- 1, w) Onto A'(2x + 1, 4) and B(8y – 1, 1) Onto B'(3, 3z).
\n(-1, w) \u2192 (-1 + 4, w + 1) = (3, w + 1)
\nA'(2x + 1, 4)
\n(3, w + 1) = (2x + 1, 4)
\n3 = 2x + 1
\n2x = 2
\nx = 1
\nw + 1 = 4
\nw = 4 – 1
\nw = 3
\nWe know that vector PQ = (4, 1) describes the translation of B(8y – 1, 1)
\n(8y – 1, 1) \u2192 (8y – 1 + 4, 1 + 1) = (8y + 3, 2)
\nB'(3, 3z)
\n(8y + 3, 2) = (3, 3z)
\n8y + 3 = 3
\n8y = 3 – 3
\n8y = 0
\ny = 0
\n3z = 2
\nz = 2\/3<\/p>\n
\nMAKING AN ARGUMENT<\/strong>
\nA translation maps \\(\\overline{G H}\\) to \\(\\overline{G’ H’}\\). Your friend claims that if you draw segments connecting G to G’ and H to H’, then the resulting quadrilateral is a parallelogram. Is your friend correct? Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-39.png\")
<\/p>\n
\nTHOUGHT PROVOKING<\/strong>
\nYou are a graphic designer for a company that manufactures floor tiles. Design a floor tile in a coordinate plane. Then use translations to show how the tiles cover an entire floor. Describe the translations that map the original tile to four other tiles.
\nAnswer:
\nLet the tile by a square of side length of 2 units on a coordinate grid.
\nIf one of its vertex lies at the origin, then its other vertices, in that order are (2, 0), (2, 2) and (0, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_147-300x225.jpg\")
\nThis design can then be used to cover the entire floor by horizontal and vertical translations of 2 units right and upwards.
\nA horizontal translation of 2 units towards the right maps the tile onto X.
\nA vertical translation of 2 units upwards maps the tile onto Y and a horizontal translation of Y by 2 units towards the right maps the tile onto Z.
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_148-300x260.jpg\")
<\/p>\n
\nREASONING<\/strong>
\nThe vertices of \u2206ABC are A(2, 2), B(4, 2), and C(3, 4). Graph the image of \u2206ABC after the transformation (x, y) \u2192 (x + y, y). Is this transformation a translation? Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.1-Question-41.png\")
<\/p>\n
\nPROOF<\/strong>
\n\\(\\overline{M N}\\) is perpendicular to line l. \\(\\overline{M’ N’}\\) is the translation of \\(\\overline{M N}\\) 2 units to the left. Prove that \\(\\overline{M’ N’}\\) is perpendicular to l.
\nAnswer:
\nM’N’ is a horizontal translation of MN by 2 units to the left.
\nMN and M’N’ are parallel to each other and so their slopes must be equal.
\nIf a line l is perpendicular to MN, then it must be perpendicular to M’N’<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-24.png\")
\nAnswer:
\nYes; The figure can be folded in half in several ways so that one side matches the other.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-25.png\")
\nAnswer: The shape is symmetrical along a diagnol.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-26.png\")
\nAnswer:
\nThe shape is asymmetrical<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-27.png\")
\nAnswer: The shape is symmetrical along its x-axis.<\/p>\n
\n– (- x)
\nAnswer:
\nWhen we multiply two negative numbers we get the positive number.
\n-1 \u00d7 -x = x<\/p>\n
\n– (x + 3)
\nAnswer:
\nIn this expression minus is multiplied with the whole term
\n-1 \u00d7 x + -1 \u00d7 3 = -x – 3<\/p>\n
\nx – (12 – 5x)
\nAnswer:
\nWhen we multiply two negative numbers we get the positive number.
\nx – 12 + 5x<\/p>\n
\nx – (- 2x + 4)
\nAnswer:
\nWhen we multiply two negative numbers we get the positive number.
\nx + 2x – 4<\/p>\n<\/a>4.2 Reflections<\/h3>\n
\nUse a straightedge to draw any triangle on paper. Label if \u2206ABC.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-28.png\")
<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-29.png\")
\na. Reflect \u2206ABC in the y-axis to form \u2206A’B’C’.
\nAnswer:<\/p>\n
\nthose of \u2206A’B’C’?
\nLOOKING FOR STRUCTURE<\/strong>
\nTo be proficient in math, you need to look closely to discern a pattern or structure.
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nHow can you reflect a figure in a coordinate plane?
\nAnswer:<\/p>\n<\/a>Lesson 4.2 Reflections<\/h3>\n
\nQuestion 1.
\nx = 4
\nAnswer:<\/p>\n
\nx = – 3
\nAnswer:<\/p>\n
\ny = 2
\nAnswer:<\/p>\n
\ny = – 1
\nAnswer:<\/p>\n
\nGraph \u2206JKL and its image after a reflection in the x-axis.
\nAnswer:<\/p>\n
\nGraph \u2206JKL and its image after a reflection in the y-axis.
\nAnswer:<\/p>\n
\nGraph \u2206JKL and its image after a reflection in the line y = x.
\nAnswer:<\/p>\n
\nGraph \u2206JKL and its image aIter a reflection in the line y = – x.
\nAnswer:<\/p>\n
\nIn Example 3. verify that \\(\\overline{F F’}\\) is perpendicular to y = – x.
\nAnswer:<\/p>\n
\nWHAT IF?<\/strong>
\nIn Example 4, \u2206ABC is translated 4 units down and then reflected in the y-axis. Graph \u2206ABC and its image after the glide reflection.
\nAnswer:<\/p>\n
\nIn Example 4. describe a glide reflection from \u2206A”B”C\u201d to \u2206ABC.
\nAnswer:<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-30.png\")
\nAnswer:<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-31.png\")
\nAnswer:<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-32.png\")
\nAnswer:<\/p>\n
\nDraw a hexagon with no lines of symmetry.
\nAnswer:<\/p>\n
\nLook back at Example 6. Answer the question by Using a reflection of point A instead of point B.
\nAnswer:<\/p>\n<\/a>Exercise 4.2 Reflections<\/h3>\n
\nVOCABULARY<\/strong>
\nA glide reflection is a combination of which two transformations?
\nAnswer:
\nA glide reflection is a combination of a translation and reflection.<\/p>\n
\nWHICH ONE DOESN’T BELONG?<\/strong>
\nWhich transformation does not belong with the other three? Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-33.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-34.png\")
\nAnswer:
\nReflection in the y-axis<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-35.png\")
\nAnswer:
\nWe see the triangle ABC with vertices A(-2, 0), B(-1, 2) and C(-2, 3)
\nWe also see the triangle DEF with vertices D(1, 0), E(2, -1) and F(1, -3)
\nApply reflection in the x-axis to the triangle ABC, we do not have the points D, E and F in the same place on opposite sides.
\nApply reflection in the y-axis to the triangle ABC, we do not have the points D, E and F in the same place on opposite sides.
\nThere is no reflection.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-36.png\")
\nAnswer:
\nWe observe the triangles ABC and DEF, we can say that there is no reflection.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-37.png\")
\nAnswer:
\nThe triangle ABC with vertices A(2, 1), B(2, 3) and C(6, 1)
\nApply reflection in the x-axis to the triangle ABC.
\nBy seeing the above graph we say that the point which is in the same place on opposite sides x-axis with respect to the point A.
\nThe point is D(2, -1)
\nBy seeing the above graph we say that the point which is in the same place on opposite sides x-axis with respect to the point B.
\nThe point is E(2, -3)
\nBy seeing the above graph we say that the point which is in the same place on opposite sides x-axis with respect to the point C.
\nThe point is F(6, -1)
\nWe see that the triangle DEF with vertices D(2, -1), E(2, -3) and F(6, -1).
\nWhen we apply reflection in the y-axis to the triangle ABC, we do not have the points D, E, F in the same place on opposite sides.<\/p>\n
\nQuestion 7.
\nJ(2, – 4), K(3, 7), L(6, – 1); x-axis
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-7.png\")
<\/p>\n
\nJ(5, 3), K(1, – 2), L(- 3, 4); y-axis
\nAnswer:
\nJ(5, 3), K(1, – 2), L(- 3, 4)
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_104-300x192.png\")
\nWe will apply reflection in the y-axis to the triangle JKL.
\nJ’ which is in the same place on opposite sides y-axis with respect to the point J(5, 3)
\nThe coordinates of this point are J'(-5, 3)
\nK’ which is in the same place on opposite sides y-axis with respect to the point K(1, -2)
\nThe coordinates of this point are K'(-1, -2)
\nL’ which is in the same place on opposite sides y-axis with respect to the point L(-3, 4)
\nThe coordinates of this point are L'(3, 4)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_103-300x201.png\")
<\/p>\n
\nJ(2, – 1), K(4, – 5), L(3, 1); x = – 1
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-9.png\")
<\/p>\n
\nJ(1, – 1), K(3, 0), L(0, – 4); x = 2
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_104-300x249.png\")
\nApply reflection in the line x = 2 to the triangle JKL.
\nJ = (1, -1) J’ = (3, -1)
\nK = (3, 0) K’= (1, 0)
\nL(0, – 4) L’ = (4, -4)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_105-300x244.png\")
<\/p>\n
\nJ(2, 4), K(- 4, – 2), L(- 1, 0); y = 1
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-11.png\")
<\/p>\n
\nJ(3, – 5), K(4, – 1), L(0, – 3); y = – 3
\nAnswer:
\nGraph the triangle by using the given vertices J(3, – 5), K(4, – 1), L(0, – 3)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_104-300x276.png\")
\nJ(3, – 5) \u2192 J'(3, -1)
\nK(4, – 1) \u2192 K'(4, -5)
\nL(0, – 3) \u2192 L'(0, 3)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_105-300x272.png\")
<\/p>\n
\ny = x
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-38.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-13.png\")
<\/p>\n
\ny = x
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-39.png\")
\nAnswer:
\nThe vertices of the above polygon are A(2, -1), B(-1, 2), C(2, 3) and D(4, 2)
\nApply reflection in the line y = x to the polygon ABCD.
\nSo, A'(-1, 2)
\nB(-1, 2) \u2192 B'(2, -1)
\nC(2, 3) \u2192 C'(3, 2)
\nD(4, 2) \u2192 D'(2, 4)
\nWhen the point is reflected in the line y = x the coordinates of the point image change place.
\nPoint A’ and B have the same coordinates, and points A and B have the same coordinates.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_108-300x238.png\")
<\/p>\n
\ny = -x
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-40.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-15.png\")
<\/p>\n
\ny = -x
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-41.png\")
\nAnswer:
\nApply reflection in the line y = x to the triangle ABC.
\nThe vertices are A(1, 2), B(4, 2), C(3, -2)
\nA(1, 2) \u2192 A'(-2, -1)
\nB(4, 2) \u2192 B'(-2, -4)
\nC(3, -2) \u2192 C'(2, -3)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_109-300x261.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x, y – 1)
\nReflection: in the y-axis
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-17.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x – 3,y)
\nReflection: in the line y = – 1
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_110-300x195.png\")
\nTranslation: (x, y) \u2192 (x – 3,y)
\nR(4, 1) to find R’
\n(4, 1) \u2192 (4 – 3,1) = R'(1, 1)
\nS(7, 3) to find S’
\n(7, 3) \u2192 (7 – 3,3) = S'(4, 3)
\nT(6, 4) to find T’
\n(6, 4) \u2192 (6 – 3,4) = T'(3, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_111-300x199.png\")
\nApply reflection in the line y = -1 to the triangle R’S’T’
\nR'(1, 1) \u2192 R”(1, -3)
\nS'(4, 3) \u2192 S”(4, -5)
\nT'(3, 4) \u2192 T”(3, -6)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_112-300x293.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x, y + 4)
\nReflection: in the line x = 3
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-19.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x + 2, y + 2),
\nReflection: in the line y = x
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_113-300x195.png\")
\nTranslation: (x, y) \u2192 (x + 2, y + 2),
\nR(4, 1) to find R’
\n(4, 1) \u2192 (4 + 2,1 + 2) = R'(6, 3)
\nS(7, 3) to find S’
\n(7, 3) \u2192 (7 + 2,3 + 2) = S'(9, 5)
\nT(6, 4) to find T’
\n(6, 4) \u2192 (6 + 2,4 + 2) = T'(8, 6)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Solution-Key-Geometry-Chapter-4-transformations-img_114-300x243.png\")
\nR'(6, 3) \u2192 R”(3, 6)
\nS'(9, 5) \u2192 S”(5, 9)
\nT'(8, 6) \u2192 T”(6, 8)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_115-300x265.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-42.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-21.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-43.png\")
\nAnswer:
\nThe figure has 4 lines of symmetry.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-44.png\")
\nAnswer:
\nThere are no lines of symmetry.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-45.png\")
\nAnswer: The figure has 5 lines of symmetry.<\/p>\n
\nUSING STRUCTURE<\/strong>
\nIdentify the line symmetry (if any) of each word.
\na. LOOK
\nb. MOM
\nc. OX
\nd. DAD
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-25.png\")
<\/p>\n
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in describing the transformation.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-46.png\")
\nAnswer:
\nBy drawing \\(\\overline{A B}\\)
\nA(5, -1) and B(2, -2)
\nA(5, -1) \u2192 A'(7, 2)
\nB(2, -2) \u2192 B'(4, 1)
\nTranslating \\(\\overline{A B}\\) with the rule (x, y) \u2192 (x + 2, y + 3)
\nReflecting \\(\\overline{A’ B’}\\) over the y-axis
\nA'(7, 2) \u2192 A”(-7, 2)
\nB'(4, 1) \u2192 B”(-4, 1)
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_116-300x269.png\")
\nThus the transformation is a glide reflection.<\/p>\n
\nMODELING WITH MATHEMATICS<\/strong>
\nYou park at some point K on line n. You deliver a pizza to House H, go back to your car. and deliver a pizza to House J. Assuming that you can cut across both lawns, how can you determine the parking location K that minimizes the distance HK + KJ?
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-47.png\")
\nAnswer:
\nReflect H in line n to obtain H’. Connect H’ to J to draw \\(\\overline{J H’}\\).
\nLabel the intersection of JH’ and n as K. Because JH’ is the shortest distance between J and H’ and HK = H’K, park at point K.<\/p>\n
\nATTENDING TO PRECISION<\/strong>
\nUse the numbers and symbols to create the glide reflection resulting in the image shown.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-48.png\")
\nAnswer:
\nWe will find the preimage of point A”(5, 6)
\nThat is point A'(6, 5)
\nWe will find the preimage of point B”(4, 2)
\nThat is point B'(2, 4)
\nWe will find the preimage of point C”(-1, 5)
\nThat is point C'(5, -1)
\nTranslation is: (x, y) \u2192 (x + 3, y + 3)
\nA(3, 2) in the translation to find A’
\n(x, y) \u2192 (x + 3, y + 3)
\n(3, 2) \u2192 (3 + 3, 2 + 3) = (6, 5)
\nB(-1, 1) in the translation to find B’
\n(x, y) \u2192 (x + 3, y + 3)
\n(-1, 1) \u2192 (-1 + 3, 1 + 3) = (2, 4)
\nC(2, -4) in the translation to find C’
\n(x, y) \u2192 (x + 3, y + 3)
\n(2, -4) \u2192 (2 + 3, -4 + 3) = (5, -1)<\/p>\n
\nA(1, 4), B(6, 1)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-29.png\")
<\/p>\n
\nA(4, – 5), B(12, 3)
\nAnswer:
\nGiven,
\nA(4, – 5), B(12, 3)
\nWe have to find the point C on the x-axis.
\nAc + BC is minimum.
\nFirst we will plot points in the coordinate system.
\nThe shortest distance from point A to point B gives the line that join these two points.
\nThe line intersects the x-axis at point C.
\nx = 9 and y = 0
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_117-300x212.png\")
<\/p>\n
\nA(- 8, 4), B(- 1, 3)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-31.png\")
<\/p>\n
\nA(- 1, 7), B(5, – 4)
\nAnswer:
\nA(- 1, 7), B(5, – 4)
\nWe have to find the point C on the x-axis.
\nAC + BC is minimum.
\nFirst we will plot points in the coordinate system.
\nThe shortest distance from point A to point B gives the line that join these two points.
\nThe line intersects the x-axis at point C.
\nx = 2.8 and y = 0
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_118-280x300.png\")
<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nThe line y = 3x + 2 is reflected in the line y = – 1. What is the equation of the image?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-33.png\")
<\/p>\n
\nHOW DO YOU SEE IT?<\/strong>
\nUse Figure A.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-49.png\")
<\/p>\n
\nAnswer:
\nThe reflection of Figure A in the line x =a is Fig 2.
\nPoints of Fig 2 is in the same place on opposite sides x = a with respect to the points of Fig A.<\/p>\n
\nAnswer:
\nThe reflection of Figure A in the line y = b is Fig 4.
\nPoints of Fig 4 is in the same place on opposite sides y = b with respect to the points of Fig A.<\/p>\n
\nAnswer:
\nThe reflection of Figure A in the line y = x is Fig 1.
\nPoints of Fig 1 is in the same place on opposite sides y = x-axis with respect to the points of Fig A.<\/p>\n
\nAnswer: The figure that represent glide reflection is Fig 4.<\/p>\n
\nCONSTRUCTION<\/strong>
\nFollow these steps to construct a reflection of \u25b3 ABC in line m. Use a compass and straightedge.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-50.png\")
\nStep 1 Draw \u25b3ABC and line m.
\nStep 2 Use one compass setting to find two points that are equidistant from A on line m. Use the same compass setting to find a point on the other side of m that is the same distance from these two points. Label that point as A’.
\nStep 3 Repeat Step 2 to find points B’ and C’.
\nDraw \u25b3A’B’C.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-35.png\")
<\/p>\n
\nUSING TOOLS<\/strong>
\nUse a reflective device to verify your construction in Exercise 35.
\nAnswer:
\n![\"Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-35\"](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-35-1.png\")
<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nReflect \u25b3MNQ in the line y = -2x.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-51.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-37.png\")
<\/p>\n
\nTHOUGHT PROVOKING<\/strong>
\nIs the composition of a translation and a reflection commutative? (In other words, do you obtain the same image regardless of the order in which you perform the transformations?) Justify your answer.
\nAnswer:
\nWhen the direction of the translation is parallel to the line of reflection, that is only case when the composition of a translation and a reflection is commutative.<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nPoint B (1, 4) is the image of B(3, 2) after a reflection in line c. Write an equation for line c.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.2-Question-39.png\")
<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-Chapter-4-Transformations-52.png\")
<\/p>\n
\nm\u2220AOC
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220AOC = 60\u00b0<\/p>\n
\nm\u2220AOD
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220AOD = 130\u00b0<\/p>\n
\nm\u2220BOE
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220BOE = 20\u00b0<\/p>\n
\nm\u2220AOE
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220AOE = 160\u00b0<\/p>\n
\nm\u2220COD
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220COD = 70\u00b0<\/p>\n
\nm\u2220EOD
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220EOD = 30\u00b0<\/p>\n
\nm\u2220COE
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220COE = 100\u00b0<\/p>\n
\nm\u2220AOB
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220AOB = 180\u00b0<\/p>\n
\nm\u2220COB
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220COB = 120\u00b0<\/p>\n
\nm\u2220BOD
\nAnswer:
\nPlace the center of the protractor at point O parallel to the longer AB, and then read the measure of the angle from the protractor.
\nm\u2220BOD = 50\u00b0<\/p>\n<\/a>4.3 Rotations<\/h3>\n
![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-53.png\")
<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\na. The point (x, y) is rotated 90\u00b0 counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y).
\nCONSTRUCTING VIABLE ARGUMENTS<\/strong>
\nTo be proficient in math, you need to use previous established results in constructing arguments.
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-54.png\")
<\/p>\n
\ndetermine the coordinates of the image of (x, y). Explain how you found the rule.
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nHow can you rotate a figure in a coordinate plane?
\nAnswer:<\/p>\n
\nIn Exploration 3. rotate A’B’C’ 180\u00b0 counterclockwise about the origin. What are the coordinates of the vertices of the image. \u2206A”B”C”? How are these coordinates related to the coordinates of the vertices of the original triangle, \u2206ABC?
\nAnswer:<\/p>\n<\/a>Lesson 4.3 Rotations<\/h3>\n
\nTrace \u2206DEF and point P. Then draw a 50\u00b0 rotation of \u2206DEF about point P.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-55.png\")
\nAnswer:<\/p>\n
\nGraph \u2206JKL with vertices J(3, 0), K(4, 3), and L(6, 0) and its image after a 90\u00b0 rotation about the origin.
\nAnswer:<\/p>\n
\nGraph \\(\\overline{R S}\\) from Example 3. Perform the rotation first, followed by the reflection. Does the order of the transformations matter? Explain.
\nAnswer:<\/p>\n
\nWHAT IF?<\/strong>
\nIn Example 3. \\(\\overline{R S}\\) is reflected in the x-axis and rotated 180\u00b0 about the origin. Graph \\(\\overline{R S}\\) and its image after the composition.
\nAnswer:<\/p>\n
\nGraph \\(\\overline{A B}\\) with endpoints A(- 4, 4) and B(- 1, 7) and its image after the composition.
\nTranslation: (x, y) \u2192 (x – 2, y – 1)
\nRotation: 90\u00b0 about the origin
\nAnswer:<\/p>\n
\nGraph \u2206TUV with vertices T(1, 2), U(3. 5), and V(6, 3) and its image after the composition.
\nRotation: 180\u00b0 about the origin
\nReflection: in the x-axis
\nAnswer:<\/p>\n
\nrhombus
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-56.png\")
\nAnswer:
\nSo within 360\u2218 rhombus turns 2 times into itself as 360\u2218 divided by 180\u2218 is 2.
\nHence the order of rotational symmetry of the rhombus is 2.<\/p>\n
\noctagon
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-57.png\")
\nAnswer:
\nA regular octagon has 8 lines of symmetry and a rotational symmetry of order 8. This means that it can be rotated in such a way that it will look the same as the original shape 8 times in 360\u00b0.<\/p>\n
\nright triangle
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-58.png\")
\nAnswer: A right triangle does not have rotational symmetry.<\/p>\n<\/a>Exercise 4.3 Rotations<\/h3>\n
\nCOMPLETE THE SENTENCE<\/strong>
\nWhen a point (a, b) is rotated counterclockwise about the origin. (a, b) \u2192 (b, – a) is the result of a rotation of _________ .
\nAnswer:
\nWhen a point (a, b) is rotated counterclockwise about the origin. (a, b) \u2192 (b, – a) is the result of a rotation of 270\u00b0<\/p>\n
\nDIFFERENT WORDS, SAME QUESTION<\/strong>
\nWhich is different? Find “both” answers.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-59.png\")
\nWhat are the coordinates of the vertices of the image after a 90\u00b0 counterclockwise rotation about the origin?
\nAnswer:
\nWhen rotating the figure 90\u00b0 counterclockwise the coordinates will be
\nA(1, 2) \u2192 A'(-2, 1)
\nB(2, 4) \u2192 B'(-4, 2)
\nC(4, 2) \u2192 C'(-2, 4)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_87-300x145.jpg\")
<\/p>\n
\nAnswer:
\nWhen rotating the figure 270\u00b0 counterclockwise the coordinates will be
\nA(1, 2) \u2192 A'(-2, 1)
\nB(2, 4) \u2192 B'(-4, 2)
\nC(4, 2) \u2192 C'(-2, 4)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_88-300x173.png\")
<\/p>\n
\nAnswer:
\nRotating 90\u00b0 to the left is equivalent to rotating the figure 90\u00b0 counterclockwise the coordinates will be
\nA(1, 2) \u2192 A'(-2, 1)
\nB(2, 4) \u2192 B'(-4, 2)
\nC(4, 2) \u2192 C'(-2, 4)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Chapter-4-Transformations-Answer-Key-img_89-300x145.jpg\")
<\/p>\n
\nAnswer:
\nWhen rotating the figure 270\u00b0clockwise the coordinates will be
\nA(1, 2) \u2192 A'(-2, 1)
\nB(2, 4) \u2192 B'(-4, 2)
\nC(4, 2) \u2192 C'(-2, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_90-233x300.jpg\")
<\/p>\n
\n30\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-60.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-3.png\")
<\/p>\n
\n80\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-61.png\")
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_91-300x190.png\")
<\/p>\n
\n150\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-62.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-5.png\")
<\/p>\n
\n130\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-63.png\")
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_92-300x168.png\")
<\/p>\n
\n90\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-64.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-7.png\")
<\/p>\n
\n180\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-65.png\")
\nAnswer:
\nThe vertices of DEF are D(-3, -1), E(-1, 2) and F(4, -2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_93-300x249.png\")
\nRotate D(-3, -1) through an angle 180\u00b0 about the origin, we will get the point D'(3, 1)
\nRotate E(-1, 2) through an angle 180\u00b0 about the origin, we will get the point E'(1, -2)
\nRotate F(4, -2) through an angle 180\u00b0 about the origin, we will get the point F'(-4, 2)
\nWe will rotate the point of triangle through an angle 180\u00b0 about the origin, the coordinates of the point image change sign.
\nThen we will graph the triangle D’E’F’ with vertices D'(3, 1), E'(1, -2) and F'(-4, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_94-300x196.png\")
<\/p>\n
\n180\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-66.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-9.png\")
<\/p>\n
\n270\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-67.png\")
\nAnswer:
\nRotation 270\u00b0
\nQ(-6, -3) \u2192 Q'(-3, 6)
\nR(-5, 0) \u2192 R'(0, 5)
\nS(-3, 0) \u2192 S'(0, 3)
\nT(-1, -3) \u2192 T'(-3, 1)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_91-233x300.jpg\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x, y + 2)
\nRotation: 90\u00b0 about the origin
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-11.png\")
<\/p>\n
\nRotation: 180\u00b0 about the origin
\nTranslation: (x, y) \u2192 (x – 1, y + 1)
\nAnswer:
\nX(-3, 1) and Y(4, -5)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_93-300x193.png\")
\nTranslation: (x, y) \u2192 (x – 1, y + 1)
\nx = -3 and y = 1 in the translation to find X’
\n(x, y) \u2192 (x – 1, y + 1)
\n(-3, 1) \u2192 (-3 – 1, 1 + 1) = (-4, 2)
\nx = 4 and y = -5 in the translation to find Y’
\n(x, y) \u2192 (x – 1, y + 1)
\n(4, -5) \u2192 (4 – 1, -5 + 1) = (3, -4)
\nX'(-4, 2) and Y'(3, -4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_94-300x210.png\")
\nRotate 180\u00b0 about the origin
\nX'(-4, 2) = X”(4, -2)
\nY'(3, -4) = Y”(-3, 4)
\nX”(4, -2) and Y”(-3, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_95-300x224.png\")
<\/p>\n
\nRotation: 270\u00b0 about the origin
\nReflection: in the y-axis
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-13.png\")
<\/p>\n
\nReflection: in the line y = x
\nRotation: 180\u00b0 about the origin
\nAnswer:
\nX(-3, 1) and Y(4, -5)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_95-300x223.png\")
\nReflection in the line y = x to the \\(\\overline{X Y}\\)
\nX'(1, -3)
\nY'(-5, 4)
\n\\(\\overline{X’ Y’}\\) with end points X'(1, -3) and Y'(-5, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_96-300x243.png\")
\nRotate X'(1, -3) through an angle 180\u00b0 we get X”(-1, 3)
\nRotate Y'(-5, 4) through an angle 180\u00b0 we get Y”(5, -4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_97-300x241.png\")
<\/p>\n
\nRotation: 90\u00b0 about the origin
\nTranslation: (x, y) \u2192 (x – 3, y + 2)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-15.png\")
<\/p>\n
\nReflection: in the x-axis
\nRotation: 270\u00b0 about the origin
\nAnswer:
\nThe vertices are L(1, 6), M(-2, 4) and N(3, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_95-300x222.png\")
\nL’ which is in the same place on opposite sides x-axis with respect to the point L(1, 6).
\nThe coordinates of this point are L'(1, -6)
\nM’ which is in the same place on opposite sides x-axis with respect to the point M(-2, 4).
\nThe coordinates of this point are M'(-2, -4)
\nN’ which is in the same place on opposite sides x-axis with respect to the point N(3, 2).
\nThe coordinates of this point are N'(3, -2)
\nL'(1, -6), M'(-2, -4) and N'(3, -2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_96-300x282.png\")
\nL'(1, -6) \u2192 L”(-6, -1)
\nM'(-2, -4) \u2192 M”(-4, 2)
\nN'(3, -2) \u2192 N”(-2, -3)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_97-300x226.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-68.png\")
\nAnswer:
\nThe rotations of 90\u00b0 and 180\u00b0 about the center will map this figure onto itself.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-69.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_98-300x281.jpg\")
\nThe figure has rotational symmetry.
\nThe rotation center is the center of the shape.
\nRotation of 72\u00b0, 144\u00b0, 216\u00b0, 288\u00b0 and 360\u00b0 about the center all map the shape onto itself.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-70.png\")
\nAnswer:
\nThe rotations of 45\u00b0, 90\u00b0, 135\u00b0 and 180\u00b0 about the center will map this figure onto itself.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-71.png\")
\nAnswer:
\nA rectangle has two equal sides and all equal angles.
\nWe have to determine which angles of rotation map the figure onto itself.
\n![\"Geometry](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Geometry-Big-Ideas-Math-Answers-Chapter-4-Transformations-img_99-300x170.png\")
\nThe interior angles are 90\u00b0
\nThe angle of rotational symmetry for the rectangle is 180\u00b0<\/p>\n
\nIn Exercises 21-24, select the angles of rotational symmetry for the regular polygon. Select all that apply.<\/p>\n
\n(E) 90\u00b0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(F) 120\u00b0\u00a0 \u00a0 \u00a0(G) 144\u00b0\u00a0 \u00a0 \u00a0 (H) 180\u00b0<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-72.png\")
\nAnswer: Option F
\nThe angle of rotational symmetry of this figure is 120\u00b0<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-73.png\")
\nAnswer:
\nThe above figure has equal lengths of sides and angles.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_100-300x288.png\")
\nThe angles of rotational symmetry for the regular polygon are 90\u00b0 and 180\u00b0.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-74.png\")
\nAnswer:
\nThe above figure has equal lengths of sides and angles.
\nThe angles of rotational symmetry for the regular polygon are 72\u00b0 and 144\u00b0.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-75.png\")
\nAnswer:
\nThe above figure has equal lengths of sides and angles.
\nThe interior angles in the hexagon are 60\u00b0
\nThe angles of rotational symmetry for the regular polygon are 60\u00b0, 120\u00b0 and 180\u00b0<\/p>\n
\nIn Exercises 25 and 26, the endpoints of \\(\\overline{C D}\\) are C(- 1, 1) and D(2, 3). Describe and correct the error in finding the coordinates of the vertices of the image after a rotation of 270\u00b0 about the origin.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-76.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-25.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-77.png\")
\nAnswer:
\nRotating 270\u00b0 (a, b) \u2192 (b, -a)
\nReflection about y = x
\nThe correct answer is
\nC(-1, 1) = C'(1, 1)
\nD(2, 3) = D'(3, -2)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_101-278x300.jpg\")
<\/p>\n
\nCONSTRUCTION<\/strong>
\nFollow these Steps to construct a rotation of \u2206ABC by angle D around a point O. Use a compass and straightedge.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-78.png\")
\nStep 1 Draw \u2206ABC, \u2220D, and O, the center of rotation.
\nStep 2 Draw \\(\\overline{O A}\\). Use the construction for copying an angle to copy \u2220D at O. as shown. Then use distance OA and center O to find A’.
\nStep 3 Repeat Step 2 to find points B’ and C’. Draw \u2206 A’B\u2019C\u2019.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-27.png\")
<\/p>\n
\nREASONING<\/strong>
\nYou enter the revolving door at a hotel.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-79.png\")
\na. You rotate the door 180\u00b0. what does this mean in the context of the situation? Explain.
\nAnswer:
\nThe revolving door is composed of 3 rectangular glass parts intersecting at the door axis.
\nIn the context of our lesson, Rotating the revolving door 180\u00b0 means rotating the short sides of the rectangles 180\u00b0 about the origin.<\/p>\n
\nAnswer:
\nRotating the revolving door 360\u00b0 means rotating the short sides of the rectangles 360\u00b0 about the origin.<\/p>\n
\nMATHEMATICAL CONNECTIONS<\/strong>
\nUse the graph of Y = 2X – 3.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-80.png\")
\na. Rotate the line 90\u00b0, 180\u00b0 270\u00b0, and 360\u00b0 about the origin. Write the equation of the line for each image. Describe the relationship between the equation of the preimage and the equation of each image.
\nb. Do you think that the relationships you described in part (a) are true for any line? Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-29.png\")
<\/p>\n
\nMAKING AN ARGUMENT<\/strong>
\nYour friend claims that rotating a figure by 180\u00b0 is the same as reflecting a figure in the y-axis arid then reflecting it in the x-axis. Is your friend correct? Explain your reasoning.
\nAnswer:
\nrotating a figure by 180\u00b0 (a, b) \u2192 (-a, -b)
\nreflecting a figure in the y-axis (a, b) \u2192 (-a, b)
\nreflecting a figure in the x-axis (a, b) \u2192 (a, -b)
\nFor reflecting a figure in the y-axis and then reflecting in the x-axis
\nthe first step will result (a, b) \u2192 (a, -b)
\nReflecting (-a, b) in the x-axis (-a, b) \u2192 (-a, -b)
\nThus your friend is correct.<\/p>\n
\nDRAWING CONCLUSIONS<\/strong>
\nA figure only has point symmetry. How many times can you rotate the figure before it is back where it started?
\nAnswer:
\nIf a figure only has point symmetry, then it has 180\u00b0 rotational symmetry. One rotation will rotate the figure 180\u00b0. A second rotation will rotate the figure another 180\u00b0. The two rotations combined result in a 360\u00b0 rotation and therefore the rotations map the figure onto itself. So, you can rotate the figure 2 times before it is back where it started.<\/p>\n
\nANALYZING RELATIONSHIPS<\/strong>
\nIs it possible for a figure to have 90\u00b0 rotational symmetry but not 180\u00b0 rotational symmetry? Explain your reasoning.
\nAnswer:
\nNo, it is not possible.
\nWhen figure has 90\u00b0 rotational symmetry, it must has 180\u00b0 rotational symmetry.
\nWhen we rotate the figure twice by 90\u00b0, we get 180\u00b0<\/p>\n
\nANALYZING RELATIONSHIPS<\/strong>
\nIs it possible for a figure to have 180\u00b0 rotational symmetry hut not 90\u00b0 rotational symmetry? Explain your reasoning.
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-33.png\")
<\/p>\n
\nTHOUGHT PROVOKING<\/strong>
\nCan rotations of 90\u00b0, 180\u00b0, 270\u00b0, and 360\u00b0 be written as the composition of two reflections? Justify your answer.
\nAnswer:
\n1. 180\u00b0
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_102-300x259.jpg\")
\nThe 1st image is original. The vertical lines in the above image are mirrors.
\nSo, if I use two reflections I get image 3 eventually from two possible paths.
\nImage 3 is the 180\u00b0 rotation of original object 1.
\n2. 360\u00b0 rotation
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_103.jpg\")
\nThe 1st image is original. The horizontal lines in the above image are mirrors.
\nSo, if I use two reflections – First take the reflection of 1st image to get it flipped.
\nThen again take reflection of this flipped image again to get original image.<\/p>\n
\nUSING AN EQUATION<\/strong>
\nInside a kaleidoscope. two mirrors are placed next to each other to form a V. The angle between the mirrors determines the number of lines of symmetry in the image. Use the formula n(m\u22201) = 180\u00b0 to find the measure of \u22201, the angle between the mirrors, for the number n of lines of symmetry.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-81.png\")
\na.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-82.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-83.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-35.png\")
<\/p>\n
\nREASONING<\/strong>
\nUse the coordinate rules for counterclockwise rotations about the origin to write coordinate rules 11w clockwise rotations of 9o\u00b0. 180\u00b0, or 270\u00b0 about the origin.
\nAnswer:
\nFor a rotation of 90\u00b0 (a, b) = (b, -a)
\nFor a rotation of 180\u00b0 (a, b) = (-a, -b)
\nFor a rotation of 270\u00b0 (a, b) = (-b, a)<\/p>\n
\nUSING STRUCTURE<\/strong>
\n\u2206XYZ has vertices X(2, 5). Y(3, 1), and Z(0, 2). Rotate \u2206XYZ 90\u00b0 about the point P(- 2, – 1).
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-37.png\")
<\/p>\n
\nHOW DO YOU SEE IT?<\/strong>
\nYou are finishing the puzzle. The remaining two pieces both have rotational symmetry.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-84.png\")
\na. Describe the rotational symmetry of Piece 1 and of Piece 2.
\nAnswer: The rotational symmetry of the two pieces are 180\u00b0<\/p>\n
\nAnswer:
\nThere are 2 different ways piece 1 can fit in the puzzle.
\nIf we fit it in one of the two ways, we rotate it 180\u00b0 to get the second way.<\/p>\n
\nAnswer:
\nIf we connect piece 1 to piece 2 before putting them in the puzzle, there is only one way we can fit it in the puzzle.<\/p>\n
\nUSING STRUCTURE<\/strong>
\nA polar coordinate system locates a point in a plane by its distance from the origin O and by the measure of an angle with its vertex at the origin. For example, the point A(2, 30\u00b0) is 2 units from the origin and m\u2220XOA = 30\u00b0. What are the polar coordinates of the image of point A after a 90\u00b0 rotation? a 180\u00b0 rotation? a 270\u00b0 rotation? Explain.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-85.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-39.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-86.png\")
\nAnswer:
\nEach of the figures is a reflection to the other on a vertical line between them
\n\\(\\overline{V W}\\) \u2245 \\(\\overline{Q P}\\) and \u2220V \u2245 \u2220Q
\n\\(\\overline{W X}\\) \u2245 \\(\\overline{P T}\\) and \u2220W \u2245 \u2220P
\n\\(\\overline{X Y}\\) \u2245 \\(\\overline{T S}\\) and \u2220X \u2245 \u2220T
\n\\(\\overline{Y Z}\\) \u2245 \\(\\overline{R S}\\) and \u2220Y \u2245 \u2220S
\n\\(\\overline{Z V}\\) \u2245 \\(\\overline{R Q}\\) and \u2220Z \u2245 \u2220R
\nThus QPTSR \u2245 VWXYZ<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-87.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.3-Question-41.png\")
<\/p>\n<\/a>4.1 – 4.3 Quiz<\/h3>\n
\nimage alter the translation.<\/p>\n
\n(x, y) \u2192 (x + 4, y – 2)
\nAnswer:
\nGiven the vertices of the quadrilateral are A(- 4, 1), B(- 3, 3), C(0, 1), and D(- 2, 0)
\ntranslation: (x, y) \u2192 (x + 4, y – 2)
\nA(- 4, 1) \u2192 (-4 + 4, 1 – 2) \u2192 A'(0, -1)
\nB(- 3, 3) \u2192 (-3 + 4, 3 – 2) \u2192 B'(1, 1)
\nC(0, 1) \u2192 (0 + 4, 1 – 2) \u2192 C'(4, -1)
\nD(- 2, 0) \u2192 (-2 + 4, 0 – 2) \u2192 D'(2, -2)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-4th-Chapter-Transformations-img_61-300x224.jpg\")
<\/p>\n
\n(x, y) \u2192 (x – 1, y – 5)
\nAnswer:
\nGiven the vertices of the quadrilateral are A(- 4, 1), B(- 3, 3), C(0, 1), and D(- 2, 0)
\nTranslation: (x, y) \u2192 (x – 1, y – 5)
\nA(- 4, 1) \u2192 (-4 – 1, 1 – 5) \u2192 A'(-5, -4)
\nB(- 3, 3) \u2192 (-3 – 1, 3 – 5) \u2192 B'(-4, -2)
\nC(0, 1) \u2192 (0 – 1, 1 – 5) \u2192 C'(-1, -4)
\nD(- 2, 0) \u2192 (-2 – 1, 0 – 5) \u2192 D'(-3, -5)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_62-270x300.png\")
<\/p>\n
\n(x, y) \u2192 (x + 3, y + 6)
\nAnswer:
\nGiven the vertices of the quadrilateral are A(- 4, 1), B(- 3, 3), C(0, 1), and D(- 2, 0)
\nTranslation: (x, y) \u2192 (x + 3, y + 6)
\nA(- 4, 1) \u2192 (-4 + 3, 1 + 6) = A’ (-1, 7)
\nB(- 3, 3) \u2192 (-3 + 3, 3 + 6) = B'(0, 9)
\nC(0, 1) \u2192 (0 + 3, 1 + 6) = C'(3, 7)
\nD(- 2, 0) \u2192 (-2 + 3, 0 + 6) = D'(1, 6)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_63-300x275.png\")
<\/p>\n
\nA(- 5, 6), B(- 7, 8), c(- 3, 11); x – axis
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_64-280x300.png\")
\nNow we will apply reflection in the x-axis to the \u0394ABC.
\nA(-5, 6) \u2192 A'(-5, -6)
\nB(-7, 8) \u2192 B'(-7, -8)
\nC(-3, 11) \u2192 C'(-3, -11)
\nThe vertices are A'(-5, -6), B'(-7, -8) and C'(-3, -11)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_65-187x300.png\")
<\/p>\n
\nD(- 5, – 1), E(- 2, 1), F(- 1, – 3); y = x
\nAnswer:
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_66-300x247.png\")
\nApply reflection in the line y = x to the \u0394DEF
\nD(- 5, – 1) \u2192 D'(-1, -5)
\nE(- 2, 1) \u2192 E'(1, -2)
\nF(- 1, – 3) \u2192 F'(-3, -1)
\nD'(-1, -5), E'(1, -2) and F'(-3, -1).
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_69-300x262.png\")
<\/p>\n
\nJ(- 1, 4), K(2, 5), L(5, 2), M(4, – 1); x = 3
\nAnswer:
\nJ(- 1, 4), K(2, 5), L(5, 2), M(4, – 1); x = 3
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_70-300x267.png\")
\nJ(- 1, 4) \u2192 J'(7, 4)
\nK(2, 5) \u2192 K'(4, 5)
\nL(5, 2) \u2192 L'(1, 2)
\nM(4, – 1) \u2192 M'(2, -1)
\nJ'(7, 4), K'(4, 5), L'(1, 2) and M'(2, -1)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_71-300x235.png\")
<\/p>\n
\nP(2, – 4), Q(6, – 1), R(9, – 4), S(6, – 6); y = – 2
\nAnswer:
\nP(2, – 4), Q(6, – 1), R(9, – 4), S(6, – 6)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_72-300x189.png\")
\nP(2, – 4) \u2192 P'(2, 0)
\nQ(6, – 1) \u2192 Q'(6, -3)
\nR(9, – 4) \u2192 R'(9, 0)
\nS(6, – 6) \u2192 S'(6, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_73-300x228.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x, y + 6)
\nReflection: in the y – axis
\nAnswer:
\nA(2, – 1), B(5, 2), and C(8, – 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_74-300x193.png\")
\nTranslation: (x, y) \u2192 (x, y + 6)
\nA(2, -1) \u2192 (2, -1 + 6) = A'(2, 5)
\nB(5, 2) \u2192 (5, 2 + 6) = B'(5, 8)
\nC(8, – 2) \u2192 (8, -2 + 6) = C'(8, 4)
\nA'(2, 5), B'(5, 8) and C'(8, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_75-300x272.png\")
\nA'(2, 5) \u2192 A”(-2, 5)
\nB'(5, 8) \u2192 B”(-5, 8)
\nC'(8, 4) \u2192 C”(-8, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_76-300x198.png\")
<\/p>\n
\nTranslation: (x, y) \u2192 (x – 9, y)
\nReflection: in the line y = 1
\nAnswer:
\nA(2, – 1), B(5, 2), and C(8, – 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Answer-Key-Geometry-Chapter-4-Transformations-img_77-300x193.png\")
\nTranslation: (x, y) \u2192 (x – 9, y)
\nA(2, – 1) \u2192 (2 – 9, -1) = A'(-7, -1)
\nB(5, 2) \u2192 (5 – 9, 2) = B'(-4, 2)
\nC(8, – 2) \u2192 (8 – 9, -2) = C'(-1, -2)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-4th-Chapter-Transformations-img_77-300x124.png\")
\nApply reflection in the line y = 1 to the triangle A’B’C’
\nA'(-7, -1) \u2192 A”(-7, 3)
\nB'(-4, 2) \u2192 B”(-4, 0)
\nC'(-1, -2) \u2192 C”(-1, 4)
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_78-300x161.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-88.png\")
\nAnswer: The hexagon has 6 lines of symmetry.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-89.png\")
\nAnswer: The figure has no lines of symmetry. It cannot be reflected anyway.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-90.png\")
\nAnswer: The figure has two lines of symmetry.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-91.png\")
\nAnswer: The figure has only 1 line of symmetry.<\/p>\n
\n90\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-92.png\")
\nAnswer: ABC with the vertices are A(1, 1), B(2, 4) and C(4, 1)
\nA(1, 1) through an angle 90\u00b0 about the origin, A'(-1, 1)
\nB(2, 4) through an angle 90\u00b0 about the origin, B'(-4, 2)
\nC(4, 1) through an angle 90\u00b0 about the origin, C'(-1, 4)
\n![\"Bigideas](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Bigideas-Math-Answers-Geometry-Chapter-4-Transformations-img_79-300x192.png\")
<\/p>\n
\n270\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-93.png\")
\nAnswer:
\nRotate D(-3, 2) through an angle 270\u00b0 about the origin, we get D'(2, 3)
\nRotate E(-1, 4) through an angle 270\u00b0 about the origin, we get E'(4, 1)
\nRotate F(1, 2) through an angle 270\u00b0 about the origin, we get F'(2, -1)
\nRotate G(1, -1) through an angle 270\u00b0 about the origin, we get G'(-1, -1)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_80-300x205.png\")
<\/p>\n
\n180\u00b0
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-94.png\")
\nAnswer:
\nRotating 180\u00b0 (a, b) \u2192 (-a, -b)
\nH(-4, 1) \u2192 H'(4, -1)
\nI(-2, 2) \u2192 I'(2, -2)
\nJ(-1, -2) \u2192 J'(1, 2)
\nK(-4, -4) \u2192 K'(4, 4)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_81-300x273.jpg\")
<\/p>\n
\nthe composition.<\/p>\n
\nTranslation: (x, y) \u2192 (x – 4, y + 3)
\nRotation: 180\u00b0 about the origin
\nAnswer:
\nL(- 3, – 2), M (- 1, 1), and N(2, – 3)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-4th-Chapter-Transformations-img_81-300x217.png\")
\nTranslation: (x, y) \u2192 (x – 4, y + 3)
\nL(- 3, – 2) \u2192 (-3 – 4, -2 + 3) = L'(-7, 1)
\nM (- 1, 1) \u2192 (-1 – 4, 1 + 3) = M'(-5, 4)
\nN(2, – 3) \u2192 (2 – 4, -3 + 3) = N'(-2, 0)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_82-300x203.png\")
\nL'(-7, 1) through an angle 180\u00b0 about the origin, we will get L”(7, -1).
\nM'(-5, 4) through an angle 180\u00b0 about the origin, we will get M”(5, -4).
\nN'(-2, 0) through an angle 180\u00b0 about the origin, we will get N”(2, 0).
\nL”(7, -1), M”(5, -4) and N”(2, 0)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_83-300x171.png\")
<\/p>\n
\nRotation: 90\u00b0 about the origin
\nReflection: in the y-axis
\nAnswer:
\nL(- 3, – 2), M (- 1, 1), and N(2, – 3)
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answer-Key-Geometry-Chapter-4-Transformations-img_84-300x217.png\")
\nL(- 3, – 2) through an angle 90\u00b0 about the origin, we will get L'(2, -3)
\nM (- 1, 1) through an angle 90\u00b0 about the origin, we will get M'(-1, -1)
\nN(2, – 3) through an angle 90\u00b0 about the origin, we will get N'(3, 2)
\nL'(2, -3), M'(-1, -1) and N'(3, 2)
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Solution-Key-for-Geometry-Chapter-4-Transformations-img_86-300x200.png\")
\nL” which is in the same place on opposite sides y-axis with respect to the point L'(2, -3)
\nL”(-2, -3)
\nM'(-1, -1) = M”(1, -1)
\nN'(3, 2) = N”(-3, 2)
\nL”(-2, -3), M”(1, -1), and N”(-3, 2) will be vertices of the triangle L”M”N”<\/p>\n
\nThe figure shows a game in which the object is to create solid rows using the pieces given. Using only translations and rotations, describe the transformations for each piece at the top that will form two solid rows at the bottom.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answer-Key-Chapter-4-Transformations-95.png\")
\nAnswer:
\nFigure A can be moved on to its location A at the bottom of the chart by a rotation 90\u00b0 anticlockwise followed by a translation of 4 units horizontally towards the right and 6 units downwards.
\nFigure B can be moved on to its location B at the bottom of the chart by translation of 3 units horizontally towards the right and 7 units downwards.
\nFigure C can be moved on to its location C at the bottom of the chart by a rotation 180\u00b0 anticlockwise followed by a translation of 5 units horizontally towards the right and 7 units downwards.<\/p>\n<\/a>4.4 Congruence and Transformations<\/h3>\n
![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-96.png\")
<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-97.png\")
. Reflect \u2206 ABC in to form \u2206A’B’C’.
\nAnswer:<\/p>\n. Reflect \u2206A’B’C’ in the new line to form \u2206A”B”C”.
\nAnswer:<\/p>\n. What do you notice?
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nCONSTRUCTING VIABLE ARGUMENTS<\/strong>
\nTo be proficient in math, you need to make conjectures and justify your conclusions.
\nAnswer:<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-99.png\")
<\/p>\n. Reflect \u2206ABC in to form \u2206A’B’C’.
\nAnswer:<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-98.png\")
so that angle EDF is less than or equal to 90\u00b0. Reflect \u2206A’B’C’ in to form \u2206A”B”C”.
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nAnswer:<\/p>\n
\nWhat conjectures can you make about a figure reflected in two lines?
\nAnswer:<\/p>\n
\nPoint Q is reflected in two parallel lines, ![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-100.png\")
and ![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-101.png\")
. to form Q’ and The distance from to is 3.2 inches. What is the distance QQ”?
\nAnswer:<\/p>\n<\/a>Lesson 4.4 Congruence and Transformations<\/h3>\n
\nIdentify any congruent figures in the coordinate plane. Explain.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-102.png\")
\nAnswer:<\/p>\n
\nIn Example 2. describe another congruence transformation that maps \u25b1ABCD to \u25b1EFGH.
\nAnswer:
\nReflected over the x-axis and then translate 5 units left.<\/p>\n
\nDescribe a congruence transformation that maps \u25b3JKL to \u25b3MNP.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-103.png\")
\nAnswer:
\nJ(-1, 2)
\nK(-3, 4)
\nL(-4, 2)
\nM(4, -2)
\nN(2, -4)
\nP(1, -2)
\nReflect over the y-axis and translate 5 units right.<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-104.png\")
<\/p>\n
\nThe preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure to the green figure.
\nAnswer: Translation 3.2 cm to the right.<\/p>\n
\nWhat is the relationship between \\(\\overline{P P’}\\) and line k? Explain.
\nAnswer: \\(\\overline{P P’}\\) is perpendicular to line k by reflections in parallel lines.<\/p>\n
\nWhat is the distance between P and P”?
\nAnswer: 3.2 cm<\/p>\n
\nIn the diagram. the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure.
\nAnswer:\u00a0 160\u00b0 rotation about point P.<\/p>\n
\nA rotation of 76\u00b0 maps C to C’. To map C to C’ Using two reflections, what is the measure of the angle formed by the intersecting lines of reflection?
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-105.png\")
\nAnswer: 38\u00b0<\/p>\n<\/a>Exercise 4.4 Congruence and Transformations<\/h3>\n
\nCOMPLETE THE SENTENCE<\/strong>
\nTwo geometric figures are __________ if and only if there is a rigid motion or a composition of rigid motions that moves one of the figures onto the other.
\nAnswer:
\nTwo geometric figures are congruent if and only if there is a rigid motion or a composition of rigid motions that moves one of the figures onto the other.<\/p>\n
\nVOCABULARY<\/strong>
\nWhy is the term congruence transformation used to refer to a rigid motion?
\nAnswer: preimage and image are congruent.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-106.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-3.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-107.png\")
\nAnswer:
\nRectangle ABCD \u2245 Rectangle HJKL
\nBoth rectangle are 3 units length and 2 units width.
\nRectangle HJKL is the image Rectangle ABCD after translating the second 7 units right and 4 units Down.
\n\u0394MNP \u2245 \u0394TUV
\nBoth triangles are isosceles with 2 units base and 2 units altitude.
\n\u0394MNP is the image of \u0394TUV after rotating the second 90\u00b0
\nRotating 90\u00b0 is (a, b) \u2192 (-b, a)
\nT(2, 0) \u2192 M(0, 2)
\nU(3, 2) \u2192 N(-2, 3)
\nV(4, 0) \u2192 P(0, 4)
\n\u0394QRS \u2245 \u0394EFG
\nBoth triangles are right isosceles with 3 units length.
\n\u0394EFG is the image of \u0394QRS after rotating the second 180\u00b0
\nRotating 180\u00b0 is (a, b) \u2192 (-a, -b)
\nQ(5, 5) \u2192 E(-5, -5)
\nR(5, 2) \u2192 F(-5, -2)
\nS(2, 5) \u2192 G(-2, -5)<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-108.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-5.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-109.png\")
\nAnswer:
\nThe congruent transformation is 180\u00b0 rotation about the origin.
\n180\u00b0 rotation: (a, b) \u2192 (-a, -b)
\nP(5, -4) \u2192 W(-5, 4)
\nQ(3, -4) \u2192 X(-3, 4)
\nR(1, -2) \u2192 Y(-1, 2)
\nS(5, -2) \u2192 Z(-5, 2)
\nPQRS \u2245 WXYZ<\/p>\n
\nQ(2, 4), R(5, 4), S(4, 1) and T(6, 4), U(9, 4), V(8, 1)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-7.png\")
<\/p>\n
\nW(- 3, 1), X(2, 1), Y(4, -,4),,Z(- 5, – 4) and C(- 1, – 3) D(- 1, 2), E(4, 4), F(4, – 5)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Answers-Geometry-4th-Chapter-Transformations-img_58-300x298.png\")
\nWe have to check whether two polygons are congruent.
\nNow Rotate W(-3, 1) through an angle 90\u00b0 about the origin, we will get C(-1, -3)
\nNow Rotate Y(4, -4) through an angle 90\u00b0 about the origin, we will get E(4, 4)
\nNow Rotate X(2, 1) through an angle 90\u00b0 about the origin, we will get D(-1, 2)
\nNow Rotate Z(-5, -4) through an angle 90\u00b0 about the origin, we will get F(4, -5)
\nThese two polygons are congruent when we apply rotation.<\/p>\n
\nJ(1, 1), K(3, 2), L(4, 1) and M(6, 1), N(5, 2), P(2, 1)
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-9.png\")
<\/p>\n
\nA(0, 0), B(1, 2), C(4, 2), D(3, 0) and E(0, – 5), F( – 1, – 3), G(- 4, – 3), H(- 3, – 5)
\nAnswer:
\nA(0, 0), B(1, 2), C(4, 2), D(3, 0)
\n(x, y) \u2192 (x, y – 5)
\nA(0, 0) \u2192 A'(0, -5)
\nB(1, 2) \u2192 B'(1, -3)
\nC(4, 2) \u2192 C'(4, -3)
\nD(3, 0) \u2192 D'(3, -5)
\nThen reflecting the image A’B’C’D’ in the y-axis with the rule
\n(x, y) \u2192 (-x, y)
\nA'(0, -5) \u2192 A”(0, -5) \u2245 E
\nB'(1, -3) \u2192 B”(-1, -3) \u2245 F
\nC'(4, -3) \u2192 C”(-4, -3) \u2245 G
\nD'(3, -5) \u2192 D”(-3, -5) \u2245 H
\nSo, ABCD \u2245 EFGH
\n![\"BIM](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/BIM-Geometry-Answer-Key-for-Chapter-4-Transformations-img_59-300x271.png\")
<\/p>\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-110.png\")
<\/p>\n
\nA translation maps \u2206ABC onto which triangle?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-11.png\")
<\/p>\n
\nWhich lines are perpendicular to \\(\\overline{A A”}\\)?
\nAnswer: The lines k and m are perpendicular<\/p>\n
\nIf the distance between k and m is 2.6 inches. what is the length of \\(\\overline{C C”}\\)?
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-13.png\")
<\/p>\n
\nIs the distance from B’ to in the same as the distance from B” to m? Explain.
\nAnswer:
\nThe lines k and m are parallel.
\nThe triangle ABC is reflected in the line k.
\nThe triangle ABC is reflected in the line m.
\nLet us apply reflection in the line m to the triangle A’B’C’
\nPoint A’ which is in the same place on opposite sides from the line m with respect to the point A”
\nPoint B’ which is in the same place on opposite sides from the line m with respect to the point B”
\nPoint C’ which is in the same place on opposite sides from the line m with respect to the point C”
\nWhen we join point B and point B”, then we get \\(\\overline{B B”}\\)”
\nThe distance from B’ to m is the same as the distance from B” to m.<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-111.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-15.png\")
<\/p>\n
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-112.png\")
\nAnswer:
\nIn the intersection of the line m and the line n there is a point that we will mark with M.
\nWhen we join point A and point M, then we get \\(\\overline{A M}\\)
\nWhen we join point A” and point M, then we get \\(\\overline{A” M}\\)
\nNow observe at the angle \u2220AMA’
\n\u2220AMA” = 2 \u00d7 15 = 30<\/p>\n
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in describing the congruence transformation.
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-113.png\")
\nAnswer:
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Answers-Chapter-4-Transformations-4.4-Question-17.png\")
<\/p>\n
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in using the Reflections in Intersecting Lines Theorem
\n![\"Big](\"https:\/\/ccssmathanswers.com\/wp-content\/uploads\/2021\/02\/Big-Ideas-Math-Geometry-Solutions-Chapter-4-Transformations-114.png\")
\nAnswer:
\nWhen a figure is reflected in two intersecting lines, then the image is a rotation of the preimage about the point of intersection of the 2 reflecting lines through an angle between the intersecting lines.
\nThis means that the blue figure should be rotated about the point of intersection by
\n72\u00b0 \u00d7 2 = 144\u00b0 anticlockwise to obtain the image of given preimage.
\nThe solution shown here is incorrect because the reflection is done in the first line first, followed by the reflection in the second line, which is incorrect.<\/p>\n