{"id":50181,"date":"2022-04-06T13:42:56","date_gmt":"2022-04-06T08:12:56","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=50181"},"modified":"2022-04-07T11:11:19","modified_gmt":"2022-04-07T05:41:19","slug":"big-ideas-math-geometry-answers-chapter-10","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/big-ideas-math-geometry-answers-chapter-10\/","title":{"rendered":"Big Ideas Math Geometry Answers Chapter 10 Circles"},"content":{"rendered":"

Gain more subject knowledge on Circles concepts with the help of Big Ideas Math Geometry Answers Chapter 10 Circles<\/strong> guide. Shared guide of BIM Textbook Solutions Geometry Ch 10 Circles helps students understand the concepts quickly and concentrate on other math concepts too. Want to get a good grip on the high school-related circle topics? Then, make use of this BigIdeas Math Geometry Answers<\/a> Ch 10 Circles Pdf for free. Subject experts provided this BIM geometry 10th chapter solution key based on the latest syllabus and common core standards curriculum guidelines.<\/p>\n

Big Ideas Math Book Geometry Answer Key Chapter 10 Circles<\/h2>\n

Presented Geometry Chapter 10 Circles Big Ideas Math Answers are easy to understand and free to access at any time and anywhere. So, students are advised to access online or download the lesson-wise BIM Textbook Answers of Geometry Ch 10 Circles<\/strong> in Pdf<\/strong><\/em> format. Check out each and every lesson of circles as a part of your preparation and clear your queries within no time.<\/p>\n

Common Core Curriculum High School Big Ideas Math Book Geometry Ch 10 Solution Key<\/b> are prepared as per the latest textbooks. Questions included in the BIM Modeling Real Life Geometry Answers are from Exercises, Chapter Tests, Review Tests, Cumulative Practice, Assessment Tests, etc.<\/p>\n

Circles Maintaining Mathematical Proficiency<\/h3>\n

Find the Product.<\/p>\n

Question 1.
\n(x + 7) (x + 4)<\/p>\n

Answer:
\n(x + 7) (x + 4) = x\u00b2 + 14x + 28<\/p>\n

Explanation:
\n(x + 7) (x + 4) = x(x + 7) + 7(x + 4)
\n= x\u00b2 + 7x + 7x + 28
\n= x\u00b2 + 14x + 28<\/p>\n

Question 2.
\n(a + 1) (a – 5)<\/p>\n

Answer:
\n(a + 1) (a – 5) = a\u00b2 – 4a – 5<\/p>\n

Explanation:
\n(a + 1) (a – 5) = a(a – 5) + 1(a – 5)
\n= a\u00b2 – 5a + a – 5
\n= a\u00b2 – 4a – 5<\/p>\n

Question 3.
\n(q – 9) (3q – 4)<\/p>\n

Answer:
\n(q – 9) (3q – 4) = 3q\u00b2 – 31q + 32<\/p>\n

Explanation:
\n(q – 9) (3q – 4) = q(3q – 4) – 9(3q – 4)Exercise 10.3 Using Chords<\/p>\n

= 3q\u00b2 – 4q – 27q + 32
\n= 3q\u00b2 – 31q + 32<\/p>\n

Question 4.
\n(2v – 7) (5v + 1)<\/p>\n

Answer:
\n(2v – 7) (5v + 1) = 10v\u00b2 – 33v – 7<\/p>\n

Explanation:
\n(2v – 7) (5v + 1) = 2v(5v + 1)- 7(5v + 1)
\n= 10v\u00b2 + 2v – 35v – 7
\n= 10v\u00b2 – 33v – 7<\/p>\n

Question 5.
\n(4h + 3) (2 + h)<\/p>\n

Answer:
\n(4h + 3) (2 + h) = 4h\u00b2 + 11h + 6<\/p>\n

Explanation:
\n(4h + 3) (2 + h) = 4h(2 + h) + 3(2 + h)
\n= 8h + 4h\u00b2 + 6 + 3h
\n= 4h\u00b2 + 11h + 6<\/p>\n

Question 6.
\n(8 – 6b) (5 – 3b)<\/p>\n

Answer:
\n(8 – 6b) (5 – 3b) = 18b\u00b2 – 54b + 40<\/p>\n

Explanation:
\n(8 – 6b) (5 – 3b) = 8(5 – 3b) – 6b(5 – 3b)
\n= 40 – 24b – 30b + 18b\u00b2
\n= 18b\u00b2 – 54b + 40<\/p>\n

Solve the equation by completing the square. Round your answer to the nearest hundredth, if necessary.<\/p>\n

Question 7.
\nx2<\/sup> – 2x = 5<\/p>\n

Answer:
\nThe solutions are x = \u221a6 + 1, x = 1 – \u221a6<\/p>\n

Explanation:
\nx\u00b2 – 2x = 5
\nx\u00b2 – 2x + 1\u00b2 = 5 + 1\u00b2
\n(x – 1)\u00b2 = 6
\nx – 1 = \u00b1\u221a6
\nx = \u00b1\u221a6 + 1
\nThe solutions are x = \u221a6 + 1, x = -\u221a6 + 1<\/p>\n

Question 8.
\nr2<\/sup> + 10r = -7<\/p>\n

Answer:
\nThe solutions are r = \u221a18 – 5, r = 5 – \u221a18<\/p>\n

Explanation:
\nr2<\/sup> + 10r = -7
\nr\u00b2 + 10r + 5\u00b2 = -7 + 5\u00b2
\n(r + 5)\u00b2 = -7 + 25 = 18
\nr + 5 = \u00b1\u221a18
\nr = \u00b1\u221a18 – 5
\nThe solutions are r = \u221a18 – 5, r = 5 – \u221a18<\/p>\n

Question 9.
\nw2<\/sup> – 8w = 9<\/p>\n

Answer:
\nThe solutions are w = 9, w = -1<\/p>\n

Explanation:
\nw2<\/sup> – 8w = 9
\nw2<\/sup> – 8w + 4\u00b2 = 9 + 4\u00b2
\n(w – 4)\u00b2 = 9 + 16 = 25
\nw – 4 = \u00b15
\nw = 5 + 4, w = -5 + 4
\nw = 9, w = -1
\nThe solutions are w = 9, w = -1<\/p>\n

Question 10.
\np2<\/sup> + 10p – 4 = 0<\/p>\n

Answer:
\nThe solutions are p = \u221a29 – 5, p = 5 – \u221a29<\/p>\n

Explanation:
\np2<\/sup>\u00a0+ 10p = 4
\np\u00b2 + 10p + 5\u00b2 = 4 + 5\u00b2
\n(p + 5)\u00b2 = 4 + 25
\n(p + 5)\u00b2 = 29
\np + 5 = \u00b1\u221a29
\np = \u00b1\u221a29 – 5
\nThe solutions are p = \u221a29 – 5, p = 5 – \u221a29<\/p>\n

Question 11.
\nk2<\/sup> – 4k – 7 = 0<\/p>\n

Answer:
\nThe solutions are k = \u221a11 + 2, k = 2 – \u221a11<\/p>\n

Explanation:
\nk\u00b2 – 4k= 7
\nk\u00b2 – 4k + 2\u00b2 = 7 + 4
\n(k – 2)\u00b2 = 11
\nk – 2 = \u00b1\u221a11
\nk = \u221a11 + 2, k = 2 – \u221a11
\nThe solutions are k = \u221a11 + 2, k = 2 – \u221a11<\/p>\n

Question 12.
\n– z2<\/sup> + 2z = 1<\/p>\n

Answer:
\nThe solutions are z = 1<\/p>\n

Explanation:
\n-z\u00b2 + 2z = 1
\nz\u00b2 – 2z = -1
\nz\u00b2 – 2z + 1 = -1 + 1
\n(z – 1)\u00b2 = 0
\nz = 1
\nThe solutions are z = 1<\/p>\n

Question 13.
\nABSTRACT REASONING<\/strong>
\nwrite an expression that represents the product of two consecutive positive odd integers. Explain your reasoning.<\/p>\n

Answer:
\nLet us take two consecutive odd integers are x and (x + 2)
\nThe product of two consecutive odd integers is x \u2022 (x + 2)
\nx(x + 2) = x\u00b2 + 2x<\/p>\n

Circles Mathematical Practices<\/strong><\/p>\n

Monitoring progress<\/p>\n

Let \u2299A, \u2299B, and \u2299C consist of points that are 3 units from the centers.<\/p>\n

\"Circles<\/p>\n

Question 1.
\nDraw \u2299C so that it passes through points A and B in the figure at the right. Explain your reasoning.<\/p>\n

Answer:
\n\"Big<\/p>\n

Question 2.
\nDraw \u2299A, \u2299B, and OC so that each is tangent to the other two. Draw a larger circle, \u2299D, that is tangent to each of the other three circles. Is the distance from point D to a point on \u2299D less than, greater than, or equal to 6? Explain.<\/p>\n

Answer:<\/p>\n

10.1 Lines and Segments that Intersect Circles<\/h3>\n

Exploration 1<\/strong><\/p>\n

Lines and Line Segments That Intersect Circles<\/p>\n

\"Big<\/p>\n

Work with a partner: The drawing at the right shows five lines or segments that intersect a circle. Use the relationships shown to write a definition for each type of line or segment. Then use the Internet or some other resource to verify your definitions.
\nChord: _________________
\nSecant: _________________
\nTangent: _________________
\nRadius: _________________
\nDiameter: _________________<\/p>\n

Answer:
\nChord: A chord of a circle is a straight line segment whose endpoints both lie on a circular arc.
\nSecant: A straight line that intersects a circle in two points is called a secant line.
\nTangent: Tangent line is a line that intersects a curved line at exactly one point.
\nRadius: It is the distance from the centre of the circle to any point on the circle.
\nDiameter: It the straight that joins two points on the circle and passes through the centre of the circle.<\/p>\n

Exploration 2<\/strong><\/p>\n

Using String to Draw a Circle<\/p>\n

Work with a partner: Use two pencils, a piece of string, and a piece of paper.<\/p>\n

a. Tie the two ends of the piece of string loosely around the two pencils.
\nAnswer:<\/p>\n

b. Anchor one pencil of the paper at the center of the circle. Use the other pencil to draw a circle around the anchor point while using slight pressure to keep the string taut. Do not let the string wind around either pencil.
\n\"Big
\nAnswer:<\/p>\n

c. Explain how the distance between the two pencil points as you draw the circle is related to two of the lines or line segments you defined in Exploration 1.
\nREASONING ABSTRACTLY<\/strong>
\nTo be proficient in math, you need to know and flexibly use different properties of operations and objects.
\nAnswer:<\/p>\n

Communicate Your Answer<\/p>\n

Question 3.
\nWhat are the definitions of the lines and segments that intersect a circle?
\nAnswer:<\/p>\n

Question 4.
\nOf the five types of lines and segments in Exploration 1, which one is a subset of another? Explain.
\nAnswer:<\/p>\n

Question 5.
\nExplain how to draw a circle with a diameter of 8 inches.
\nAnswer:<\/p>\n

Lesson 10.1 Lines and Segments that Intersect Circles<\/h3>\n

Monitoring progress<\/strong><\/p>\n

Question 1.
\nIn Example 1, What word best describes \\(\\overline{A G}\\)? \\(\\overline{C B}\\)?<\/p>\n

Answer:
\n\\(\\overline{A G}\\) is secant because it is a line that intersects the circle at two points.
\n\\(\\overline{C B}\\) is the radius as it is the distance from the centre to the point of a circle.<\/p>\n

Question 2.
\nIn Example 1, name a tangent and a tangent segment.<\/p>\n

Answer:
\n\\(\\overline{D E}\\) is the tangent of the circle
\n\\(\\overline{D E}\\) is the tangent segment of the circle.<\/p>\n

Tell how many common tangents the circles have and draw them. State whether the tangents are external tangents or internal tangents.<\/p>\n

Question 3.
\n\"Big<\/p>\n

Answer:
\n4 tangents.
\n\"Big
\nA tangent is a line segment that intersects the circle at exactly one point. Internal tangents are the lines that intersect the segments joining the centres of two circles. External tangents are the lines that do not cross the segment joining the centres of the circles.
\nBlue lines represent the external tangents and red lines represent the internal tangents.<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nOne tangent.
\n\"Big
\nOne external tangent.<\/p>\n

Question 5.
\n\"Big<\/p>\n

Answer:
\nNo tangent.
\nIt is not possible to draw a common tangent for this type of circle.<\/p>\n

Question 6.
\nIs \\(\\overline{D E}\\) tangent to \u2299C?
\n\"Big<\/p>\n

Answer:
\nUse the converse of Pythagorean theorem i.e 2\u00b2 = 3\u00b2 + 4\u00b2
\n4 = 9 + 16
\nBy the tangent line to the circle theorem, \\(\\overline{D E}\\) is not a tangent to \u2299C<\/p>\n

Question 7.
\n\\(\\overline{S T}\\) is tangent to \u2299Q.
\nFind the radius of \u2299Q.
\n\"Big<\/p>\n

Answer:
\nThe radius of \u2299Q is 7 units.<\/p>\n

Explanation:
\nBy using the Pythagorean theorem
\n(18 + r)\u00b2 = r\u00b2 + 24\u00b2
\n324 + 36r + r\u00b2 = r\u00b2 + 576
\n36r = 576 – 324
\n36r = 252
\nr = 7<\/p>\n

Question 8.
\nPoints M and N are points of tangency. Find the value(s) of x.
\n\"Big<\/p>\n

Answer:
\nThe values of x are 3 or -3.<\/p>\n

Explanation:
\nx\u00b2 = 9
\nx = \u00b13<\/p>\n

Exercise 10.1 Lines and Segments that Intersect Circles<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nWRITING<\/strong>
\nHow are chords and secants alike? How are they different?
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWRITING<\/strong>
\nExplain how you can determine from the context whether the words radius and diameter are referring to segments or lengths.<\/p>\n

Answer:
\nRadius and diameter are the lengths of the line segments that pass through the centre of a circle. Radius is half of the diameter.<\/p>\n

Question 3.
\nCOMPLETE THE SENTENCE<\/strong>
\nCoplanar circles that have a common center are called ____________ .
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\nWHICH ONE DOESNT BELONG?<\/strong>
\nWhich segment does not belong with the other three? Explain your reasoning.
\nchord radius tangent diameter<\/p>\n

Answer:
\nA chord, a radius and a diamter are segments and they intersect a circle in two points. A tangent is a line that intersects a circle at one point.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 5 – 10, use the diagram.<\/p>\n

\"Big<\/p>\n

Question 5.
\nName the circle.
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\nName two radii.<\/p>\n

Answer:
\nThe name of the two radii is CD and AC.<\/p>\n

Question 7.
\nName two chords.
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\nName a diameter.<\/p>\n

Answer:
\nThe name of diameter is AD<\/p>\n

Question 9.
\nName a secant.
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\nName a tangent and a point of tangency<\/p>\n

Answer:
\nGE is the tangent and F is the point of tangency.<\/p>\n

In Exercises 11 – 14, copy the diagram. Tell how many common tangents the circles have and draw them.<\/p>\n

Question 11.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\n\"Big<\/p>\n

Answer:
\nNo common tangent because two circles do not intersect at one point.<\/p>\n

Question 13.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\"Big<\/p>\n

Answer:
\nOne common tangent.
\n\"Big<\/p>\n

In Exercises 15 – 18, tell whether the common tangent is internal or external.<\/p>\n

Question 15.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\n\"Big<\/p>\n

Answer:
\nThe common tangent is the internal tangent because it intersects the segment that joins the centres of two circles.<\/p>\n

Question 17.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\n\"Big
\nAnswer:
\nThe common tangent is the internal tangent because it intersects the segment that joins the centres of two circles.<\/p>\n

In Exercises 19 – 22, tell whether \\(\\overline{A B}\\) is tangent to \u2299C. Explain your reasoning.<\/p>\n

Question 19.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\n\"Big<\/p>\n

Answer:
\nUse the converse of the Pythagorean theorem
\n18\u00b2 _____________ 15\u00b2 + 9\u00b2
\n324 _____________ 225 + 81
\n324 \u2260 304
\n\u25b3 ACB is not a right angled triangle.
\nSo, \\(\\overline{A B}\\) is not tangent to \u2299C at B.<\/p>\n

Question 21.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\n\"Big<\/p>\n

Answer:
\nUse the converse of the Pythagorean theorem
\n8\u00b2 _____________ 12\u00b2 + 16\u00b2
\n64 _____________ 144 + 256
\n64 \u2260 400
\n\u25b3 ACB is not a right angled triangle.
\nSo, \\(\\overline{A B}\\) is not tangent to \u2299C at B.<\/p>\n

In Exercises 23 – 26, point B is a point of tangency. Find the radius r of \u2299C.<\/p>\n

Question 23.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\n\"Big<\/p>\n

Answer:
\n(r + 6)\u00b2 = r\u00b2 + 9\u00b2
\nr\u00b2 + 12r + 36 = r\u00b2 + 81
\n12r = 81 – 36
\n12r = 45
\nr = \\(\\frac { 15 }{ 4 } \\)
\nTherefore, the radius of \u2299C is \\(\\frac { 15 }{ 4 } \\)<\/p>\n

Question 25.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\n\"Big<\/p>\n

Answer:
\n(r + 18)\u00b2 = r\u00b2 + 30\u00b2
\nr\u00b2 + 36r + 324 = r\u00b2 + 900
\n36r = 900 – 324
\n36r = 576
\nr = 16
\nTherefore, the radius of \u2299C is 16<\/p>\n

CONSTRUCTION<\/strong>
\nIn Exercises 27 and 28. construct \u2299C with the given radius and point A outside of \u2299C. Then construct a line tangent to \u2299C that passes through A.<\/p>\n

Question 27.
\nr = 2 in.
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nr = 4.5 cm<\/p>\n

Answer:
\n\"Big<\/p>\n

In Exercises 29 – 32, points B and D are points of tangency. Find the value(s) of x.<\/p>\n

Question 29.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\n\"Big<\/p>\n

Answer:
\n3x + 10 = 7x – 6
\n7x – 3x = 10 + 6
\n4x = 16
\nx = 4<\/p>\n

Question 31.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 32.
\n\"Big<\/p>\n

Answer:
\n2x + 5 = 3x\u00b2 + 2x – 7
\n3x\u00b2 = 5 + 7
\n3x\u00b2 = 12
\nx\u00b2 = 4
\nx = \u00b12<\/p>\n

Question 33.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in determining whether \\(\\overline{X Y}\\) is tangent to \u2299Z.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 34.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in finding the radius of \u2299T.
\n\"Big<\/p>\n

Answer:
\n39\u00b2 = 36\u00b2 + 15\u00b2
\nSo, 15 is the diameter.
\nThe radius is \\(\\frac { 15 }{ 2 } \\).<\/p>\n

Question 35.
\nABSTRACT REASONING<\/strong>
\nFor a point outside of a circle, how many lines exist tangent to the circle that pass through the point? How many such lines exist for a point on the circle? inside the circle? Explain your reasoning.
\nAnswer:
\n\"Big<\/p>\n

Question 36.
\nCRITICAL THINKING<\/strong>
\nWhen will two lines tangent to the same circle not intersect? Justify your answer.<\/p>\n

Answer:
\n\"Big
\nUsing tangent line to circle theorem, it follow that the angle between tangent and radius is a right angle. Let’s draw these tangents at the two ends of the same diameter. We can observe a diameter AD like a transverzal of these tangents.<\/p>\n

Question 37.
\nUSING STRUCTURE<\/strong>
\nEach side of quadrilateral TVWX is tangent to \u2299Y. Find the perimeter of the quadrilateral.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 38.
\nLOGIC<\/strong>
\nIn \u2299C, radii \\(\\overline{C A}\\) and \\(\\overline{C B}\\) are perpendicular. \"Big are tangent to \u2299C.<\/p>\n

a. Sketch \u2299C, \\(\\overline{C A}\\), \\(\\overline{C B}\\), \"Big.
\nAnswer:<\/p>\n

b. What type of quadrilateral is CADB? Explain your reasoning.
\nAnswer:<\/p>\n

Question 39.
\nMAKING AN ARGUMENT<\/strong>
\nTwo hike paths are tangent to an approximately circular pond. Your class is building a nature trail that begins at the intersection B of the bike paths and runs between the bike paths and over a bridge through the center P of the pond. Your classmate uses the Converse of the Angle Bisector Theorem (Theorem 6.4) to conclude that the trail must bisect the angle formed by the bike paths. Is your classmate correct? Explain your reasoning.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 40.
\nMODELING WITH MATHEMATICS<\/strong>
\nA bicycle chain is pulled tightly so that \\(\\overline{M N}\\) is a common tangent of the gears. Find the distance between the centers of the gears.
\n\"Big<\/p>\n

Answer:
\nheight h = 4.3 – 1.8
\nh = 2.5
\nx\u00b2 = MN\u00b2 + h\u00b2
\nx\u00b2 = 17.6\u00b2 + 2.5\u00b2
\nx\u00b2 = 316.01
\nx = 17.8
\nTherefore, the distance between the centre of the gear is 17.8 in.<\/p>\n

Question 41.
\nWRITING<\/strong>
\nExplain why the diameter of a circle is the longest chord of the circle.
\nAnswer:
\n\"Big<\/p>\n

Question 42.
\nHOW DO YOU SEE IT?<\/strong>
\nIn the figure, \\(\\vec{P}\\)A is tangent to the dime. \\(\\vec{P}\\)C is tangent to the quarter, and \\(\\vec{P}\\)B is a common internal tangent. How do you know that \\(\\overline{P A} \\cong \\overline{P B} \\cong \\overline{P C}\\)
\n\"Big
\nAnswer:<\/p>\n

Question 43.
\nPROOF<\/strong>
\nIn the diagram, \\(\\overline{R S}\\) is a common internal tangent to \u2299A and \u2299B. Prove that \\(\\frac{\\Lambda C}{B C}=\\frac{R C}{S C}\\)
\n\"Big
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 44.
\nTHOUGHT PROVOKING<\/strong>
\nA polygon is circumscribed about a circle when every side of the polygon is tangent to the circle. In the diagram. quadrilateral ABCD is circumscribed about \u2299Q. Is it always true that AB + CD = AD + BC? Justify your answer.
\n\"Big
\nAnswer:<\/p>\n

Question 45.
\nMATHEMATICAL CONNECTIONS<\/strong>
\nFind the values of x and y. Justify your answer.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 46.
\nPROVING A THEOREM<\/strong>
\nProve the External Tangent Congruence Theorem (Theorem 10.2).
\n\"Big
\nGiven \\(\\overline{S R}\\) and \\(\\overline{S T}\\) are tangent to \u2299P.
\nProve \\(\\overline{S R} \\cong \\overline{S T}\\)<\/p>\n

Answer:
\n\"Big
\n\u2220PRS and \u2220PTS are the right angles. So the legs of circles are congruent.
\nTherefore, \\(\\overline{S R} \\cong \\overline{S T}\\)<\/p>\n

Question 47.
\nPROVING A THEOREM<\/strong>
\nUse the diagram to prove each part of the biconditional in the Tangent Line to Circle Theorem (Theorem 10.1 ).
\n\"Big
\na. Prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. (Hint: If you assume line m is not perpendicular to \\(\\overline{Q P}\\), then the perpendicular segment from point Q to line m must intersect line m at some other point R.)
\nGhen Line m is tangent to \u2299Q at point P.
\nProve m \u22a5 \\(\\overline{Q P}\\)
\nb. Prove indirectly that if a line is perpendicular to a radius at its endpoint, then the line is tangent to the circle.
\nGien m \u22a5 \\(\\overline{Q P}\\)
\nProve Line m is tangent to \u2299Q.
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 48.
\nREASONING<\/strong>
\nIn the diagram, AB = AC = 12, BC = 8, and all three segments are Langent to \u2299P. What is the radius of \u2299P? Justify your answer.
\n\"Big<\/p>\n

Answer:<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Find the indicated measure.<\/p>\n

Question 49.
\nm\u2220JKM
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 50.
\nAB
\n\"Big<\/p>\n

Answer:
\nAC = AB + BC
\n10 = AB + 7
\nAB = 10 – 7
\nAB = 3<\/p>\n

10.2 Finding Arc Measures<\/h3>\n

Exploration 1<\/strong><\/p>\n

Measuring Circular Arcs<\/p>\n

Work with a partner: Use dynamic geometry software to find the measure of \\(\\widehat{B C}\\). Verify your answers using trigonometry.<\/p>\n

a.
\n\"Big
\nPoints
\nA(0, 0)
\nB(5, 0)
\nC(4, 3)<\/p>\n

Answer:
\n30 degrees<\/p>\n

b.
\n\"Big
\nPoints
\nA(0, 0)
\nB(5, 0)
\nC(3, 4)<\/p>\n

Answer:
\n60 degrees<\/p>\n

c.
\n\"Big
\nPoints
\nA(0, 0)
\nB(4, 3)
\nC(3, 4)<\/p>\n

Answer:
\n15 degrees<\/p>\n

d.
\n\"Big
\nPoints
\nA(0, 0)
\nB(4, 3)
\nC(- 4, 3)<\/p>\n

Answer:
\n90 degrees<\/p>\n

Communicate Your Answer<\/p>\n

Question 2.
\nHow are circular arcs measured?
\nAnswer:<\/p>\n

Question 3.
\nUse dynamic geometry software to draw a circular arc with the given measure.
\nUSING TOOLS STRATEGICALLY<\/strong>
\nTo be proficient in math, you need to use technological tools to explore and deepen your understanding of concepts.
\na. 30\u00b0<\/p>\n

Answer:
\n\"Big<\/p>\n

b. 45\u00b0<\/p>\n

Answer:
\n\"Big<\/p>\n

c. 60\u00b0
\nAnswer:
\n\"Big<\/p>\n

d. 90\u00b0<\/p>\n

Answer:
\n\"Big<\/p>\n

Lesson 10.2 Finding Arc Measures<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

Identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.<\/p>\n

\"Big<\/p>\n

Question 1.
\n\\(\\widehat{T Q}\\)<\/p>\n

Answer:
\n\\(\\widehat{T Q}\\) is a minor arc.
\n\\(\\widehat{T Q}\\) = 120\u00b0<\/p>\n

Question 2.
\n\\(\\widehat{Q R T}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q R T}\\)<\/p>\n

Question 3.
\n\\(\\widehat{T Q R}\\) is a major arc.
\n\\(\\widehat{Q R T}\\) = QR + RS + ST
\nRS = 360\u00b0 – (60 + 120 + 80)
\n= 360 – 260 = 100\u00b0
\nSo, \\(\\widehat{Q R T}\\) = 60\u00b0 + 100\u00b0 + 80\u00b0
\n\\(\\widehat{Q R T}\\) = 240\u00b0<\/p>\n

Answer:<\/p>\n

Question 4.
\n\\(\\widehat{Q S}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q S}\\) = QR + RS
\n= 60 + 100 = 160\u00b0
\nTherefore, \\(\\widehat{Q S}\\) = 160\u00b0 and it is a minor arc.<\/p>\n

Question 5.
\n\\(\\widehat{T S}\\)<\/p>\n

Answer:
\n\\(\\widehat{T S}\\) = 80\u00b0 and it is a minor arc.<\/p>\n

Question 6.
\n\\(\\widehat{R S T}\\)<\/p>\n

Answer:
\n\\(\\widehat{R S T}\\) = 100 + 80 = 180
\nTherefore, \\(\\widehat{R S T}\\) = 180\u00b0 and it is a minor arc.<\/p>\n

Tell whether the red arcs are congruent. Explain why or why not.<\/p>\n

Question 7.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{A B}\\), \\(\\widehat{C D}\\) are congruent as they measure same radius and same arc length.<\/p>\n

Question 8.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{M N}\\), \\(\\widehat{P Q}\\) are not congruent as they measure different radius.<\/p>\n

Exercise 10.2 Finding Arc Measures<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nVOCABULARY<\/strong>
\nCopy and complele:
\nIf \u2220ACB and \u2220DCE are congruent central angles of \u2299C, then \\(\\widehat{A B}\\) and \\(\\widehat{D E}\\) arc.
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWHICH ONE DOESNT BELONG?<\/strong>
\nWhich circle does not belong with the other three? Explain your reasoning.
\n\"Big<\/p>\n

Answer:
\nWe know that 1 ft = 12 in
\nSo, the fourth circle does not belong to the other three as its diameter is different.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 6, name the red minor arc and find its measure. Then name the blue major arc and find its measure.<\/p>\n

Question 3.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nThe minor arc \\(\\widehat{E F}\\) = 68\u00b0 and major arc \\(\\widehat{F G E}\\) = 360\u00b0 – 68\u00b0 = 292\u00b0.<\/p>\n

Question 5.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\n\"Big<\/p>\n

Answer:
\nThe minor arc is \\(\\widehat{M N}\\) = 170\u00b0, major arc \\(\\widehat{N P M}\\) = 360\u00b0 – 170\u00b0 = 190\u00b0.<\/p>\n

In Exercises 7 – 14. identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.<\/p>\n

\"Big<\/p>\n

Question 7.
\n\\(\\widehat{B C}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\n\\(\\widehat{D C}\\)<\/p>\n

Answer:
\n\\(\\widehat{D C}\\) is a minor arc and it has a measure of 65\u00b0.<\/p>\n

Question 9.
\n\\(\\widehat{E D}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\n\\(\\widehat{A E}\\)<\/p>\n

Answer:
\n\\(\\widehat{A E}\\) is a minor arc and it has a measure of 70\u00b0.<\/p>\n

Question 11.
\n\\(\\widehat{E A B}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\n\\(\\widehat{A B C}\\)<\/p>\n

Answer:
\n\\(\\widehat{A B C}\\) is a semicircle and it has a measure of 180\u00b0.<\/p>\n

Question 13.
\n\\(\\widehat{B A C}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\\(\\widehat{E B D}\\)<\/p>\n

Answer:
\n\\(\\widehat{E B D}\\) is a major arc and it has a measure of 315\u00b0.<\/p>\n

In Exercises 15 and 16, find the measure of each arc.<\/p>\n

Question 15.
\n\"Big<\/p>\n

a. \\(\\widehat{J L}\\)<\/p>\n

b. \\(\\widehat{K M}\\)<\/p>\n

c. \\(\\widehat{J L M}\\)<\/p>\n

d. \\(\\widehat{J M}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\n\"Big<\/p>\n

a. \\(\\widehat{R S}\\)<\/p>\n

Answer:
\n\\(\\widehat{R S}\\) = \\(\\widehat{Q R S}\\) – \\(\\widehat{Q R}\\)
\n= 180 – 42
\n= 138\u00b0
\nSo, \\(\\widehat{R S}\\) = 138\u00b0<\/p>\n

b. \\(\\widehat{Q R S}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q R S}\\) = 180\u00b0<\/p>\n

c. \\(\\widehat{Q S T}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q S T}\\) = \\(\\widehat{Q R S}\\) + \\(\\widehat{S T}\\)
\n= 180 + 42 = 222
\nSo, \\(\\widehat{Q S T}\\) = 222\u00b0<\/p>\n

d. \\(\\widehat{Q T}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q T}\\) = 360 – (42 + 138 + 42)
\n= 360 – (222)
\n= 138\u00b0
\n\\(\\widehat{Q T}\\) = 138\u00b0<\/p>\n

Question 17.
\nMODELING WITH MATHEMATICS<\/strong>
\nA recent survey asked high school students their favorite type of music. The results are shown in the circle graph. Find each indicated arc measure.
\n\"Big
\na. m\\(\\widehat{A E}\\)
\nb. m\\(\\widehat{A C E}\\)
\nc. m\\(\\widehat{G D C}\\)
\nd. m\\(\\widehat{B H C}\\)
\ne. m\\(\\widehat{F D}\\)
\nf. m\\(\\widehat{F B D}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nABSTRACT REASONING<\/strong>
\nThe circle graph shows the percentages of students enrolled in fall Sports at a high school. Is it possible to find the measure of each minor arc? If so, find the measure 0f the arc for each category shown. If not, explain why it is not possible.
\n\"Big<\/p>\n

Answer:
\nSoccer angle = 30% of 360 = 108\u00b0
\nVolleyball angle = 15% of 360 = 54\u00b0
\nCross-country angle = 20% of 360 = 72\u00b0
\nNone angle = 15% of 360 = 54\u00b0
\nFootball angle = 20% of 360 = 72\u00b0<\/p>\n

In Exercises 19 – 22, tell whether the red arcs are congruent. Explain why or why not.<\/p>\n

Question 19.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{L P}\\) and \\(\\widehat{M N}\\) are not congruet because they are not in the same circle.<\/p>\n

Question 21.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{R S Q}\\), \\(\\widehat{F G H}\\) are not congruent because those two circles have different radii.<\/p>\n

MATHEMATICAL CONNECTIONS<\/strong>
\nIn Exercises 23 and 24. find the value of x. Then find the measure of the red arc.<\/p>\n

Question 23.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\n\"Big<\/p>\n

Answer:
\n4x + 6x + 7x + 7x = 360
\n24x = 360
\nx = 15
\nm\\(\\widehat{R S T}\\) = 6(15) + 7(15)
\n= 90 + 105 = 195\u00b0
\nSo, m\\(\\widehat{R S T}\\) = 195\u00b0<\/p>\n

Question 25.
\nMAKING AN ARGUMENT<\/strong>
\nYour friend claims that any two arcs with the same measure are similar. Your cousin claims that an two arcs with the same measure are congruent. Who is correct? Explain.
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nMAKING AN ARGUMENT<\/strong>
\nYour friend claims that there is not enough information given to find the value of x. Is your friend correct? Explain your reasoning.
\n\"Big<\/p>\n

Answer:
\nMy friend is wrong.
\n4x + x + x + 4x = 360
\n10x = 360
\nx = 36\u00b0<\/p>\n

Question 27.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in naming the red arc.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in naming congruent arc.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{J K}\\), \\(\\widehat{N P}\\) are not congruent because those two arcs are from different circles.<\/p>\n

Question 29.
\nATTENDING TO PRECISION<\/strong>
\nTwo diameters of \u2299P are \\(\\widehat{A B}\\) and \\(\\widehat{C D}\\). Find m\\(\\widehat{A C D}\\) and m\\(\\widehat{A C}\\) when m\\(\\widehat{A D}\\) = 20\u00b0.
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\nREASONING<\/strong>
\nIn \u2299R, m\\(\\widehat{A B}\\) = 60\u00b0, m\\(\\widehat{B C}\\) = 25\u00b0. m\\(\\widehat{C D}\\) = 70\u00b0, and m\\(\\widehat{D E}\\) = 20\u00b0. Find two possible measures of \\(\\widehat{A E}\\).<\/p>\n

Answer:
\n\\(\\widehat{A E}\\) = 360 – (\\(\\widehat{A B}\\) + \\(\\widehat{B C}\\) + \\(\\widehat{C D}\\) + \\(\\widehat{D E}\\))
\n= 360 – (60 + 25 + 70 + 20)
\n= 360 – (175)
\n= 185
\n\\(\\widehat{A E}\\) = \\(\\widehat{A B}\\) + \\(\\widehat{B C}\\) + \\(\\widehat{C D}\\) + \\(\\widehat{D E}\\)
\n= 60 + 25 + 70 + 20 = 175
\nSo, the two possibilities of \\(\\widehat{A E}\\) are 185\u00b0, 175\u00b0<\/p>\n

Question 31.
\nMODELING WITH MATHEMATICS<\/strong>
\nOn a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle?
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 32.
\nMODELING WITH MATHEMATICS<\/strong>
\nYou can use the time zone wheel to find the time in different locations across the world. For example, to find the time in Tokyo when it is 4 P.M. in San Francisco, rotate the small wheel until 4 P.M. and San Francisco line up, as shown. Then look at Tokyo to see that it is 9 A.M. there.
\n\"Big
\na. What is the arc measure between each time zone 0n the wheel?<\/p>\n

Answer:
\nAs the circle is divided into 24 sectors, each time zone angle = \\(\\frac { 360 }{ 24 } \\) = 15\u00b0<\/p>\n

b. What is the measure of the minor arc from the Tokyo zone to the Anchorage zone?<\/p>\n

Answer:
\nThe measure of the minor arc from the Tokyo zone to the Anchorage zone = 15 + 15 + 15 + 15 + 15 + 15
\n= 90\u00b0<\/p>\n

c. If two locations differ by 180\u00b0 on the wheel, then it is 3 P.M. at one location when it is _________ at the other location.<\/p>\n

Answer:
\nKuwaiti city.<\/p>\n

Question 33.
\nPROVING A THEOREM<\/strong>
\nWrite a coordinate proof of the Similar Circles Theorem (Theorem 10.5).
\nGiven \u2299O with center O(0, 0) and radius r.
\n\u2299A with center A(a, 0) and radius s
\nProve \u2299O ~ \u2299A
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 34.
\nABSTRACT REASONING<\/strong>
\nIs there enough information to tell whether \u2299C \u2245 \u2299D? Explain your reasoning.
\n\"Big<\/p>\n

Answer:
\nBoth circles \u2299C and \u2299D have the same radius so those circles are congruent.<\/p>\n

Question 35.
\nPROVING A THEOREM<\/strong>
\nUse the diagram to prove each part of the biconditional in the Congruent Circles Theorem (Theorem 10.3).
\n\"Big
\na. Given \\(\\overline{A C} \\cong \\overline{B D}\\)
\nProve \u2299A \u2245 \u2299B
\nb. Given \u2299A \u2245 \u2299B
\nprove \\(\\overline{A C} \\cong \\overline{B D}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 36.
\nHOW DO YOU SEE IT?<\/strong>
\nAre the circles on the target similar or congruent? Explain your reasoning.
\n\"Big
\nAnswer:<\/p>\n

Question 37.
\nPROVING A THEOREM<\/strong>
\nUse the diagram to prove each part of the biconditional in the Congruent Central Angles Theorem (Theorem 10.4).
\n\"Big
\na. Given \u2220ABC \u2245 \u2220DAE
\nProve \\(\\widehat{B C}\\) \u2245 \\(\\widehat{D E}\\)
\nb. Given \\(\\widehat{B C}\\) \u2245 \\(\\widehat{D E}\\)
\nProve \u2220ABC \u2245 \u2220DAE
\nAnswer:
\n\"Big<\/p>\n

Question 38.
\nTHOUGHT PROVOKING<\/strong>
\nWrite a formula for the length of a circular arc. Justify your answer.<\/p>\n

Answer:
\nThe formula to find the length of a circular arc is radius x angle.<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Find the value of x. Tell whether the side lengths form a Pythagorean triple.<\/p>\n

Question 39.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 40.
\n\"Big<\/p>\n

Answer:
\nx\u00b2 = 13\u00b2 + 13\u00b2
\n= 169 + 169
\n= 338
\nx = 13\u221a2<\/p>\n

Question 41
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 42.
\n\"Big<\/p>\n

Answer:
\n14\u00b2 = x\u00b2 + 10\u00b2
\n196 = x\u00b2 + 100
\nx\u00b2 = 196 – 100
\nx\u00b2 = 96
\nx = 4\u221a6<\/p>\n

10.3 Using Chords<\/h3>\n

Exploration 1<\/strong><\/p>\n

Drawing Diameters<\/p>\n

Work with a partner: Use dynamic geometry software to construct a circle of radius 5 with center at the origin. Draw a diameter that has the given point as an endpoint. Explain how you know that the chord you drew is a diameter.
\na. (4, 3)
\nb. (0, 5)
\nc. (-3, 4)
\nd. (-5, 0)<\/p>\n

Answer:
\n\"Big<\/p>\n

Exploration 2<\/strong><\/p>\n

Writing a Conjecture about Chords<\/p>\n

Work with a partner. Use dynamic geometry software to construct a chord \\(\\overline{B C}\\) of a circle A. Construct a chord on the perpendicular bisector of \\(\\overline{B C}\\). What do you notice? Change the original chord and the circle several times. Are your results always the same? Use your results to write a conjecture.
\nLOOKING FOR STRUCTURE<\/strong>
\nTo be proficient in math, you need to look closely to discern a pattern or structure.
\n\"Big
\nAnswer:<\/p>\n

Exploration 3<\/strong><\/p>\n

A Chord Perpendicular to a Diameter<\/p>\n

Work with a partner. Use dynamic geometry software to construct a diameter \\(\\overline{B C}\\) of a circle A. Then construct a chord \\(\\overline{D E}\\) perpendicular to \\(\\overline{B C}\\) at point F. Find the lengths DF and EF. What do you notice? Change the chord perpendicular to \\(\\overline{B C}\\) and the circle several times. Do you always get the same results? Write a conjecture about a chord that is perpendicular to a diameter of a circle.
\n\"Big
\nAnswer:<\/p>\n

Communicate Your Answer<\/p>\n

Question 4.
\nWhat are two ways to determine when a chord is a diameter of a circle?<\/p>\n

Answer:
\nIf a chord passes through the centre of the circle, then it is the diameter of a circle.
\nThe longest chord of the circle is the diameter of a circle.<\/p>\n

Lesson 10.3 Using Chords<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

In Exercises 1 and 2, use the diagram of \u2299D.<\/p>\n

\"Big<\/p>\n

Question 1.
\nIf m\\(\\widehat{A B}\\) = 110\u00b0. find m\\(\\widehat{B C}\\).<\/p>\n

Answer:
\nBecause AB and BC are congruent chords in congruent circles, the corresponding minor arcs \\(\\widehat{A B}\\), \\(\\widehat{B C}\\) are congruent by the congruent corresponding chords theorem.
\nSo, \\(\\widehat{A B}\\) = \\(\\widehat{B C}\\)
\n\\(\\widehat{B C}\\) = 110\u00b0<\/p>\n

Question 2.
\nIf m\\(\\widehat{A C}\\) = 150\u00b0 find m\\(\\widehat{A B}\\).<\/p>\n

Answer:
\n\\(\\widehat{A C}\\) = 360 – (\\(\\widehat{A B}\\) + \\(\\widehat{B C}\\))
\n150 = 360 – 2(\\(\\widehat{A B}\\))
\n2(\\(\\widehat{A B}\\)) = 360 – 150 = 210
\n\\(\\widehat{A B}\\) = 105\u00b0<\/p>\n

In Exercises 3 and 4. find the indicated length or arc measure.<\/p>\n

\"Big<\/p>\n

Question 3.
\nCE<\/p>\n

Answer:
\nCE = 5 + 5
\n= 10 units<\/p>\n

Question 4.
\nm\\(\\widehat{C E}\\)<\/p>\n

Answer:
\nm\\(\\widehat{C E}\\) = 9x + 180 – x = 180 – 8x
\nm\\(\\widehat{C E}\\) = 180 – 8x<\/p>\n

Question 5.
\nIn the diagram, JK = LM = 24, NP = 3x, and NQ = 7x – 12. Find the radius of \u2299N
\n\"Big<\/p>\n

Answer:<\/p>\n

Exercise 10.3 Using Chords<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nWRITING<\/strong>
\nDescribe what it means to bisect a chord.
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWRITING<\/strong>
\nTwo chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning.<\/p>\n

Answer:
\n\"Big
\nImagine a line segment of length 3 units, AB.
\n\"Big
\nA second congruent segment of length 3 that is perpendicular to AB called CD.
\n\"Big
\nCircumscribe both these line segments and note that AB and CD are now chords.
\nWhile both chords are perpendicular and congruent, neither chord is a diameter. Thus, it is possible to have two chords of this type with neither one diameter of the circle.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 6, find the measure of the red arc or chord in \u2299C.<\/p>\n

Question 3.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nArc length = radius x angle
\n= 5 x 34 = 170<\/p>\n

Question 5.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\n\"Big<\/p>\n

Answer:
\nAs the two circles radius is the same and the angle is the same so the chord length is 11 units.<\/p>\n

In Exercise 7-10, find the value of x.<\/p>\n

Question 7.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\n\"Big<\/p>\n

Answer:
\nBy the perpendicular bisector theorem RS = ST
\nx = 40\u00b0<\/p>\n

Question 9.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\n\"Big<\/p>\n

Answer:
\n5x + 2 = 7x – 12
\n7x – 5x = 2 + 12
\n2x = 14
\nx = 7<\/p>\n

Question 11.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in reasoning.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\nPROBLEM SOLVING<\/strong>
\nIn the cross section of the submarine shown, the control panels are parallel and the same length. Describe a method you can use to find the center of the cross section. Justify your method.
\n\"Big<\/p>\n

Answer:<\/p>\n

In Exercises 13 and 14, determine whether \\(\\overline{A B}\\) is a diameter of the circle. Explain your reasoning.<\/p>\n

Question 13.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\"Big<\/p>\n

Answer:
\n5\u00b2 = 3\u00b2 + x\u00b2
\n25 = 9 + x\u00b2
\nx\u00b2 = 25 – 9
\nx = 4
\nSo, AB is not diameter of the circle.<\/p>\n

In Exercises 15 and 16, find the radius of \u2299Q.<\/p>\n

Question 15.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\n\"Big<\/p>\n

Answer:
\nAD = BC
\n4x + 4 = 6x – 6
\n6x – 4x = 4 + 6
\n2x = 10
\nx = 5
\nBC = 6(5) – 6 = 30 – 6 = 24
\nQC\u00b2 = 5\u00b2 + 12\u00b2
\n= 25 + 144 = 169
\nQC = 13
\nTherefore, the radius is 13.<\/p>\n

Question 17.
\nPROBLEM SOLVING<\/strong>
\nAn archaeologist finds part of a circular plate. What was the diameter of the plate to the nearest tenth of an inch? Justify your answer.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nHOW DO YOU SEE IT?<\/strong>
\nWhat can you conclude from each diagram? Name a theorem that justifies your answer.
\na.
\n\"Big<\/p>\n

Answer:
\nPerpendicular chord bisector converse theorem.<\/p>\n

b.
\n\"Big<\/p>\n

Answer:
\nCongruent Corresponding Chords theorem<\/p>\n

c.
\n\"Big<\/p>\n

Answer:
\nPerpendicular chord bisector theorem<\/p>\n

d.
\n\"Big<\/p>\n

Answer:
\nEquidistant chords theorem<\/p>\n

Question 19.
\nPROVING A THEOREM<\/strong>
\nUse the diagram to prove each part of the biconditional in the Congruent Corresponding Chords Theorem (Theorem 10.6).
\n\"Big
\na. Given \\(\\overline{A B}\\) and \\(\\overline{C D}\\) are congruent chords.
\nProve \\(\\widehat{A B} \\cong \\widehat{C D}\\)
\nb. Given \\(\\widehat{A B} \\cong \\widehat{C D}\\)
\nProve \\(\\overline{A B}\\) \u2245 \\(\\overline{C D}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\nMATHEMATICAL CONNECTIONS<\/strong>
\nIn \u2299P, all the arcs shown have integer measures. Show that x must be even.
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 21.
\nREASONING<\/strong>
\nIn \u2299P. the lengths of the parallel chords are 20, 16, and 12. Find m\\(\\widehat{A B}\\). Explain your reasoning.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\nPROVING A THEOREM<\/strong>
\nUse congruent triangles to prove the Perpendicular Chord Bisector Theorem (Theorem 10.7).
\n\"Big
\nGiven \\(\\overline{E G}\\) is a diameter of \u2299L.
\n\\(\\overline{E G}\\) \u22a5 \\(\\overline{D F}\\)
\nProve \\(\\overline{D C}\\) \u2245 \\(\\overline{F C}\\), \\(\\widehat{D G} \\cong \\widehat{F G}\\)<\/p>\n

Answer:
\nLet L be the centre of the circle
\ndraw any chord DF on the circle
\nAs DF passes through LG.
\nThe length of DC is the same as FC.<\/p>\n

Question 23.
\nPROVING A THEOREM<\/strong>
\nWrite a proof of the Perpendicular Chord Bisector Converse (Theorem 10.8).
\n\"Big
\nGiven \\(\\overline{Q S}\\) is a perpendicular bisector of \\(\\overline{R T}\\).
\nProve \\(\\overline{Q S}\\) is a diameter of the circle L.
\n(Hint: Plot the center L and draw \u25b3LPT and \u25b3LPR.)
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\nTHOUGHT PROVOKING<\/strong>
\nConsider two chords that intersect at point P. Do you think that \\(\\frac{A P}{B P}=\\frac{C P}{D P}\\)? Justify your answer.
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 25.
\nPROVING A THEOREM<\/strong>
\nUse the diagram with the Equidistant Chords Theorem (Theorem 10.9) to prove both parts of the biconditional of this theorem.
\n\"Big
\n\\(\\overline{A B}\\) \u2245 \\(\\overline{C D}\\) if and only if EF = EG
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nMAKING AN ARGUMENT<\/strong>
\nA car is designed so that the rear wheel is only partially visible below the body of the car. The bottom edge of the panel is parallel [o the ground. Your friend claims that the point where the tire touches the ground bisects \\(\\widehat{A B}\\). Is your friend correct? Explain your reasoning.
\n\"Big<\/p>\n

Answer:<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Find the missing interior angle measure.<\/p>\n

Question 27.
\nQuadrilateral JKLW has angle measures m\u2220J = 32\u00b0, m\u2220K = 25\u00b0, and m\u2220L = 44\u00b0. Find m\u2220M.
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nPentagon PQRST has angle measures m\u2220P = 85\u00b0, m\u2220Q = 134\u00b0, m\u2220R = 97\u00b0, and m\u2220S =102\u00b0.
\nFind m\u2220T.<\/p>\n

Answer:
\nThe sum of interior angles of a pentagon = 540\u00b0
\nm\u2220T = 540 – (85 + 134 + 97 + 102)
\n= 540 – 418 = 122
\nm\u2220T = 122\u00b0.<\/p>\n

10.1 – 10.3 Quiz<\/h3>\n

In Exercises 1 – 6, use the diagram. (Section 10.1)<\/p>\n

\"Big<\/p>\n

Question 1.
\nName the circle.<\/p>\n

Answer:
\nThe circle has a chord, diameter and tangent.<\/p>\n

Question 2.
\nName a radius.<\/p>\n

Answer:
\nNP is the radius of the circle.<\/p>\n

Question 3.
\nName a diameter.<\/p>\n

Answer:
\nKN is the diameter of the circle.<\/p>\n

Question 4.
\nName a chord.<\/p>\n

Answer:
\nJL is the chord<\/p>\n

Question 5.
\nName a secant.<\/p>\n

Answer:
\nSN is the secant<\/p>\n

Question 6.
\nName a tangent.<\/p>\n

Answer:
\nQR is the tangent.<\/p>\n

Find the value of x.<\/p>\n

Question 7.
\n\"Big<\/p>\n

Answer:
\n(9 + x)\u00b2 = x\u00b2 + 15\u00b2
\n81 + 18x + x\u00b2 = x\u00b2 + 225
\n18x = 225 – 81
\n18x = 144
\nx = 8<\/p>\n

Question 8.
\n\"Big
\nAnswer:
\n6x – 3 = 3x + 18
\n6x – 3x = 18 + 3
\n3x = 21
\nx = 7<\/p>\n

Identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.<\/p>\n

\"Big<\/p>\n

Question 9.
\n\\(\\widehat{A E}\\)<\/p>\n

Answer:
\n\\(\\widehat{A E}\\) = 180 – 36
\n= 144
\nSo, \\(\\widehat{A E}\\) = 144\u00b0<\/p>\n

Question 10.
\n\\(\\widehat{B C}\\)<\/p>\n

Answer:
\n\\(\\widehat{B C}\\) = 180 – (67 + 70)
\n= 180 – 137 = 43
\nSo, \\(\\widehat{B C}\\) = 43\u00b0<\/p>\n

Question 11.
\n\\(\\widehat{A C}\\)<\/p>\n

Answer:
\n\\(\\widehat{A C}\\) = 43 + 67 = 110\u00b0<\/p>\n

Question 12.
\n\\(\\widehat{A C D}\\)<\/p>\n

Answer:
\n\\(\\widehat{A C D}\\) = 180\u00b0<\/p>\n

Question 13.
\n\\(\\widehat{A C E}\\)<\/p>\n

Answer:
\n\\(\\widehat{A C E}\\) = 180 + 36 = 216\u00b0<\/p>\n

Question 14.
\n\\(\\widehat{B E C}\\)<\/p>\n

Answer:
\n\\(\\widehat{B E C}\\) = 70 + 36 + 43 = 149\u00b0<\/p>\n

Tell whether the red arcs are congruent. Explain why or why not.<\/p>\n

Question 15.
\n\"Big<\/p>\n

Answer:
\nAs two chords pass through the centre of the circle. Those two red arcs are congruent.<\/p>\n

Question 16.
\n\"Big
\nAnswer:
\nRed arcs are not congruent because the radius of the two circles is different.<\/p>\n

Question 17.
\nFind the measure of the red arc in \u2299Q.
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 18.
\nIn the diagram. AC = FD = 30, PG = x + 5, and PJ = 3x – 1. Find the radius of \u2299P.
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 19.
\nA circular clock can be divided into 12 congruent sections.
\n\"Big<\/p>\n

a. Find the measure of each arc in this circle.
\nAnswer:
\nThe measure of each arc = \\(\\frac { 360 }{ 12 } \\) = 30\u00b0<\/p>\n

b. Find the measure of the minor arc formed by the hour and minute hands when the times is 7:00.
\nAnswer:
\nWhen the time is 7:00 the minute hand is at 12 and hour hand is at 7 and so the minor arc is subtended by 12 – 7 = 5 of these sections and so the angle subtended is 30 x 5 = 150\u00b0<\/p>\n

c. Find a time at which the hour and minute hands form an arc that is congruent to the arc in part (b).
\nAnswer:
\nA minor arc is equal to 150\u00b0 can be formed by multiple placements of the hour and the minute hand. One of them can be the time 5:00 when the minute hand is at 12 and the hour hand is at 5.<\/p>\n

10.4 Inscribed Angles and Polygons<\/h3>\n

Exploration 1<\/strong><\/p>\n

Inscribed Angles and Central Angles<\/p>\n

work with a partner: Use dynamic geometry software.<\/p>\n

Sample
\n\"Big<\/p>\n

a. Construct an inscribed angle in a circle. Then construct the corresponding central angle.
\nAnswer:<\/p>\n

b. Measure both angles. How is the inscribed angle related to its intercepted arc?
\nAnswer:<\/p>\n

c. Repeat parts (a) and (b) several times. Record your results in a table. Write a conjecture about how an inscribed angle is related to its intercepted arc.
\nATTENDING TO PRECISION<\/strong>
\nTo be proficient in math, you need to communicate precisely with others.
\nAnswer:<\/p>\n

Exploration 2<\/strong><\/p>\n

A Quadrilateral with Inscribed Angles<\/p>\n

work with a partner: Use dynamic geometry software.<\/p>\n

Sample
\n\"Big<\/p>\n

a. Construct a quadrilateral with each vertex on a circle.
\nAnswer:<\/p>\n

b. Measure all four angles. What relationships do you notice?
\nAnswer:<\/p>\n

c. Repeat parts (a) and (b) several times. Record your results in a table. Then write a conjecture that summarizes the data.
\nAnswer:<\/p>\n

Communicate Your Answer<\/p>\n

Question 3.
\nHow are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other?
\nAnswer:<\/p>\n

Question 4.
\nQuadrilateral EFGH is inscribed in \u2299C. and m \u2220 E = 80\u00b0. What is m \u2220 G? Explain.<\/p>\n

Answer:
\nm \u2220 E + m \u2220 H = 80 + 80 = 160\u00b0
\nm \u2220 E + m \u2220 H + m \u2220 G + m \u2220 F = 360
\n160\u00b0 + m \u2220 G + m \u2220 F = 360
\nm \u2220 G + m \u2220 F = 360 – 160 = 200
\nm \u2220 G = 100\u00b0<\/p>\n

Lesson 10.4 Inscribed Angles and Polygons<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

Find the measure of the red arc or angle.<\/p>\n

Question 1.
\n\"Big<\/p>\n

Answer:
\nm\u2220G = \\(\\frac { 90 }{ 2 } \\) = 45\u00b0<\/p>\n

Question 2.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{T V}\\) = 2 \u2022 38 = 76\u00b0<\/p>\n

Question 3.
\n\"Big<\/p>\n

Answer:
\nm\u2220W = 72\u00b0<\/p>\n

Find the value of each variable.<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nx\u00b0 = 90\u00b0
\ny\u00b0 = 180 – (40 + 90) = 180 – 130
\ny\u00b0 = 50\u00b0<\/p>\n

Question 5.
\n\"Big<\/p>\n

Answer:
\n\u2220B + \u2220D = 180
\n\u2220B + 82 = 180
\nx\u00b0 = 98\u00b0
\n\u2220C + \u2220A = 180
\n68 + y\u00b0 = 180
\ny\u00b0 = 112\u00b0<\/p>\n

Question 6.
\n\"Big<\/p>\n

Answer:
\n\u2220S + \u2220U = 180\u00b0
\nc + 2c – 6 = 180
\n3c = 186
\nc = 62\u00b0
\n\u2220T + \u2220V = 180\u00b0
\n10x + 8x = 180
\n18x = 180
\nx = 10\u00b0<\/p>\n

Question 7.
\nIn Example 5, explain how to find locations where the left side of the statue is all that appears in your camera\u2019s field of vision.
\nAnswer:<\/p>\n

Exercise 10.4 Inscribed Angles and Polygons<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nVOCABULARY<\/strong>
\nIf a circle is circumscribed about a polygon, then the polygon is an ___________ .
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nDIFFERENT WORDS, SAME QUESTION<\/strong>
\nWhich is different?
\nFind “both\u201d answers.
\n\"Big<\/p>\n

Find m\u2220ABC.
\nAnswer:
\nm\u2220ABC = 60\u00b0<\/p>\n

Find m\u2220AGC.
\nAnswer:
\nm\u2220AGC = 180 – (25 + 25)
\n= 180 – 50 = 130\u00b0<\/p>\n

Find m\u2220AEC.
\nAnswer:
\nm\u2220AEC = 180 – (50 + 50)
\n= 180 – 100 = 80\u00b0<\/p>\n

Find m\u2220ADC.
\nAnswer:
\nm\u2220ADC = 180 – (25 + 50)
\n= 180 – 75 = 105\u00b0<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 8, find the indicated measure.<\/p>\n

Question 3.
\nm\u2220A
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\nm\u2220G
\n\"Big
\nAnswer:
\nm\u2220G = 360 – (70 + 120)
\n= 360 – 190 = 170\u00b0<\/p>\n

Question 5.
\nm \u2220 N
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\nm\\(\\widehat{R S}\\)
\n\"Big
\nAnswer:
\nm\\(\\widehat{R S}\\) = 2 \u2022 67 = 134\u00b0<\/p>\n

Question 7.
\nm\\(\\widehat{V U}\\)
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\nm\\(\\widehat{W X}\\)
\n\"Big
\nAnswer:
\nm\\(\\widehat{W X}\\) = \\(\\frac { 75 }{ 2 } \\) = 37.5<\/p>\n

In Exercises 9 and 10, name two pairs of congruent angles.<\/p>\n

Question 9.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\n\"Big
\nAnswer:
\nm\u2220W = m\u2220Z, m\u2220X = m\u2220Y<\/p>\n

In Exercises 11 and 12, find the measure of the red arc or angle.<\/p>\n

Question 11.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\n\"Big
\nAnswer:
\n\\(\\widehat{P S}\\) = 2 \u2022 40 = 80<\/p>\n

In Exercises 13 – 16, find the value of each variable.<\/p>\n

Question 13.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\"Big
\nAnswer:
\nm\u2220E + m\u2220G = 180
\nm + 60 = 180
\nm = 120\u00b0
\nm\u2220D + m\u2220F = 180
\n60 + 2k = 180
\nk = 60\u00b0<\/p>\n

Question 15.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\n\"Big
\nAnswer:
\n3x\u00b0 = 90\u00b0
\nx\u00b0 = 30\u00b0
\n2y\u00b0 + 90\u00b0 + 34\u00b0 = 180\u00b0
\n2y\u00b0 + 124\u00b0 = 180\u00b0
\n2y\u00b0 = 56\u00b0
\ny\u00b0 = 28\u00b0<\/p>\n

Question 17.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in finding m\\(\\widehat{B C}\\).
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nMODELING WITH MATHEMATICS<\/strong>
\nA carpenter’s square is an L-shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use the carpenter\u2019s square to draw a diameter on the circular piece of wood?
\n\"Big
\nAnswer:
\nRecall that when a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Simply use the carpenter’s square to inscribe it into the circle. The hypotenuse formed by both legs of the square should provide a diameter for the circle.<\/p>\n

MATHEMATICAL CONNECTIONS<\/strong>
\nIn Exercises 19 – 21, find the values of x and y. Then find the measures of the interior angles of the polygon.<\/p>\n

Question 19.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\n\"Big
\nAnswer:
\n\u2220B + \u2220C = 180
\n14x + 4x = 180
\n18x = 180\u00b0
\nx = 10\u00b0
\n\u2220A + \u2220D = 180
\n9y + 24y = 180
\n33y = 180\u00b0
\ny = 5.45\u00b0
\n\u2220A = 130.9\u00b0, \u2220B = 40\u00b0, \u2220C = 140\u00b0, \u2220D = 49\u00b0<\/p>\n

Question 21.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\nMAKING AN ARGUMENT<\/strong>
\nYour friend claims that \u2220PTQ \u2245 \u2220PSQ \u2245 \u2220PRQ. Is our friend correct? Explain your reasoning.
\n\"Big
\nAnswer:
\nYes, my friend is correct.
\n\u2220PTQ \u2245 \u2220PSQ \u2245 \u2220PRQ is correct according to the inscribed angles of a circle theorem.<\/p>\n

Question 23.
\nCONSTRUCTION<\/strong>
\nConstruct an equilateral triangle inscribed in a circle.
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\nCONSTRUTION<\/strong>
\nThe side length of an inscribed regular hexagon is equal to the radius of the circumscribed circle. Use this fact to construct a regular hexagon inscribed in a circle.<\/p>\n

Answer:
\nAs the side length is equal to the radius. Draw a line representing the radius and draw a chord different chords in the form of hexagons of the radius of the circle.<\/p>\n

REASONING<\/strong>
\nIn Exercises 25 – 30, determine whether a quadrilateral of the given type can always be inscribed inside a circle. Explain your reasoning.<\/p>\n

Question 25.
\nSquare
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nrectangle
\nAnswer:
\nyes, angles are right angles.<\/p>\n

Question 27.
\nparallelogram
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nkite
\nAnswer:
\nNo.<\/p>\n

Question 29.
\nrhombus
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\nisosceles trapezoid
\nAnswer:
\nYes, the opposite angles are always supplementary.<\/p>\n

Question 31.
\nMODELING WITH MATHEMATICS<\/strong>
\nThree moons, A, B, and C, are in the same circular orbit 1,00,000 kilometers above the surface of a planet. The planet is 20,000 kilometers in diameter and m\u2220ABC = 90\u00b0. Draw a diagram of the situation. How far is moon A from moon C?
\nAnswer:
\n\"Big<\/p>\n

Question 32.
\nMODELING WITH MATHEMATICS<\/strong>
\nAt the movie theater. you want to choose a seat that has the best viewing angle, so that you can be close to the screen and still see the whole screen without moving your eyes. You previously decided that seat F7 has the best viewing angle, but this time someone else is already sitting there. Where else can you sit so that your seat has the same viewing angle as seat F7? Explain.
\n\"Big
\nAnswer:<\/p>\n

Question 33.
\nWRITING<\/strong>
\nA right triangle is inscribed in a circle, and the radius of the circle is given. Explain how to find the length of the hypotenuse.
\nAnswer:
\n\"Big<\/p>\n

Question 34.
\nHOW DO YOU SEE IT?<\/strong>
\nLet point Y represent your location on the soccer field below. What type of angle is \u2220AYB if you stand anywhere on the circle except at point A or point B?
\n\"Big
\nAnswer:<\/p>\n

Question 35.
\nWRITING<\/strong>
\nExplain why the diagonals of a rectangle inscribed in a circle are diameters of the circle.
\nAnswer:
\n\"Big<\/p>\n

Question 36.
\nTHOUGHT PROVOKING<\/strong>
\nThe figure shows a circle that is circumscribed about \u2206ABC. Is it possible to circumscribe a circle about any triangle? Justify your answer.
\n\"Big
\nAnswer:
\nYes.<\/p>\n

Question 37.
\nPROVING A THEOREM<\/strong>
\nIf an angle is inscribed in \u2299Q. the center Q can be on a side of the inscribed angle, inside the inscribed angle, or outside the inscribed angle. Prove each case of the Measure of an Inscribed Angle Theorem (Theorem 10. 10).<\/p>\n

a. Case 1
\nGiven \u2220ABC is inscribed in \u2299Q
\nLet m\u2220B = x\u00b0
\nCenter Q lies on \\(\\overline{B C}\\).
\nProve m\u2220ABC = \\(\\frac{1}{2}\\)m\\(\\widehat{A C}\\)
\n(Hint: Show that \u2206AQB is isosceles. Then write m\\(\\widehat{A C}\\) in terms of x.)
\n\"Big
\nb. Case 2
\nUse the diagram and auxiliary line to write Given and Prove statements for Case 2. Then write a proof
\n\"Big
\nc. Case 3
\nUse the diagram and auxiliary line to write Given and Prove statements for Case 3. Then write a proof.
\n\"Big
\nAnswer:
\n\"Big
\n\"Big
\n\"Big<\/p>\n

Question 38.
\nPROVING A THEOREM<\/strong>
\nWrite a paragraph proof of the Inscribed Angles of a Circle Theorem (Theorem 10.11). First, draw a diagram and write Given and Prove statements.
\nAnswer:
\nIf two inscribed angles of a circle intercept the same arc, then the angles are congruent.<\/p>\n

Question 39.
\nPROVING A THEOREM<\/strong>
\nThe Inscribed Right Triangle Theorem (Theorem 10.12) is written as a conditional statement and its converse. Write a plan for proof for each statement.
\nAnswer:
\n\"Big<\/p>\n

Question 40.
\nPROVING A THEOREM<\/strong>
\nCopy and complete the paragraph proof for one part of the Inscribed Quadrilateral Theorem (Theorem 10. 13).
\nGiven \u2299C with inscribed quadrilateral DEFG
\nProve m \u2220 D + m \u2220 F = 180\u00b0,
\nm \u2220 E + m \u2220 G = 180\u00b0
\n\"Big
\nBy the Arc Addition Postulate (Postulate 10. 1),
\nm\\(\\widehat{E F G}\\) + ________ = 360\u00b0 and m\\(\\widehat{F G D}\\) + m\\(\\widehat{D E F}\\) = 360\u00b0.
\nUsing the ___________ Theorem. m\\(\\widehat{E D G}\\) = 2m \u2220 F, m\\(\\widehat{E F G}\\) = 2m\u2220D, m\\(\\widehat{D E F}\\) = 2m\u2220G, and m\\(\\widehat{F G D}\\) = 2m \u2220 E. By the Substitution Property of Equality, 2m\u2220D + ________ = 360\u00b0, So _________ . Similarly, __________ .
\nAnswer:
\nm\\(\\widehat{E F G}\\) + m\\(\\widehat{E D F}\\) = 360\u00b0 and m\\(\\widehat{F G D}\\) + m\\(\\widehat{D E F}\\) = 360\u00b0.
\nUsing the the measure of an inscribed angle Theorem. m\\(\\widehat{E D G}\\) = 2m \u2220 F, m\\(\\widehat{E F G}\\) = 2m\u2220D, m\\(\\widehat{D E F}\\) = 2m\u2220G, and m\\(\\widehat{F G D}\\) = 2m \u2220 E. By the Substitution Property of Equality, 2m\u2220D + 2m\u2220G = 360\u00b0.<\/p>\n

Question 41.
\nCRITICAL THINKING<\/strong>
\nIn the diagram, \u2220C is a right angle. If you draw the smallest possible circle through C tangent to \\(\\overline{A B}\\), the circle will intersect \\(\\overline{A C}\\) at J and \\(\\overline{B C}\\) at K. Find the exact length of \\(\\overline{J K}\\).
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 42.
\nCRITICAL THINKING<\/strong>
\nYou are making a circular cutting board. To begin, you glue eight 1-inch boards together, as shown. Then you draw and cut a circle with an 8-inch diameter from the boards.
\n\"Big
\na. \\(\\overline{F H}\\) is a diameter of the circular cutting board. Write a proportion relating GJ and JH. State a theorem in to justify your answer.
\nAnswer:
\nEach board is 1 inch and FJ spans 6 boards.
\n\\(\\overline{F H}\\) = 6 inches<\/p>\n

b. Find FJ, JH, and GJ. What is the length of the cutting board seam labeled \\(\\overline{G K}\\)?
\nAnswer:
\nEach board is 1 inch and JH spans 2 boards.
\nJH = 2 inches
\nEquation is \\(\\frac { JH }{ GJ } \\) = \\(\\frac { GJ }{ FJ } \\)
\n\\(\\frac { 2 }{ GJ } \\) = \\(\\frac { GJ }{ 6 } \\)
\n12 = GJ\u00b2
\nGJ = 2\u221a3
\nGK = 2(GJ)
\nGK = 4\u221a3
\nSo, FJ = 6, JH = 2, JG = 2\u221a3, GK = 4\u221a3<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Solve the equation. Check your solution.<\/p>\n

Question 43.
\n3x = 145
\nAnswer:
\n\"Big<\/p>\n

Question 44.
\n\\(\\frac{1}{2}\\)x = 63
\nAnswer:
\nx = 63 \u2022 2
\nx = 126<\/p>\n

Question 45.
\n240 = 2x
\nAnswer:
\n\"Big<\/p>\n

Question 46.
\n75 = \\(\\frac{1}{2}\\)(x – 30)
\nAnswer:
\n75 \u2022 2 = x – 30
\n150 + 30 = x
\nx = 180<\/p>\n

10.5 Angle Relationships in Circles<\/h3>\n

Exploration 1<\/strong><\/p>\n

Angles Formed by a Chord and Tangent Line<\/p>\n

Work with a partner: Use dynamic geometry software.<\/p>\n

Sample
\n\"Big<\/p>\n

a. Construct a chord in a circle. At one of the endpoints of the chord. construct a tangent line to the circle.
\nAnswer:<\/p>\n

b. Find the measures of the two angles formed by the chord and the tangent line.
\nAnswer:<\/p>\n

c. Find the measures of the two circular arcs determined by the chord.
\nAnswer:<\/p>\n

d. Repeat parts (a) – (c) several times. Record your results in a table. Then write a conjecture that summarizes the data.
\nAnswer:<\/p>\n

Exploration 2<\/strong><\/p>\n

Angles Formed by Intersecting Chords<\/p>\n

Work with a partner: Use dynamic geometry software.<\/p>\n

sample
\n\"Big<\/p>\n

a. Construct two chords that intersect inside a circle.
\nAnswer:<\/p>\n

b. Find the measure of one of the angles formed by the intersecting chords.
\nAnswer:<\/p>\n

c. Find the measures of the arcs intercepted h the angle in part (b) and its vertical angle. What do you observe?
\nAnswer:<\/p>\n

d. Repeat parts (a) – (c) several times. Record your results in a table. Then write a conjecture that summarizes the data.
\nCONSTRUCTING VIABLE ARGUMENTS<\/strong>
\nTo be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.
\nAnswer:<\/p>\n

Communicate Your Answer<\/p>\n

Question 3.
\nWhen a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed?
\nAnswer:<\/p>\n

Question 4.
\nLine m is tangent to the circle in the figure at the left. Find the measure of \u22201.
\n\"Big
\nAnswer:
\nm\u22201 = \\(\\frac { 1 }{ 2 } \\) \u2022 148
\nm\u22201 = 74\u00b0<\/p>\n

Question 5.
\nTwo chords intersect inside a circle to form a pair of vertical angles with measures of 55\u00b0. Find the sum of the measures of the arcs intercepted by the two angles.
\nAnswer:
\nThe sum of the measures of the arcs intercepted by the two angles = \\(\\frac { 1 }{ 2 } \\) \u2022 55
\n= 27.5<\/p>\n

Lesson 10.5 Angle Relationships in Circles<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

Line m is tangent to the circle. Find the indicated measure.<\/p>\n

Question 1.
\nm \u2220 1
\n\"Big
\nAnswer:
\nm \u2220 1 = \\(\\frac { 1 }{ 2 } \\) \u2022 210
\nm \u2220 1 = 105\u00b0<\/p>\n

Question 2.
\nm\\(\\widehat{R S T}\\)
\n\"Big
\nAnswer:
\nm\\(\\widehat{R S T}\\) = 2 \u2022 98 = 196\u00b0
\nm\\(\\widehat{R S T}\\) = 196\u00b0<\/p>\n

Question 3.
\nm\\(\\widehat{X Y}\\)
\n\"Big
\nAnswer:
\nm\\(\\widehat{X Y}\\) = \\(\\frac { 1 }{ 2 } \\) \u2022 80
\nm\\(\\widehat{X Y}\\) = 40\u00b0<\/p>\n

Find the value of the variable.<\/p>\n

Question 4.
\n\"Big
\nAnswer:
\ny\u00b0 = \\(\\frac { 1 }{ 2 } \\) \u2022 (102 + 95)
\n= 98.5\u00b0<\/p>\n

Question 5.
\n\"Big
\nAnswer:
\na\u00b0 = 2 \u2022 30\u00b0 + 44\u00b0
\n= 60\u00b0 + 44\u00b0 = 104\u00b0
\nSo, a\u00b0 = 104\u00b0.<\/p>\n

Find the value of x.<\/p>\n

Question 6.
\n\"Big
\nAnswer:
\nx\u00b0 = 180\u00b0 – 120\u00b0
\nx\u00b0 = 60\u00b0<\/p>\n

Question 7.
\n\"Big
\nAnswer:
\n50\u00b0 = 180\u00b0 – x\u00b0
\nx\u00b0 = 180\u00b0 – 50\u00b0
\nx\u00b0 = 130\u00b0<\/p>\n

Question 8.
\nYou are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level at point B. Find m\\(\\widehat{C D}\\), which represents the part of Earth that you can see.
\n\"Big
\nAnswer:
\nCB and BD are tangents, CB is perpendicular to AB and CD is perpendicular to AD by the tangent line to circle theorem.
\n\u25b3ABC is similar to \u25b3ABD by the hypotenuese leg congruence theorem.
\n\u2220CBA is similar to \u2220ABD. So, m\u2220CBA = 74.5\u00b0, m\u2220CBD = 2 \u2022 74.5\u00b0 = 149\u00b0
\nm\u2220CBD = 180\u00b0 – m\u2220CAD
\nm\u2220CBD = 180\u00b0 – CD
\n149\u00b0 = 180\u00b0 – CD
\nCD = 31\u00b0
\nThe part of earth you can see<\/p>\n

Exercise 10.5 Angle Relationships in Circles<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nCOMPLETE THE SENTENCE<\/strong>
\nPoints A, B, C, and D are on a circle, and \"Big intersects \"Big at point P.
\nIf m\u2220APC = \\(\\frac{1}{2}\\)(m\\(\\widehat{B D}\\) – m\\(\\widehat{A C}\\)). then point P is _________ the circle.
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWRITING<\/strong>
\nExplain how to find the measure of a circumscribed angle.
\nAnswer:
\nA circumscribed angle is the angle made by two intersecting tangent lines to a circle. Draw lines from the circle centre to the point of tangency. The angle between the radius and tangent line is 90\u00b0. The sum of angles of a quadrilateral is 360\u00b0. Angles between radii and tangent lines is 180\u00b0. The angle at two tangent lines meet is circumscribed angle.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 6, line t is tangent to the circle. Find the indicated measure.<\/p>\n

Question 3.
\nm\\(\\widehat{A B}\\)
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\nm\\(\\widehat{D E F}\\)
\n\"Big
\nAnswer:
\nm\\(\\widehat{D E F}\\) = 2(117\u00b0) = 234\u00b0<\/p>\n

Question 5.
\nm < 1
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\nm \u2220 3
\n\"Big
\nAnswer:
\nm \u2220 3 = \u00bd \u2022 140 = 70\u00b0<\/p>\n

In Exercises 7 – 14, find the value of x.<\/p>\n

Question 7.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\n\"Big
\nAnswer:
\nx\u00b0 = \u00bd \u2022 (30 + 2x – 30)<\/p>\n

Question 9.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\n\"Big
\nAnswer:
\n34\u00b0 = \u00bd (3x – 2 – (x + 6))
\n34\u00b0 = \u00bd (3x – 2 – x – 6)
\n34\u00b0 = \u00bd (2x – 8)
\n34\u00b0 = x – 4
\nx\u00b0 = 34 + 4
\nx\u00b0 = 38\u00b0<\/p>\n

Question 11.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\n\"Big
\nAnswer:
\n6x – 11 = 2\u00a0 \u2022\u00a0125
\n6x = 250 + 11
\n6x = 261
\nx\u00b0 = 43.5\u00b0<\/p>\n

Question 13.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\"Big<\/p>\n

Answer:
\n17x\u00b0 = 75\u00b0
\nx\u00b0 = 4.41\u00b0<\/p>\n

ERROR ANALYSIS<\/strong>
\nIn Exercises 15 and 16, describe and correct the error in finding the angle measure.<\/p>\n

Question 15.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\n\"Big
\nAnswer:
\nm\u22201 = \u00bd (122 – 70)
\n= \u00bd (52) = 26
\nSo, m\u22201 = 26\u00b0<\/p>\n

In Exercises 17 – 22, find the indicated angle measure. justify your answer.<\/p>\n

\"Big<\/p>\n

Question 17.
\nm \u2220 1
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nm \u2220 2
\nAnswer:
\nm \u2220 2 = 60\u00b0<\/p>\n

Explanation:
\nm \u2220 3 =30\u00b0, So, m \u2220 2 = 180\u00b0 – (90\u00b0 + 30\u00b0)
\n= 180\u00b0 – 120\u00b0 = 60\u00b0
\nTherefore, m \u2220 2 = 60\u00b0<\/p>\n

Question 19.
\nm \u2220 3
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\nm \u2220 4
\nAnswer:
\nm \u2220 4 = 90\u00b0<\/p>\n

Question 21.
\nm \u2220 5
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\nm \u2220 6
\nAnswer:
\nm \u2220 6 = 180\u00b0 – (60\u00b0 + 30\u00b0 + 30\u00b0) = 180\u00b0 – 120\u00b0
\nm \u2220 6 = 60\u00b0<\/p>\n

Question 23.
\nPROBLEM SOLVING<\/strong>
\nYou are flying in a hot air balloon about 1.2 miles above the ground. Find the measure of the arc that represents the part of Earth you can see. The radius of Earth is about 4000 miles.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\nPROBLEM SOLVING<\/strong>
\nYou are watching fireworks over San Diego Bay S as you sail away in a boat. The highest point the fireworks reach F is about 0.2 mile above the bay. Your eyes E are about 0.01 mile above the water. At point B you can no longer see the fireworks because of the curvature of Earth. The radius of Earth is about 4000 miles, and \\(\\overline{F E}\\) is tangent to Earth at point T. Find m\\(\\widehat{s B}\\). Round your answer to the nearest tenth.
\n\"Big
\nAnswer:<\/p>\n

Question 25.
\nMATHEMATICAL CONNECTIONS<\/strong>
\nIn the diagram, \\(\\vec{B}\\)A is tangent to \u2299E. Write an algebraic expression for m\\(\\widehat{C D}\\) in terms of x. Then find m\\(\\widehat{C D}\\).
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nMATHEMATICAL CONNECTIONS<\/strong>
\nThe circles in the diagram are concentric. Write an algebraic expression for c in terms of a and b.
\n\"Big
\nAnswer:
\na\u00b0 = \u00bd(c\u00b0 – b\u00b0)<\/p>\n

Question 27.
\nABSTRACT REASONING<\/strong>
\nIn the diagram. \\(\\vec{P}\\)L is tangent to the circle, and \\(\\overline{K J}\\) is a diameter. What is the range of possible angle measures of \u2220LPJ? Explain your reasoning.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nABSTRACT REASONING<\/strong>
\nIn the diagram, \\(\\overline{A B}\\) is an chord that is not a diameter of the circle. Line in is tangent to the circle at point A. What is the range of possible values of x? Explain your reasoning. (The diagram is not drawn to scale.)
\n\"Big
\nAnswer:
\nThe possible values of x are less than 180\u00b0.<\/p>\n

Question 29.
\nPROOF<\/strong>
\nIn the diagram \"Big and \"Big are secant lines that intersect at point L. Prove that m\u2220JPN > m\u2220JLN.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\nMAKING AN ARGUMENT<\/strong>
\nYour friend claims that it is possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? Explain your reasoning.<\/p>\n

Answer:
\nYes, when the circumscribed angle measures 90\u00b0, the central angle measures 90\u00b0, so the intercepted arc also measures 90\u00b0.<\/p>\n

Question 31.
\nREASONING<\/strong>
\nPoints A and B are on a circle, and t is a tangent line containing A and another point C.
\na. Draw two diagrams that illustrate this situation.
\nb. Write an equation for m\\(\\widehat{A B}\\) in terms of m\u2220BAC for each diagram.
\nc. For what measure of \u2220BAC can you use either equation to find m\\(\\widehat{A B}\\)? Explain.
\nAnswer:
\n\"Big<\/p>\n

Question 32.
\nREASONING<\/strong>
\n\u2206XYZ is an equilateral triangle inscribed in \u2299P. AB is tangent to \u2299P at point X, \\(\\overline{B C}\\) is tangent to \u2299P at point Y. and \\(\\overline{A C}\\) is tangent to \u2299P at point Z. Draw a diagram that illustrates this situation. Then classify \u2206ABC by its angles and sides. Justify your answer.
\nAnswer:<\/p>\n

Question 33.
\nPROVING A THEOREM<\/strong>
\nTo prove the Tangent and Intersected Chord Theorem (Theorem 10. 14), you must prove three cases.
\na. The diagram shows the case where \\(\\overline{A B}\\) contains the center of the circle. Use the Tangent Line to Circle Theorem (Theorem 10.1) to write a paragraph proof for this case.
\n\"Big
\nb. Draw a diagram and write a proof for the case where the center of the circle is in the interior of \u2220CAB.
\nc. Draw a diagram and write a proof for the case where the center of the circle is in the exterior of \u2220CAB.
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 34.
\nHOW DO YOU SEE IT?<\/strong>
\nIn the diagram, television cameras are Positioned at A and B to record what happens on stage. The stage is an arc of \u2299A. You would like the camera at B to have a 30\u00b0 view of the stage. Should you move the camera closer or farther away? Explain your reasoning.
\n\"Big<\/p>\n

Answer:
\n25\u00b0 = \u00bd(80\u00b0 – 30\u00b0) = \u00bd(50\u00b0)
\nSo, you should move the camera closer.<\/p>\n

Question 35.
\nPROVING A THEOREM<\/strong>
\nWrite a proof of the Angles Inside the Circle Theorem (Theorem 10.15).
\n\"Big
\nGiven Chords \\(\\overline{A C}\\) and \\(\\overline{B D}\\) intersect inside a circle.
\nProve m \u2220 1 = \\(\\frac{1}{2}\\)(m\\(\\widehat{D C}\\) + m\\(\\widehat{A B}\\))
\nAnswer:
\n\"Big<\/p>\n

Question 36.
\nTHOUGHT PROVOKING<\/strong>
\nIn the figure, \"Big and \"Big are tangent to the circle. Point A is any point on the major are formed by the endpoints of the chord \\(\\overline{B C}\\). Label all congruent angles in the figure. Justify your reasoning.
\n\"Big<\/p>\n

Answer:
\nm\u2220CPB = \u00bd(CAB – CB)<\/p>\n

Question 37.
\nPROVING A THEOREM<\/strong>
\nUse the diagram below to prove the Angles Outside the Circle Theorem (Theorem 10.16) for the case of a tangent and a secant. Then copy the diagrams for the other two cases on page 563 and draw appropriate auxiliary segments. Use your diagrams to prove each case.
\n\"Big
\nAnswer:
\n\"Big
\n\"Big
\n\"Big<\/p>\n

Question 38.
\nPROVING A THEOREM<\/strong>
\nProve that the Circumscribed Angle Theorem (Theorem 10.17) follows from the Angles Outside the Circle Theorem (Theorem 10.16).
\nAnswer:<\/p>\n

In Exercises 39 and 40, find the indicated measure(s). justify your answer<\/p>\n

Question 39.
\nFind m \u2220 P when m\\(\\widehat{W Z Y}\\) = 200\u00b0
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 40.
\nFind m\\(\\widehat{A B}\\) and m\\(\\widehat{E D}\\)
\n\"Big<\/p>\n

Answer:
\nm\\(\\widehat{E D}\\) = \u00bd (115\u00b0) = 57.5\u00b0
\n\u2220GJA = 30\u00b0<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Solve the equation.<\/p>\n

Question 41.
\nx2<\/sup> + x = 12
\nAnswer:
\n\"Big<\/p>\n

Question 42.
\nx2<\/sup> = 12x + 35<\/p>\n

Answer:<\/p>\n

Explanation:
\nx\u00b2 = 12x + 35
\nx = \\(\\frac { 12 \u00b1 \u221a(144 + 140)\u00a0 }{ 2 } \\)
\nx = \\(\\frac { 12 \u00b1 \u221a284\u00a0 }{ 2 } \\)
\nx = \\(\\frac { 12 + \u221a284\u00a0 }{ 2 } \\), \\(\\frac { 12 – \u221a284\u00a0 }{ 2 } \\)<\/p>\n

Question 43.
\n– 3 = x2<\/sup> + 4x
\nAnswer:
\n\"Big<\/p>\n

10.6 Segment Relationships in Circles<\/h3>\n

Exploration 1<\/strong><\/p>\n

Segments Formed by Two Intersecting Chords<\/p>\n

Work with a partner: Use dynamic geometry software.<\/p>\n

Sample
\n\"Big<\/p>\n

a.
\nConstruct two chords \\(\\overline{B C}\\) and \\(\\overline{D E}\\) that intersect in the interior of a circle at a point F.
\nAnswer:<\/p>\n

b.
\nFind the segment lengths BE, CF, DF, and EF and complete the table. What do you observe?
\n\"Big
\nAnswer:<\/p>\n

c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
\nREASONING ABSTRACTLY<\/strong>
\nTo be proficient in math, you need to make sense of quantities and their relationships in problem situations.
\nAnswer:<\/p>\n

Exploration 2<\/strong><\/p>\n

Secants Intersecting Outside a Circle<\/p>\n

Work with a partner: Use dynamic geometry software.<\/p>\n

Sample
\n\"Big<\/p>\n

a. Construct two secant \"Big and \"Big that intersect at a point B outside a circle, as shown.
\nAnswer:<\/p>\n

b. Find the segment lengths BE, BC, BF, and BD. and complete the table. What do you observe?
\n\"Big
\nAnswer:<\/p>\n

c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
\nAnswer:<\/p>\n

Communicate Your Answer<\/p>\n

Question 3.
\nWhat relationships exist among the segments formed by two intersecting chords or among segments of two secants that intersect outside a circle?<\/p>\n

Question 4.
\nFind the segment length AF in the figure at the left.
\n\"Big<\/p>\n

Answer:<\/p>\n

Explanation:
\nEA \u2022 AF = AD \u2022 AC
\n18 \u2022 AF = 9 \u2022 8
\nAF = 4<\/p>\n

Lesson 10.6 Segment Relationships in Circles<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

Find the value of x.<\/p>\n

Question 1.
\n\"Big<\/p>\n

Answer:
\nx = 8<\/p>\n

Explanation:
\n4 \u2022 6 = 3 \u2022 x
\n3x = 24
\nx = 8<\/p>\n

Question 2.
\n\"Big<\/p>\n

Answer:
\nx = 5<\/p>\n

Explanation:
\n2 \u2022 x + 1 = 4 \u2022 3
\nx + 1 = 6
\nx = 5<\/p>\n

Question 3.
\n\"Big<\/p>\n

Answer:
\nx = \\(\\frac { 54 }{ 5 } \\)<\/p>\n

Explanation:
\n6 \u2022 9 = 5 \u2022 x
\n54 = 5x
\nx = \\(\\frac { 54 }{ 5 } \\)<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nx = \\(\\frac { 3 \u00b1 \u221a37 }{ 2 } \\)<\/p>\n

Explanation:
\n3 \u2022 x + 2 = x + 1 \u2022 x – 1
\n3x + 6 = x\u00b2 – 1
\nx\u00b2 – 3x – 7 = 0
\nx = \\(\\frac { 3 \u00b1 \u221a(9 + 28) }{ 2 } \\)
\nx = \\(\\frac { 3 \u00b1 \u221a37 }{ 2 } \\)<\/p>\n

Question 5.
\n\"Big<\/p>\n

Answer:
\nx = \u00b1\u221a3<\/p>\n

Explanation:
\nx\u00b2 = 3 \u2022 1
\nx\u00b2 = 3
\nx = \u00b1\u221a3<\/p>\n

Question 6.
\n\"Big<\/p>\n

Answer:
\nx = \\(\\frac { 49 }{ 5 } \\)<\/p>\n

Explanation:
\n7\u00b2 = 5 \u2022 x
\n49 = 5x
\nx = \\(\\frac { 49 }{ 5 } \\)<\/p>\n

Question 7.
\n\"Big<\/p>\n

Answer:
\nx = 14.4<\/p>\n

Explanation:
\n12\u00b2 = 10x
\n144 = 10x
\nx = 14.4<\/p>\n

Question 8.
\nWHAT IF?<\/strong>
\nIn Example 4, CB = 35 feet and CE = 14 feet. Find the radius of the tank.<\/p>\n

Answer:
\nThe radius of the tank is 36.75<\/p>\n

Explanation:
\nCB\u00b2 = CE \u22c5 CD
\n35\u00b2 = 14 \u22c5 (2r + 14)
\n1225 = 28r + 196
\n28r = 1029
\nr = 36.75<\/p>\n

Exercise 10.6 Segment Relationships in Circles<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nVOCABULARY<\/strong>
\nThe part of the secant segment that is outside the circle is called a(n) ______________ .
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWRITING<\/strong>
\nExplain the difference between a tangent segment and a secant segment.<\/p>\n

Answer:
\nA tangent segment intersects the circle at only one point. It actually doesn’t go through the circle. If a ball is rolling on a table top, then it would be the tangent. A secant segment intersects the circle in two points. It goes through the circle. In a tangent, no part is in the interior of the circle. In a secant, there is a part in the interior called a chord.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 6, find the value of x.<\/p>\n

Question 3.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nx = 23<\/p>\n

Explanation:
\n10 \u2022 18 = 9 \u2022 (x – 3)
\n20 = x – 3
\nx = 23<\/p>\n

Question 5.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\n\"Big<\/p>\n

Answer:
\nx = 5<\/p>\n

Explanation:
\n2x \u2022 12 = 15 \u2022 (x + 3)
\n24x = 15x + 45
\n9x = 45
\nx = 5<\/p>\n

In Exercises 7 – 10, find the value of x.<\/p>\n

Question 7.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\n\"Big<\/p>\n

Answer:
\nx = \\(\\frac { 35 }{ 4 } \\)<\/p>\n

Explanation:
\n5 \u2022 7 = 4 \u2022 x
\n4x = 35
\nx = \\(\\frac { 35 }{ 4 } \\)<\/p>\n

Question 9.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\n\"Big<\/p>\n

Answer:
\nx = 30<\/p>\n

Explanation:
\n45 \u2022 x = 50 \u2022 27
\n45x = 1350
\nx = 30<\/p>\n

In Exercises 11 – 14. find the value of x.<\/p>\n

Question 11.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\n\"Big<\/p>\n

Answer:
\nx = 48<\/p>\n

Explanation:
\n24\u00b2 = 12x
\n576 = 12x
\nx = 48<\/p>\n

Question 13.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n\"Big<\/p>\n

Answer:
\nx = 1.5<\/p>\n

Explanation:
\n3 = 2x
\nx = 1.5<\/p>\n

Question 15.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in finding CD.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\nMODELING WITH MATHEMATICS<\/strong>
\nThe Cassini spacecraft is on a mission in orbit around Saturn until September 2017. Three of Saturn\u2019s moons. Tethys. Calypso, and Teleslo. have nearly circular orbits of radius 2,95,000 kilometers. The diagram shows the positions of the moons and the spacecraft on one of Cassini\u2019s missions. Find the distance DB from Cassini to Tethys when \\(\\overline{A D}\\) is tangent to the circular orbit.
\n\"Big<\/p>\n

Answer:
\nBD = 579493 km<\/p>\n

Explanation:
\n(203,000)\u00b2 = 83000x
\nx = 496493
\nBC = 496493
\nBD = 496493 + 83000 = 579493<\/p>\n

Question 17.
\nMODELING WITH MATHEMATICS<\/strong>
\nThe circular stone mound in Ireland called Newgrange has a diameter of 250 feet. A passage 62 feet long leads toward the center of the mound. Find the perpendicular distance x from the end of the passage to either side of the mound.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nMODELING WITH MATHEMATICS<\/strong>
\nYou are designing an animated logo for our website. Sparkles leave point C and move to the Outer circle along the segments shown so that all of the sparkles reach the outer circle at the same time. Sparkles travel from point C to point D at 2 centimeters per second. How fast should sparkles move from point C to point N? Explain.
\n\"Big<\/p>\n

Answer:
\n5.33 should sparkles move from point C to point N.<\/p>\n

Explanation:
\n4 \u2022 8 = 6 \u2022 x
\nx = 5.33<\/p>\n

Question 19.
\nPROVING A THEOREM<\/strong>
\nWrite a two-column proof of the Segments of Chords Theorem (Theorem 10.18).<\/p>\n

Plan for Proof:
\nUse the diagram to draw \\(\\overline{A C}\\) and \\(\\overline{D B}\\). Show that \u2206EAC and \u2206EDB are similar. Use the fact that corresponding side lengths in similar triangles are proportional.
\n\"Big
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 20.
\nPROVING A THEOREM<\/strong>
\nProve the Segments of Secants Theorem (Theorem 10.19). (Hint: Draw a diagram and add auxiliary line segments to form similar triangles.)
\nAnswer:<\/p>\n

Question 21.
\nPROVING A THEOREM<\/strong>
\nUse the Tangent Line to Circle Theorem (Theorem 10. 1) to prove the Segments of Secants and Tangents Theorem (Theorem 10.20) for the special case when the secant segment Contains the center of the circle.
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\nPROVING A THEOREM<\/strong>
\nProve the Segments of Secants and Tangents Theorem (Theorem 10.20). (Hint: Draw a diagram and add auxiliary line segments to form similar triangles.)
\nAnswer:<\/p>\n

Question 23.
\nWRITING EQUATIONS<\/strong>
\nIn the diagram of the water well, AB, AD, and DE are known. Write an equation for BC using these three measurements.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 24.
\nHOW DO YOU SEE IT?<\/strong>
\nWhich two theorems would you need to use to tind PQ? Explain your reasoning.
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 25.
\nCRITICAL THINKING<\/strong>
\nIn the figure, AB = 12, BC = 8, DE = 6, PD = 4, and a is a point of tangency. Find the radius of \u2299P.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nTHOUGHT PROVOKING<\/strong>
\nCircumscribe a triangle about a circle. Then, using the points of tangency, inscribe a triangle in the circle. Must it be true that the two triangles are similar? Explain your reasoning.<\/p>\n

Answer:<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Solve the equation by completing the square.<\/p>\n

Question 27.
\nx2<\/sup> + 4x = 45
\nAnswer:
\n\"Big<\/p>\n

Question 28.
\nx2<\/sup> – 2x – 1 = 8<\/p>\n

Answer:
\nx = 1 + \u221a10, x = 1 – \u221a10<\/p>\n

Explanation:
\nx\u00b2 – 2x – 1 = 8
\nx\u00b2 – 2x – 9 = 0
\nx = \\(\\frac { 2 \u00b1 \u221a(4 + 36) }{ 2 } \\)
\nx = \\(\\frac { 2 \u00b1 \u221a40 }{ 2 } \\)
\nx = 1 + \u221a10, x = 1 – \u221a10<\/p>\n

Question 29.
\n2x2<\/sup> + 12x + 20 = 34
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\n– 4x2<\/sup> + 8x + 44 = 16<\/p>\n

Answer:
\nx = 1 + \u221a8, x = 1 – \u221a8<\/p>\n

Explanation:
\n– 4x\u00b2 + 8x + 44 = 16
\n4x\u00b2 – 8x – 28 =0
\nx\u00b2 – 2x – 7 = 0
\nx = \\(\\frac { 2 \u00b1 \u221a(4 + 28) }{ 2 } \\)
\nx = \\(\\frac { 2 \u00b1 \u221a32 }{ 2 } \\)
\nx = 1 \u00b1 \u221a8<\/p>\n

10.7 Circles in the Coordinate Plane<\/h3>\n

Exploration 1<\/strong><\/p>\n

The Equation of a Circle with Center at the Origin<\/p>\n

Work with a partner: Use dynamic geometry software to Construct and determine the equations of circles centered at (0, 0) in the coordinate plane, as described below.
\n\"Big
\na. Complete the first two rows of the table for circles with the given radii. Complete the other rows for circles with radii of your choice.<\/p>\n

Answer:
\n\"Big<\/p>\n

b. Write an equation of a circle with center (0, 0) and radius r.<\/p>\n

Answer:
\nx\u00b2\u00a0 + y\u00b2\u00a0 = r\u00b2<\/p>\n

Explanation:
\n(x – 0)\u00b2 + (y – 0)\u00b2\u00a0 = r\u00b2
\nx\u00b2\u00a0 + y\u00b2\u00a0 = r\u00b2<\/p>\n

Exploration 2<\/strong><\/p>\n

The Equation of a Circle with Center (h, k)<\/p>\n

Work with a partner: Use dynamic geometry software to construct and determine the equations of circles of radius 2 in the coordinate plane, as described below.
\n\"Big
\na. Complete the first two rows of the table for circles with the given centers. Complete the other rows for circles with centers of your choice.<\/p>\n

Answer:
\n\"Big<\/p>\n

b. Write an equation of a circle with center (h, k) and radius 2.<\/p>\n

Answer:
\n(x – h)\u00b2 + (y – k)\u00b2 = 4<\/p>\n

c. Write an equation of a circle with center (h, k) and radius r.<\/p>\n

Answer:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2<\/p>\n

Exploration 3<\/strong><\/p>\n

Deriving the Standard Equation of a Circle<\/p>\n

Work with a partner. Consider a circle with radius r and center (h, k).<\/p>\n

Write the Distance Formula to represent the distance d between a point (x, y) on the circle and the center (h, k) of the circle. Then square each side of the Distance Formula equation.<\/p>\n

How does your result compare with the equation you wrote in part (c) of Exploration 2?<\/p>\n

MAKING SENSE OF PROBLEMS<\/strong>
\nTo be proficient in math, you need to explain correspondences between equations and graphs.
\n\"Big<\/p>\n

Answer:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2<\/p>\n

Communicate Your Answer<\/p>\n

Question 4.
\nWhat is the equation of a circle with center (h, k) and radius r in the coordinate plane?<\/p>\n

Answer:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2<\/p>\n

Question 5.
\nWrite an equation of the circle with center (4, – 1) and radius 3.<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 – 8x + 2y + 8 = 0<\/p>\n

Explanation:
\n(x – 4)\u00b2 + (y + 1)\u00b2 = 9
\nx\u00b2 – 8x + 16 + y\u00b2 + 2y + 1 = 9
\nx\u00b2 + y\u00b2 – 8x + 2y = 9 – 17
\nx\u00b2 + y\u00b2 – 8x + 2y + 8 = 0<\/p>\n

Lesson 10.7 Circles in the Coordinate Plane<\/h3>\n

Monitoring Progress<\/strong><\/p>\n

Write the standard equation of the circle with the given center and radius.<\/p>\n

Question 1.
\ncenter: (0, 0), radius: 2.5<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 6.25<\/p>\n

Explanation:
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 2.5\u00b2
\nx\u00b2 + y\u00b2 = 6.25<\/p>\n

Question 2.
\ncenter: (- 2, 5), radius: 7<\/p>\n

Answer:
\n(x + 2)\u00b2 + (y – 5)\u00b2 = 49<\/p>\n

Explanation:
\n(x + 2)\u00b2 + (y – 5)\u00b2 = 7\u00b2
\n(x + 2)\u00b2 + (y – 5)\u00b2 = 49<\/p>\n

Question 3.
\nThe point (3, 4) is on a circle with center (1, 4). Write the standard equation of the circle.<\/p>\n

Answer:
\n(x – 1)\u00b2 + (y – 4)\u00b2 = 4<\/p>\n

Explanation:
\nr = \u221a(3 – 1)\u00b2 + (4 – 4)\u00b2
\n= \u221a(2)\u00b2
\nr = 2
\n(x – 1)\u00b2 + (y – 4)\u00b2 = 2\u00b2
\n(x – 1)\u00b2 + (y – 4)\u00b2 = 4<\/p>\n

Question 4.
\nThe equation of a circle is x2<\/sup> + y2<\/sup> – 8x + 6y + 9 = 0. Find the center and the radius of the circle. Then graph the circle.<\/p>\n

Answer:
\nThe center of the circle (4, -3) and radius is 4.<\/p>\n

Explanation:
\nx\u00b2 + y\u00b2\u00a0– 8x + 6y + 9 = 0
\nx\u00b2 – 8x + 16 + y\u00b2 + 6y + 9 = 16
\n(x – 4)\u00b2 + (y + 3)\u00b2 = 4\u00b2
\nThe center of the circle (4, -3) and radius is 4.<\/p>\n

Question 5.
\nProve or disprove that the point (1, \u221a5 ) lies on the circle centered at the origin and containing the point (0, 1).<\/p>\n

Answer:
\nDisproved.<\/p>\n

Explanation:
\nWe consider the circle centred at the origin and containing the point (0, 1).
\nTherefore, we can conclude that the radius of the circle r = 1, let O (0, 0) and B (1, \u221a5). So the distance between two points is
\nOB = \u221a(1 – 0) + (\u221a5 – 0)\u00b2 = \u221a(1 + 5) = \u221a6
\nAs the radius of the given circle is 1 and distance of the point B from its centre is \u221a6. So we can conclude that point does lie on the given circle.<\/p>\n

Question 6.
\nwhy are three seismographs needed to locate an earthquake\u2019s epicentre?<\/p>\n

Answer:<\/p>\n

Exercise 10.7 Circles in the Coordinate Plane<\/h3>\n

Vocabulary and Core Concept Check<\/p>\n

Question 1.
\nVOCABULARY<\/strong>
\nWhat is the standard equation of a circle?
\nAnswer:
\n\"Big<\/p>\n

Question 2.
\nWRITING<\/strong>
\nExplain why knowing the location of the center and one point on a circle is enough to graph the circle.<\/p>\n

Answer:
\nIf we know the location of the center and one point on the circle, we can graph a circle because the distance from the center to the point is called the radius.<\/p>\n

Monitoring Progress and Modeling with Mathematics<\/p>\n

In Exercises 3 – 8, write the standard equation of the circle.<\/p>\n

Question 3.
\n\"Big
\nAnswer:
\n\"Big<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 36<\/p>\n

Explanation:
\nThe center is (0, 0) and the radius is 6
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 6\u00b2
\nx\u00b2 + y\u00b2 = 36<\/p>\n

Question 5.
\na circle with center (0, 0) and radius 7
\nAnswer:
\n\"Big<\/p>\n

Question 6.
\na circle with center (4, 1) and radius 5<\/p>\n

Answer:
\n(x – 4)\u00b2 + (y – 1)\u00b2 = 25<\/p>\n

Explanation:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2
\n(x – 4)\u00b2 + (y – 1)\u00b2 = 5\u00b2
\n(x – 4)\u00b2 + (y – 1)\u00b2 = 25<\/p>\n

Question 7.
\na circle with center (- 3, 4) and radius 1
\nAnswer:
\n\"Big<\/p>\n

Question 8.
\na circle with center (3, – 5) and radius 7<\/p>\n

Answer:
\n(x – 3)\u00b2 + (y + 5)\u00b2 = 49<\/p>\n

Explanation:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2
\n(x – 3)\u00b2 + (y + 5)\u00b2 = 7\u00b2
\n(x – 3)\u00b2 + (y + 5)\u00b2 = 49<\/p>\n

In Exercises 9 – 11, use the given information to write the standard equation of the circle.<\/p>\n

Question 9.
\nThe center is (0, 0), and a point on the circle is (0, 6).
\nAnswer:
\n\"Big<\/p>\n

Question 10.
\nThe center is (1, 2), and a point on the circle is (4, 2).<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 9<\/p>\n

Explanation:
\nr = \u221a(x – h)\u00b2 + (y – k)\u00b2
\n= \u221a(4 – 1)\u00b2 + (2 – 2)\u00b2
\n= \u221a3\u00b2
\nr = 3
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 3\u00b2
\nx\u00b2 + y\u00b2 = 9<\/p>\n

Question 11.
\nThe center is (0, 0). and a point on the circle is (3, – 7).
\nAnswer:
\n\"Big<\/p>\n

Question 12.
\nERROR ANALYSIS<\/strong>
\nDescribe and correct the error in writing the standard equation of a circle.
\n\"Big<\/p>\n

Answer:
\n(x – h)\u00b2 + (y – k)\u00b2 = r\u00b2
\n(x + 3)\u00b2 + (y + 5)\u00b2 = 3\u00b2
\n(x + 3)\u00b2 + (y + 5)\u00b2 = 9<\/p>\n

In Exercises 13 – 18, find the center and radius of the circle. Then graph the circle.<\/p>\n

Question 13.
\nx2<\/sup> + y2<\/sup> = 49
\nAnswer:
\n\"Big<\/p>\n

Question 14.
\n(x + 5)2<\/sup> + (y – 3)2<\/sup> = 9<\/p>\n

Answer:
\nCenter is (-5, 3) and rdaius is 3.<\/p>\n

Explanation:
\nFor the equation (x + 5)2<\/sup> + (y – 3)2<\/sup> = 9, center is (-5, 3) and rdaius is 3.<\/p>\n

Question 15.
\nx2<\/sup> + y2<\/sup> – 6x = 7
\nAnswer:
\n\"Big<\/p>\n

Question 16.
\nx2<\/sup> + y2<\/sup> + 4y = 32<\/p>\n

Answer:
\nThe center is (0, -2) and radius is 6<\/p>\n

Explanation:
\nx2<\/sup> + y2<\/sup> + 4y = 32
\nx\u00b2 + y\u00b2 + 4y + 4 = 32 + 4
\nx\u00b2 + (y + 2)\u00b2 = 36
\n(x – 0)\u00b2 + (y – (-2))\u00b2 = 6\u00b2
\nThe center is (0, -2) and radius is 6
\n\"Big<\/p>\n

Question 17.
\nx2<\/sup> + y2<\/sup> – 8x – 2y = – 16
\nAnswer:
\n\"Big<\/p>\n

Question 18.
\nx2<\/sup> + y2<\/sup> + 4x + 12y = – 15<\/p>\n

Answer:
\nThe center is (-2, -6) and radius is 5<\/p>\n

Explanation:
\nx2<\/sup> + y2<\/sup> + 4x + 12y = – 15
\nx\u00b2 + 4x + 4 + y\u00b2 + 12y + 36 = -15 + 36 + 4
\n(x + 2)\u00b2 + (y + 6)\u00b2 = 5\u00b2
\nThe center is (-2, -6) and radius is 5<\/p>\n

In Exercises 19 – 22, prove or disprove the statement.<\/p>\n

Question 19.
\nThe point (2, 3) lies on the circle centered at the origin with radius 8.
\nAnswer:
\n\"Big<\/p>\n

Question 20.
\nThe point (4, \u221a5) lies on the circle centered at the origin with radius 3.<\/p>\n

Answer:
\nThe point (4, \u221a5) does not lie on the circle.<\/p>\n

Explanation:
\nr\u00b2 = (x – h)\u00b2 + (y – k)\u00b2
\n3\u00b2 ______________ (4 – 0)\u00b2 + (\u221a5 – 0)\u00b2
\n9 ______________ 16 + 5
\n9 \u2260 21
\nBecause the radius is 3 and the distance between center and the point is more than the radius. So the point does not lie on the circle.<\/p>\n

Question 21.
\nThe point (\u221a6, 2) lies on the circle centered at the origin and containing the point (3, – 1).
\nAnswer:
\n\"Big<\/p>\n

Question 22.
\nThe point (\u221a7, 5) lies on the circle centered at the origin and containing the point (5, 2).<\/p>\n

Answer:
\nThe point (\u221a7, 5) does not lie on the circle.<\/p>\n

Explanation:
\nr\u00b2 = (x – h)\u00b2 + (y – k)\u00b2
\n= (\u221a7 – 0)\u00b2 + (5 – 0)\u00b2 = 7 + 25 = 32
\nr = 5.65
\n(5.65)\u00b2 ______________ (5 – 0)\u00b2 + (2 – 0)\u00b2
\n32 ______________ 25 + 4
\n32 \u2260 29
\nBecause the radius is 3 and the distance between center and the point is more than the radius. So the point does not lie on the circle.<\/p>\n

Question 23.
\nMODELING WITH MATHEMATICS<\/strong>
\nA City’s commuter system has three zones. Zone I serves people living within 3 miles of the city\u2019s center. Zone 2 serves those between 3 and 7 miles from the center. Zone 3 serves those over 7 miles from the center.
\n\"Big
\na. Graph this Situation on a coordinate plane where each unit corresponds to 1 mile. Locate the city\u2019s center at the origin.
\nb. Determine which zone serves people whose homes are represented by the points (3, 4), (6, 5), (1, 2), (0.3). and (1, 6).
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 24.
\nMODELING WITH MATHEMATICS<\/strong>
\nTelecommunication towers can be used to transmit cellular phone calls. A graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). These towers have a range of about 3 kilometers.
\na. Sketch a graph and locate the towers. Are there any locations that may receive calls from more than one tower? Explain your reasoning.
\nAnswer:<\/p>\n

b. The center of City A is located at (- 2, 2.5), and the center of City B is located at (5, 4). Each city has a radius of 1.5 kilometers. Which city seems to have better cell phone coverage? Explain your reasoning.<\/p>\n

Answer:
\nThere are three towers at points (0, 0), (0, 5) and (6, 3) with range of about 3 km
\na. Let’s sketch the graph to locate towers. Draw the points (0, 0), (0, 5) and (6, 3). Then draw three circles with centers at these points and radii 3.
\nThere are locations that can receive calls from more that one tower because circles with centers (0, 0) and (0, 5) overlap. Locations in their intersection can receive calls from two towers.
\nThe city A has a center located at (-2, 2.5) and city B has a center located at (5, 4). Both cities have radius 1.5 km
\nLet’s draw the city A as a circle with center (-2, 2.5) and radius 1.5 and city B with center (5, 4) and radius 1.5.
\nFrom the graph we can conclude that the city B has better cell phone coverage because parts of city A do not have coverage.<\/p>\n

Question 25.
\nREASONING<\/strong>
\nSketch the graph of the circle whose equation is x2<\/sup> + y2<\/sup> = 16. Then sketch the graph of the circle after the translation (x, y) \u2192 (x – 2, y – 4). What is the equation of the image? Make a conjecture about the equation of the image of a circle centered at the origin after a translation m units to the left and n units down.
\nAnswer:
\n\"Big<\/p>\n

Question 26.
\nHOW DO YOU SEE IT?<\/strong>
\nMatch each graph with its equation.<\/p>\n\n\n\n\n\n\n
a. \"Big<\/td>\nA. x2<\/sup> + (y + 3) 2<\/sup> = 4<\/td>\n<\/tr>\n
b. \"Big<\/td>\nB. (x \u2013 3) 2<\/sup> + y2<\/sup> = 4<\/td>\n<\/tr>\n
c. \"Big<\/td>\nC. (x + 3) 2<\/sup> + y2<\/sup> = 4<\/td>\n<\/tr>\n
d. \"Big<\/td>\nD. x2<\/sup> + (y \u2013 3) 2<\/sup> = 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Answer:
\na \u279d C, b \u279d A, c \u279d D, d \u279d B<\/p>\n

Question 27.
\nUSING STRUCTURE<\/strong>
\nThe vertices of \u2206XYZ are X(4, 5), Y(4, 13), and Z(8, 9). Find the equation of the circle circumscribed about \u2206XYZ. Justify your answer.
\nAnswer:
\n\"Big
\n\"Big<\/p>\n

Question 28.
\nTHOUGHT PROVOKING<\/strong>
\nA circle has center (h, k) and contains point (a, b). Write the equation of the line tangent to the circle at point (a, b).<\/p>\n

Answer:
\ny – b = \\(\\frac { h – a }{ b – k } \\)(x – a)<\/p>\n

Explanation:
\nIt is given a circle with center C(h, k. A circle with point A(a, b). We have to write the equation of a tangent that intersects the circle at point A
\nBy the tangent line to circle theorem, a tangent is perpendicular to the radius. Two lines are perpendicular if and only if their slopes are negative reciprocals. So, find the equation of the line AC to know its slope.
\nThe equation of the line which has two points (a, b), (c, d) is y – b = \\(\\frac { b – c }{ a – d } \\)(x – a)
\nThe equation of the line which has two points A(a, b) and C(h, k) is
\ny – b = \\(\\frac { b – k }{ a – h } \\)(x – a)
\nTherefore, the slope of the line through A nad C is \\(\\frac { b – k }{ a – h } \\)
\nHence the slope of the tangent is –\\(\\frac { a – h }{ b – k } \\) = \\(\\frac { h – a }{ b – k } \\)
\nUse the equation of the line y = kx + n through the point (a, b)
\ny – b = k(x – a) to find the equation of the tangent
\nThe equation of the tangent with slope \\(\\frac { h – a }{ b – k } \\) and through the point A(a, b) is y – b = \\(\\frac { h – a }{ b – k } \\)(x – a)<\/p>\n

MATHEMATICAL CONNECTIONS<\/strong>
\nIn Exercises 29 – 32, use the equations to determine whether the line is a tangent, a secant a secant that contains the diameter, or name of these. Explain your reasoning.<\/p>\n

Question 29.
\nCircle: (x – 4)2<\/sup> + (y – 3)2<\/sup> = 9
\nLine: y = 6
\nAnswer:
\n\"Big<\/p>\n

Question 30.
\nCircle: (x + 2)2<\/sup> + (y – 2)2<\/sup> = 16
\nLine: y = 2x – 4<\/p>\n

Answer:
\nThe line is a secant line.<\/p>\n

Explanation:
\n(x + 2)2<\/sup> + (y – 2)2<\/sup> = 16, y = 2x – 4
\n(x + 2)2<\/sup> + (2x – 4 – 2)2<\/sup> = 16
\nx\u00b2 + 4x + 4 + (2x – 6)\u00b2 = 16
\nx\u00b2 + 4x + 4 + 4x\u00b2 – 24x + 36 = 16
\n5x\u00b2 – 20x + 40 – 16 = 0
\n5x\u00b2 – 20x + 24 = 0
\nx = \\(\\frac { 20 \u00b1\u221a-80 }{ 10 } \\)
\nx = 2, y = 2 \u2022 2 – 4 = 0, (2, 0)
\nThe system has two solutions and point does not lie on the line. So the line is a secant line.<\/p>\n

Question 31.
\nCircle: (x – 5)2<\/sup> + (y + 1)2<\/sup> = 4
\nLine: y = \\(\\frac{1}{5}\\)x – 3\\
\nAnswer:
\n\"Big
\n\"Big
\n\"Big<\/p>\n

Question 32.
\nCircle: (x + 3)2<\/sup> + (y – 6)2<\/sup> = 25
\nLine: y = –\\(\\frac{4}{3}\\)x + 2<\/p>\n

Answer:
\nThe line is a secant line.<\/p>\n

Explanation:
\n(x + 3)2<\/sup> + (y – 6)2<\/sup> = 25, y = –\\(\\frac{4}{3}\\)x + 2
\n(x + 3)2<\/sup> + (-\\(\\frac{4}{3}\\)x + 2 – 6)2<\/sup> = 25
\nx\u00b2 + 6x + 9 + \\(\\frac { 16x\u00b2 }{ 9 } \\) + \\(\\frac { 32x }{ 3 } \\) + 16 = 25
\n\\(\\frac { 25x\u00b2 }{ 9 } \\) + \\(\\frac { 50x }{ 3 } \\) = 0
\nx(25x + 150) = 0
\nx = 0 or x = -6
\ny = 2, y = 10
\n(0, 2) and (-6, 10)
\nd = \u221a(0 + 6)\u00b2 + (2 – 10)\u00b2
\n= \u221a(36 + 64)
\n= 10 \u2260 5
\nThe system has two solutions and point does not lie on the line. So the line is a secant line.<\/p>\n

MAKING AN ARGUMENT<\/strong>
\nQuestion 33.
\nYour friend claims that the equation of a circle passing through the points (- 1, 0) and (1, 0) is x2<\/sup> – 2yk + y2<\/sup> = 1 with center (0, k). Is your friend correct? Explain your reasoning.
\nAnswer:
\n\"Big<\/p>\n

Question 34.
\nREASONING<\/strong>
\nFour tangent circles are centered on the x-axis. The radius of \u2299A is twice the radius of \u2299O, The radius of \u2299B is three times the radius of \u2299O, The radius of \u2299C is four times the radius of \u2299O, All circles have integer radii, and the point (63, 16) is On \u2299C. What is the equation of \u2299A? Explain your reasoning.
\n\"Big
\nAnswer:<\/p>\n

Maintaining Mathematical Proficiency<\/p>\n

Identify the arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.<\/p>\n

\"Big<\/p>\n

Question 35.
\n\\(\\widehat{R S}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 36.
\n\\(\\widehat{P R}\\)<\/p>\n

Answer:
\n\\(\\widehat{P R}\\) is a right angle
\n\\(\\widehat{P R}\\) = 25 + 65 = 90\u00b0<\/p>\n

Question 37.
\n\\(\\widehat{P R T}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 38.
\n\\(\\widehat{S T}\\)<\/p>\n

Answer:
\n\\(\\widehat{S T}\\) is a major arc
\n\\(\\widehat{S T}\\) = 360 – (90 + 65 +25 + 53) = 127\u00b0<\/p>\n

Question 39.
\n\\(\\widehat{R S T}\\)
\nAnswer:
\n\"Big<\/p>\n

Question 40.
\n\\(\\widehat{Q S}\\)<\/p>\n

Answer:
\n\\(\\widehat{Q S}\\) is a minor arc
\n\\(\\widehat{Q S}\\) = 25 + 53 = 78\u00b0<\/p>\n

Circles Review<\/h3>\n

10.1 Lines and Segments That Intersect Circles<\/h3>\n

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of \u2299P.<\/p>\n

\"Big<\/p>\n

Question 1.
\n\\(\\overline{P K}\\)<\/p>\n

Answer:
\n\\(\\overline{P K}\\) is radius<\/p>\n

Question 2.
\n\\(\\overline{N M}\\)<\/p>\n

Answer:
\n\\(\\overline{N M}\\) is chord<\/p>\n

Question 3.
\n\\(\\vec{J}\\)L<\/p>\n

Answer:
\n\\(\\vec{J}\\)L is tangent<\/p>\n

Question 4.
\n\\(\\overline{K N}\\)<\/p>\n

Answer:
\n\\(\\overline{K N}\\) is diameter<\/p>\n

Question 5.
\n\"Big<\/p>\n

Answer:
\nNL is secant<\/p>\n

Question 6.
\n\\(\\overline{P N}\\)<\/p>\n

Answer:
\n\\(\\overline{P N}\\) is radius<\/p>\n

Tell whether the common tangent is internal or external.<\/p>\n

Question 7.
\n\"Big<\/p>\n

Answer:
\nInternal common tangent<\/p>\n

Question 8.
\n\"Big<\/p>\n

Answer:
\nExternal common tangent<\/p>\n

Points Y and Z are points of tangency. Find the value of the variable.<\/p>\n

Question 9.
\n\"Big<\/p>\n

Answer:
\na = \\(\\frac { 3 \u00b1 33 }{ 18 } \\)<\/p>\n

Explanation:
\n3a = 9a\u00b2 – 30
\n9a\u00b2 – 3a – 30 = 0
\na = \\(\\frac { 3 \u00b1 33 }{ 18 } \\)<\/p>\n

Question 10.
\n\"Big<\/p>\n

Answer:
\nc = 2<\/p>\n

Explanation:
\n2c\u00b2 + 9c + 6 = 9c + 14
\n2c\u00b2 – 8 = 0
\nc\u00b2 – 4 = 0
\nc = 2<\/p>\n

Question 11.
\n\"Big<\/p>\n

Answer:
\nr = 12<\/p>\n

Explanation:
\n(3 + r)\u00b2 = r\u00b2 + 9\u00b2
\n9 + 6r + r\u00b2 = r\u00b2 + 81
\n6r = 72
\nr = 12<\/p>\n

Question 12.
\nTell whether \\(\\overline{B D}\\) is tangent to \u2299C. Explain.
\n\"Big<\/p>\n

Answer:
\n\\(\\overline{B D}\\) is not tangent to \u2299C<\/p>\n

Explanation:
\n52\u00b2 = 10\u00b2 + 48\u00b2
\n2704 = 100 + 2304
\nSo, \\(\\overline{B D}\\) is not tangent to \u2299C<\/p>\n

10.2 Finding Arc Measures<\/h3>\n

Use the diagram above to find the measure of the indicated arc.<\/p>\n

Question 13.
\n\\(\\widehat{K L}\\)<\/p>\n

Answer:
\n\\(\\widehat{K L}\\) = 100\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{K L}\\) = \u2220KPL = 100\u00b0<\/p>\n

Question 14.
\n\\(\\widehat{L M}\\)<\/p>\n

Answer:
\n\\(\\widehat{L M}\\) = 60\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{L M}\\) = 180\u00b0 – 120\u00b0
\n= 60\u00b0<\/p>\n

Question 15.
\n\\(\\widehat{K M}\\)<\/p>\n

Answer:
\n\\(\\widehat{K M}\\) = 160\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{K M}\\) = 100\u00b0 + 60\u00b0
\n= 160\u00b0<\/p>\n

Question 16.
\n\\(\\widehat{K N}\\)<\/p>\n

Answer:
\n\\(\\widehat{K N}\\) = 80\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{K N}\\) = 360 – (120 + 100 + 60)
\n= 360 – 280 = 80\u00b0<\/p>\n

Tell whether the red arcs are congruent. Explain why or why not.<\/p>\n

Question 17.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{S T}\\), \\(\\widehat{Y Z}\\) are not congruent.<\/p>\n

Explanation:
\n\\(\\widehat{S T}\\), \\(\\widehat{Y Z}\\) are not congruent. Because both arcs are from different circles and having different radii.<\/p>\n

Question 18.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{A B}\\), \\(\\widehat{E F}\\) are congruent.<\/p>\n

Explanation:
\n\\(\\widehat{A B}\\), \\(\\widehat{E F}\\) are congruent. Because those circles have same radii.<\/p>\n

10.3 Using Chords<\/h3>\n

Find the measure of \\(\\widehat{A B}\\).<\/p>\n

Question 19.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{A B}\\) = 61\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{A B}\\) = 61\u00b0
\nIf ED = AB, then \\(\\widehat{A B}\\) = \\(\\widehat{E D}\\)<\/p>\n

Question 20.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{A B}\\) = 65\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{A B}\\) = \\(\\widehat{A D}\\)
\nSo, \\(\\widehat{A B}\\) = 65\u00b0<\/p>\n

Question 21.
\n\"Big<\/p>\n

Answer:
\n\\(\\widehat{A B}\\) = 91\u00b0<\/p>\n

Explanation:
\n\\(\\widehat{A B}\\) = \\(\\widehat{E D}\\)
\nSo, \\(\\widehat{A B}\\) = 91\u00b0<\/p>\n

Question 22.
\nIn the diagram. QN = QP = 10, JK = 4x, and LM = 6x – 24. Find the radius of \u2299Q.
\n\"Big<\/p>\n

Answer:
\nThe radius of \u2299Q is 26.<\/p>\n

Explanation:
\n6x – 24 = 4x
\n6x – 4x = 24
\n2x = 24
\nx = 12
\nML = 6(12) – 24 = 48
\nJN = \\(\\frac { 48 }{ 2 } \\) = 24
\nJQ\u00b2 = JN\u00b2 + NQ\u00b2
\n= 24\u00b2 + 10\u00b2 = 576 + 100
\nJQ = 26
\nTherefore, the radius of \u2299Q is 26<\/p>\n

10.4 Inscribed Angles and Polygons<\/h3>\n

Find the value(s) of the variable(s).<\/p>\n

Question 23.
\n\"Big<\/p>\n

Answer:
\nx\u00b0 = 80\u00b0<\/p>\n

Explanation:
\nx\u00b0 = 2 \u2022 40\u00b0 = 80\u00b0<\/p>\n

Question 24.
\n\"Big<\/p>\n

Answer:
\nq\u00b0 = 100\u00b0, r\u00b0 = 20\u00b0<\/p>\n

Explanation:
\nq\u00b0 + 80\u00b0 = 180\u00b0
\nq\u00b0 = 100\u00b0
\n4r\u00b0 + 100 = 180\u00b0
\n4r\u00b0 = 80\u00b0
\nr\u00b0 = 20\u00b0<\/p>\n

Question 25.
\n\"Big<\/p>\n

Answer:
\nd\u00b0 = 5\u00b0<\/p>\n

Explanation:
\n14d\u00b0 = 70\u00b0
\nd\u00b0 = 5\u00b0<\/p>\n

Question 26.
\n\"Big<\/p>\n

Answer:
\ny\u00b0 = 30\u00b0, z\u00b0 = 10\u00b0<\/p>\n

Explanation:
\n3y\u00b0 = 90\u00b0
\ny\u00b0 = 30\u00b0
\n50\u00b0 + 90\u00b0 + 4z\u00b0 = 180\u00b0
\n4z\u00b0 = 40\u00b0
\nz\u00b0 = 10\u00b0<\/p>\n

Question 27.
\n\"Big<\/p>\n

Answer:
\nm\u00b0 = 44\u00b0
\nn\u00b0 = 39\u00b0<\/p>\n

Explanation:
\nm\u00b0 = 44\u00b0
\nn\u00b0 = 39\u00b0<\/p>\n

Question 28.
\n\"Big<\/p>\n

Answer:
\nc\u00b0 = 28\u00b0<\/p>\n

Explanation:
\nc\u00b0 = \u00bd \u2022 56 = 28<\/p>\n

10.5 Angle Relationships in Circles<\/h3>\n

Find the value of x.<\/p>\n

Question 29.
\n\"Big<\/p>\n

Answer:
\nx\u00b0 = 250\u00b0<\/p>\n

Explanation:
\nx\u00b0 = 250\u00b0<\/p>\n

Question 30.
\n\"Big<\/p>\n

Answer:
\nx\u00b0 = 106\u00b0<\/p>\n

Explanation:
\nx\u00b0 = \u00bd(152 + 60)
\n= \u00bd(212) = 106\u00b0<\/p>\n

Question 31.
\n\"Big<\/p>\n

Answer:
\nx\u00b0 = 28\u00b0<\/p>\n

Explanation:
\nx\u00b0 = \u00bd(96 – 40)
\n= \u00bd(56) = 28\u00b0<\/p>\n

Question 32.
\nLine l is tangent to the circle. Find m\\(\\widehat{X Y Z}\\).
\n\"Big<\/p>\n

Answer:
\nm\\(\\widehat{X Y Z}\\) = 240\u00b0<\/p>\n

Explanation:
\nm\\(\\widehat{X Y Z}\\) = 2(120)
\n= 240\u00b0<\/p>\n

10.6 Segment Relationships in Circles<\/h3>\n

Find the value of x.<\/p>\n

Question 33.
\n\"Big<\/p>\n

Answer:
\nx = 8<\/p>\n

Explanation:
\n3 \u2022 x = 4 \u2022 6
\nx = 8<\/p>\n

Question 34.
\n\"Big<\/p>\n

Answer:
\nx = 3<\/p>\n

Explanation:
\n(x + 3) \u2022 x = (6 – x) \u2022 2x
\nx + 3 = 12 – 2x
\n3x = 9
\nx = 3<\/p>\n

Question 35.
\n\"Big<\/p>\n

Answer:
\nx = 18<\/p>\n

Explanation:
\n12\u00b2 = 8 \u2022 x
\n144 = 8x
\nx = 18<\/p>\n

Question 36.
\nA local park has a circular ice skating rink. You are standing at point A, about 12 feet from the edge of the rink. The distance from you to a point of tangency on the rink is about 20 feet. Estimate the radius of the rink.
\n\"Big<\/p>\n

Answer:
\nEstimated radius of the rink is 10 ft<\/p>\n

Explanation:
\n20\u00b2 = 12 \u2022 (2r + 12)
\n400 = 24r + 144
\n256 = 24r
\nr = 10.66
\nTherefore, estimated radius of the rink is 10 ft<\/p>\n

10.7 Circles in the Coordinate Plane<\/h3>\n

Write the standard equation of the circle shown.<\/p>\n

Question 37.
\n\"Big<\/p>\n

Answer:
\n(x – 4)\u00b2 + (y + 1)\u00b2 = 12.25<\/p>\n

Explanation:
\n(x – 4)\u00b2 + (y + 1)\u00b2 = 3.5\u00b2
\n(x – 4)\u00b2 + (y + 1)\u00b2 = 12.25<\/p>\n

Question 38.
\n\"Big<\/p>\n

Answer:
\n(x – 8)\u00b2 + (y – 5)\u00b2 = 36<\/p>\n

Explanation:
\n(x – 8)\u00b2 + (y – 5)\u00b2 = 6\u00b2
\n(x – 8)\u00b2 + (y – 5)\u00b2 = 36<\/p>\n

Question 39.
\n\"Big<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 4<\/p>\n

Explanation:
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 2\u00b2
\nx\u00b2 + y\u00b2 = 4<\/p>\n

Write the standard equation of the circle with the given center and radius.<\/p>\n

Question 40.
\ncenter: (0,0), radius: 9<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 81<\/p>\n

Explanation:
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 9\u00b2
\nx\u00b2 + y\u00b2 = 81<\/p>\n

Question 41.
\ncenter: (- 5, 2), radius: 1.3<\/p>\n

Answer:
\n(x + 5)\u00b2 + (y – 2)\u00b2 = 1.69<\/p>\n

Explanation:
\n(x + 5)\u00b2 + (y – 2)\u00b2 = 1.3\u00b2
\n(x + 5)\u00b2 + (y – 2)\u00b2 = 1.69<\/p>\n

Question 42.
\ncenter: (6, 21), radius: 4<\/p>\n

Answer:
\n(x – 6)\u00b2 + (y – 21)\u00b2 = 16<\/p>\n

Explanation:
\n(x – 6)\u00b2 + (y – 21)\u00b2 = 4\u00b2
\n(x – 6)\u00b2 + (y – 21)\u00b2 = 16<\/p>\n

Question 43.
\ncenter: (- 3, 2), radius: 16<\/p>\n

Answer:
\n(x + 3)\u00b2 + (y – 2)\u00b2 = 256<\/p>\n

Explanation:
\n(x + 3)\u00b2 + (y – 2)\u00b2 = 16\u00b2
\n(x + 3)\u00b2 + (y – 2)\u00b2 = 256<\/p>\n

Question 44.
\ncenter: (10, 7), radius: 3.5<\/p>\n

Answer:
\n(x – 10)\u00b2 + (y – 7)\u00b2 = 12.25<\/p>\n

Explanation:
\n(x – 10)\u00b2 + (y – 7)\u00b2 = 3.5\u00b2
\n(x – 10)\u00b2 + (y – 7)\u00b2 = 12.25<\/p>\n

Question 45.
\ncenter: (0, 0), radius: 5.2<\/p>\n

Answer:
\nx\u00b2 + y\u00b2 = 27.04<\/p>\n

Explanation:
\n(x – 0)\u00b2 + (y – 0)\u00b2 = 5.2\u00b2
\nx\u00b2 + y\u00b2 = 27.04<\/p>\n

Question 46.
\nThe point (- 7, 1) is on a circle with center (- 7, 6). Write the standard equation of the circle.<\/p>\n

Answer:
\n(x + 7)\u00b2 + (y – 6)\u00b2 = 25<\/p>\n

Explanation:
\nr\u00b2 = (-7 + 7)\u00b2 + (6 – 1)\u00b2
\n= 5\u00b2
\nr = 5
\nAnd, centre is (-7, 6)
\nThe standard equation of a circle is (x – (-7))\u00b2 + (y – 6)\u00b2 = 5\u00b2
\n(x + 7)\u00b2 + (y – 6)\u00b2 = 25<\/p>\n

Question 47.
\nThe equation of a circle is x2<\/sup> + y2<\/sup> – 12x + 8y + 48 = 0. Find the center and the radius of the circle. Then graph the circle.<\/p>\n

Answer:
\nThe radius of the circle is 2, the centre is (6, -4)<\/p>\n

Explanation:
\nx2<\/sup> + y2<\/sup> – 12x + 8y + 48 = 0
\nx\u00b2 – 12x + 36 + y\u00b2 + 8y + 16 = 4
\n(x – 6)\u00b2 + (y + 4)\u00b2 = 2\u00b2
\nSo, the radius of the circle is 2, the centre is (6, -4)
\n\"Big<\/p>\n

Question 48.
\nProve or disprove that the point (4, – 3) lies on the circle centred at the origin and containing
\nthe point (- 5, 0).<\/p>\n

Answer:
\nThe point (4, – 3) lies on the circle.<\/p>\n

Explanation:
\nUse the distance formula to find the radius of the circle with cente (0, 0) and a point (-5, 0)
\nr = \u221a(-5 – 0)\u00b2 + (0 – 0)\u00b2 = 5
\nThe distance from the point (4, -3) to the center (0, 0)
\nd = \u221a(4 – 0)\u00b2 + (-3 – 0)\u00b2 = \u221a(16 +9) = 5
\nSince the radius of the circle is 5, the point lies on the circle.<\/p>\n

Circles Chapter Test<\/h3>\n

Find the measure of each numbered angle in \u2299P. Justify your answer.<\/p>\n

Question 1.
\n\"Big<\/p>\n

Answer:
\nm\u22201 = 72.5\u00b0
\nm\u22202 = 145\u00b0<\/p>\n

Explanation:
\nm\u22201 = \\(\\frac { 145 }{ 2 } \\)
\n= 72.5\u00b0
\nm\u22202 = 145\u00b0<\/p>\n

Question 2.
\n\"Big<\/p>\n

Answer:
\nm\u22201 = 60\u00b0, m\u22202 = 90\u00b0<\/p>\n

Explanation:
\nA tangent is perpendicualr to diameter. So, m\u22202 = 90\u00b0
\nm\u22201 = 60\u00b0<\/p>\n

Question 3.
\n\"Big<\/p>\n

Answer:
\nm\u22201 = 48\u00b0<\/p>\n

Explanation:
\nm\u22201 = \\(\\frac { 96 }{ 2 } \\) = 48\u00b0<\/p>\n

Question 4.
\n\"Big<\/p>\n

Answer:<\/p>\n

Use the diagram.<\/p>\n

\"Big<\/p>\n

Question 5.
\nAG = 2, GD = 9, and BG = 3. Find GF.<\/p>\n

Answer:<\/p>\n

Question 6.
\nCF = 12, CB = 3, and CD = 9. Find CE.<\/p>\n

Answer:<\/p>\n

Question 7.
\nBF = 9 and CB = 3. Find CA<\/p>\n

Answer:<\/p>\n

Question 8.
\nSketch a pentagon inscribed in a circle. Label the pentagon ABCDE. Describe the relationship between each pair of angles. Explain your reasoning.<\/p>\n

a. \u2220CDE and \u2220CAE<\/p>\n

Answer:<\/p>\n

b. \u2220CBE and \u2220CAE<\/p>\n

Answer:<\/p>\n

Find the value of the variable. Justify your answer.<\/p>\n

Question 9.
\n\"Big<\/p>\n

Answer:
\nx = 5<\/p>\n

Explanation:
\n5x – 4 = 3x + 6
\n5x – 3x = 6 + 4
\n2x = 10
\nx = 5<\/p>\n

Question 10.
\n\"Big<\/p>\n

Answer:
\nr = 9<\/p>\n

Explanation:
\n(6 + r)\u00b2 = 12\u00b2 + r\u00b2
\n36 + 12r + r\u00b2 = 144 + r\u00b2
\n12r = 108
\nr = 9<\/p>\n

Question 11.
\nProve or disprove that the point (2\u221a2, – 1) lies on the circle centered at (0, 2) and containing the point (- 1, 4).<\/p>\n

Answer:
\nDisproved<\/p>\n

Explanation:
\nWe consider the circle centred at the A(0, 2) and containing the point B(-1, 4).
\nAB = \u221a(-1 – 0)\u00b2 + (4 – 2)\u00b2 = \u221a1 + 4 = \u221a5
\nThe distance between centre A(0, 2) and P(2\u221a2, – 1) is
\nAP = \u221a(2\u221a2 – 0)\u00b2 + (-1 – 2)\u00b2 = \u221a8 + 9 = \u221a17
\nAB \u2260 AP
\nSo, the point (2\u221a2, – 1) dies not lie on the circle.<\/p>\n

Prove the given statement.<\/p>\n

Question 12.
\n\\(\\widehat{S T} \\cong \\widehat{R Q}\\)
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 13.
\n\\(\\widehat{J M} \\cong \\widehat{L M}\\)
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 14.
\n\\(\\widehat{D G} \\cong \\widehat{F G}\\)
\n\"Big<\/p>\n

Answer:<\/p>\n

Question 15.
\nA bank of lighting hangs over a stage. Each light illuminates a circular region on the stage. A coordinate plane is used to arrange the lights, using a corner of the stage as the origin. The equation (x – 13)2<\/sup> + (y – 4)2<\/sup> = 16 represents the boundary of the region illuminated by one of the lights. Three actors stand at the points A(11, 4), B(8, 5), and C(15, 5). Graph the given equation. Then determine which actors are illuminated by the light.<\/p>\n

Answer:
\nThe equation (x – 13)\u00b2 + (y – 4)\u00b2= 16 represents the standard equation of the circle with center (13, 4) and radius 4
\nGraph the circle with center S(13, 4), radius 4. Then graph the points A(11,4), B (8, 5), C(15,5) which represents the places where the actors stand.
\n\"Big
\nFrom the graph, we can see that points A and C inside the circle and point B is outside the circle. Therefore, actors who stand at points A and C are illuminated by the light<\/p>\n

Question 16.
\nIf a car goes around a turn too quickly, it can leave tracks that form an arc of a circle. By finding the radius of the circle, accident investigators can estimate the speed of the car.
\nAnswer:<\/p>\n

\"Big<\/p>\n

a. To find the radius, accident investigators choose points A and B on the tire marks. Then the investigators find the midpoint C of \\(\\overline{A B}\\). Use the diagram to find the radius r of the circle. Explain why this method works.<\/p>\n

Answer:
\nThe radius r of the circle = 155.71 ft<\/p>\n

Explanation:
\nGiven that,
\nAC = 130 ft, CD = 70 ft
\nCE = (r – 70) ft
\nr\u00b2 = a\u00b2 + b\u00b2
\nr\u00b2= 130\u00b2+ (r – 70)\u00b2
\nr\u00b2 = 16900 + r\u00b2 – 140r + 4900
\n140r = 21,800
\nr = 155.71 ft<\/p>\n

b. The formula S = 3.87\u221afr can be used to estimate a car’s speed in miles per hour, where f is the coefficient of friction and r is the radius of the circle in feet. If f = 0.7, estimate the car\u2019s speed in part (a).<\/p>\n

Answer:
\nThe estimated car’s speed is 39.67 miles per hour<\/p>\n

Explanation:
\nS = 3.87\u221afr
\nS = 3.8 \u221a(0.7 x 155.71)
\nS = 3.8 \u221a108.997
\nS = 3.8 x 10.44
\nS = 39.67<\/p>\n

Circles Cumulative Assessment<\/h3>\n

Question 1.
\nClassify each segment as specifically as possible.
\n\"Big
\na. \\(\\overline{B G}\\)<\/p>\n

Answer:
\n\\(\\overline{B G}\\) is a chord<\/p>\n

b. \\(\\overline{C D}\\)<\/p>\n

Answer:
\n\\(\\overline{C D}\\) is radius.<\/p>\n

c. \\(\\overline{A D}\\)<\/p>\n

Answer:
\n\\(\\overline{A D}\\) is diameter.<\/p>\n

d. \\(\\overline{F E}\\)<\/p>\n

Answer:
\n\\(\\overline{F E}\\) is a chord<\/p>\n

Question 2.
\nCopy and complete the paragraph proof.<\/p>\n

\"Big<\/p>\n

Given Circle C with center (2, 1) and radius 1,
\nCircle D with center (0, 3) and radius 4
\nProve Circle C is similar to Circle D.<\/p>\n

Map Circle C to Circle C’ by using the _________ (x, y) \u2192 _________ so that Circle C’ and Circle D have the same center at (____, _____). Dilate Circle C’ using a cellIer of dilation (_____, _____) and a scale factor of _____ . Because there is a _________ transformation that maps Circle C to Circle D, Circle C is __________ Circle D.<\/p>\n

Answer:
\nMap Circle C to Circle C’ by using the scale factor (x, y) \u2192 (0, 3) so that Circle C’ and Circle D have the same center at (2, 1). Dilate Circle C’ using a cellIer of dilation (2, 1) and a scale factor of circles. Because there is a transformation that maps Circle C to Circle D, Circle C is similar to Circle D.<\/p>\n

Question 3.
\nUse the diagram to write a proof.
\n\"Big
\nGiven \u2206JPL \u2245 \u2206NPL
\n\\(\\overline{P K}\\) is an altitude of \u2206JPL
\n\\(\\overline{P M}\\) is an altitude \u2206NPL
\nProve \u2206PKL ~ \u2206NMP<\/p>\n

Answer:
\n\u2206JPL is similar to \u2206NPL and PK is the altitude of \u2206JPL and PM is an altitude of \u2206NPL
\nAltitude is a line drawn from one vertex to the opposite site. It is perpendicular to the side.
\nSo, \u2206PKL is similar to \u2206NMP<\/p>\n

Question 4.
\nThe equation of a circle is x\u00b2 + y\u00b2 + 14x – 16y + 77 = 0. What are the center and radius of the circle?
\n(A) center: (14, – 16). radius: 8.8
\n(B) center: (- 7, 8), radius: 6
\n(C) center (- 14, 16), radius: 8.8
\n(D) center: (7, – 8), radius: 5.2<\/p>\n

Answer:
\n(B) center: (- 7, 8), radius: 6<\/p>\n

Explanation:
\nx\u00b2 + y\u00b2 + 14x – 16y + 77 = 0
\nx\u00b2 + 14x + 49 + y\u00b2 – 16y + 64 = 36
\n(x + 7)\u00b2 + (y – 8)\u00b2 = 6\u00b2
\nSo, the centre is (-7, 8) and radius is 6.<\/p>\n

Question 5.
\nThe coordinates of the vertices of a quadrilateral are W(- 7, – 6), X(1, – 2), Y(3, – 6) and Z(- 5, – 10). Prove that quadrilateral WXYZ is a rectangle.<\/p>\n

Answer:
\nProved<\/p>\n

Explanation:
\nFind the distance of WY and ZX
\nWY = \u221a(-7 – 3)\u00b2 + (-6 + 6)\u00b2 = \u221a(-10)\u00b2 = 10
\nZX = \u221a(1 + 5)\u00b2 + (-2 + 10)\u00b2 = \u221a6\u00b2 + 8\u00b2 = 10
\nWY = ZX, the diagonals are congruent
\nUse the slope formula to find the slopes of diagonals
\nSlope of WY = \\(\\frac { -6 + 6 }{ -7 – 3 } \\) = 0
\nSlope of ZX = \\(\\frac { -2 + 10 }{ 1 + 5 } \\) = \\(\\frac { 4 }{ 3 } \\)
\nBecause the product of slopes of diagonals is 0, the diagonals are not perpendicular
\nTherefore, the quadrilateral WXYZ is a rectangle.<\/p>\n

Question 6.
\nWhich angles have the same measure as \u2220ACB? Select all that apply.
\nim – 295
\n\u2220DEF \u2220JGK \u2220KGL \u2220LGM \u2220MGJ
\n\u2220QNR \u2220STV \u2220SWV \u2220VWU \u2220XYZ
\nAnswer:
\n\u2220VWU<\/p>\n

Question 7.
\nClassify each related conditional statement based on the conditional statement
\n\u201cIf you are a soccer player. then you are an athlete.\u201d
\na. If you are not a soccer player, then you are not an athlete.<\/p>\n

Answer:
\nFalse<\/p>\n

b. If you are an athlete, then you are a soccer player.<\/p>\n

Answer:
\nFalse<\/p>\n

c. You are a soccer player if and only if you are an athlete.<\/p>\n

Answer:
\nTrue<\/p>\n

d. If you are not an athlete, then you are not a soccer player.<\/p>\n

Answer:
\nFalse<\/p>\n

Question 8.
\nYour friend claims that the quadrilateral shown can be inscribed in a circle. Is your friend correct? Explain our reasoning.
\n\"Big<\/p>\n

Answer:
\nIf the sum of any two angles is 180\u00b0, then the quadrilateral is inscribed in a circle.
\nSo, 70\u00b0 + 110\u00b0 = 180\u00b0, 110\u00b0 + 70\u00b0= 180\u00b0
\nSo, my friend is correct.<\/p>\n","protected":false},"excerpt":{"rendered":"

Gain more subject knowledge on Circles concepts with the help of Big Ideas Math Geometry Answers Chapter 10 Circles guide. Shared guide of BIM Textbook Solutions Geometry Ch 10 Circles helps students understand the concepts quickly and concentrate on other math concepts too. Want to get a good grip on the high school-related circle topics? … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":89439,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[4],"tags":[],"jetpack_featured_media_url":"https:\/\/bigideasmathanswers.com\/wp-content\/uploads\/2021\/04\/Big-Ideas-Math-Geometry-Answers-Chapter-10-Circles.png","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50181"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=50181"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50181\/revisions"}],"predecessor-version":[{"id":90291,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/50181\/revisions\/90291"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media\/89439"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=50181"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=50181"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=50181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}