ig Idea 1 H UY I I Using roperties of egments that Intersect ircles or Your otebook You learned several relationships between tangents, secants, and chords. ome of these relationships can help you determine that two chords or tangents are congruent. or eample, tangent segments from the same eterior point are congruent. } > } Other relationships allow you to find the length of a secant or chord if you know the length of related segments. or eample, with the egments of a hord heorem you can find the length of an unknown chord segment. p 5 p ig Idea pplying ngle elationships in ircles You learned to find the measures of angles formed inside, outside, and on circles. ngles formed on circles m 5 1 } m ngles formed inside circles 1 m 1 5 1 } 1 m 1 m, m 5 1 } 1 m 1 m ngles formed outside circles W 3 X Y m 3 5 1 } 1 m XY m W ig Idea 3 Using ircles in the oordinate lane y he standard equation of ( is: ( h) 1 (y k) 5 r ( ) 1 (y 1) 5 1 ( ) 1 (y 1) 5 4 hapter ummary 707
H VIW VIW Y VOUY classzone.com ulti-anguage lossary Vocabulary practice or a list of postulates and theorems, see pp. 96 931. circle, p. 651 center, radius, diameter chord, p. 651 secant, p. 651 tangent, p. 651 central angle, p. 659 minor arc, p. 659 major arc, p. 659 semicircle, p. 659 measure of a minor arc, p. 659 measure of a major arc, p. 659 congruent circles, p. 660 congruent arcs, p. 660 inscribed angle, p. 67 intercepted arc, p. 67 inscribed polygon, p. 674 circumscribed circle, p. 674 segments of a chord, p. 689 secant segment, p. 690 eternal segment, p. 690 standard equation of a circle, p. 699 VOUY XI 1. opy and complete: If a chord passes through the center of a circle, then it is called a(n)?.. raw and describe an inscribed angle and an intercepted arc. 3. WII escribe how the measure of a central angle of a circle relates to the measure of the minor arc and the measure of the major arc created by the angle. In ercises 4 6, match the term with the appropriate segment. 4. angent segment. } 5. ecant segment. } 6. ternal segment. } VIW X XI Use the review eamples and eercises below to check your understanding of the concepts you have learned in each lesson of hapter 10. 10.1 Use roperties of angents pp. 651 658 X In the diagram, and are points of tangency on (. ind the value of. Use heorem 10. to find. 5 angent segments from the same point are >. 1 5 33 1 5 5 33 ubstitute. 5 14 olve for. 708 hapter 10 roperties of ircles
classzone.com hapter eview ractice X 5 and 6 on p. 654 for s. 7 9 XI ind the value of the variable. Y and are points of tangency on (W. 7. W Y 9a 30 3a X 8. X c 1 9c 1 6 9c 1 14 Y W 9. X 3 9 r r W 10. ind rc easures pp. 659 663 X ind the measure of the arc of (. In the diagram, } is a diameter. a. b. c. 1008 108 a. is a minor arc, so m 5 m 5 108. b. is a major arc, so m 5 3608 108 5 408. c. is a semicircle, so m 5 1808. X 1 and on pp. 659 660 for s. 10 13 XI Use the diagram above to find the measure of the indicated arc. 10. 11. 1. 13. 10.3 pply roperties of hords pp. 664 670 X In the diagram, ( > (, } > }, and m 5 758. ind m. y heorem 10.3, } and } are congruent chords in congruent circles, so the corresponding minor arcs and are congruent. o, m 5 m 5 758. 758 X 1, 3, and 4 on pp. 664, 666 for s. 14 16 XI ind the measure of. 14. 618 15. 658 16. 918 hapter eview 709
Use 10.4 H VIW Inscribed ngles and olygons pp. 67 679 X ind the value of each variable. is inscribed in a circle, so by heorem 10.10, opposite angles are supplementary. m 1 m 5 1808 m 1 m 5 1808 3a8 1 3a8 5 1808 b8 1 508 5 1808 6a 5 180 b 5 130 3a8 b8 3a8 508 a 5 30 X 1,, and 5 on pp. 67 675 for s. 17 19 XI ind the value(s) of the variable(s). 17. X c8 Y 568 18. 8 408 19. q8 1008 4r 8 808 10.5 pply Other ngle elationships in ircles pp. 680 686 X ind the value of y. he tangent ] Q and secant ] intersect outside the circle, so you can use heorem 10.13 to find the value of y. 1908 y 8 608 y8 5 1 } 1m Q m Q Use heorem 10.13. y8 5 1 } (1908 608) y 5 65 ubstitute. implify. X and 3 on pp. 681 68 for s. 0 XI ind the value of. 0. 508 8 1. 408 8 968. 158 8 608 710 hapter 10 roperties of ircles
classzone.com hapter eview ractice 10.6 ind egment engths in ircles pp. 689 695 X ind the value of. he chords } and } H intersect inside the circle, so you can use heorem 10.14 to find the value of. p 5 ph Use heorem 10.14. 6 3 p 5 3p 6 ubstitute. H 5 9 olve for. X 4 on p. 69 for. 3 XI 3. I I local park has a circular ice skating rink. You are standing at point, about 1 feet from the edge of the rink. he distance from you to a point of tangency on the rink is about 0 feet. stimate the radius of the rink. r r 0 ft 1 ft 10.7 Write and raph quations of ircles pp. 699 705 X Write an equation of the circle shown. y he radius is and the center is at (, 4). ( h) 1 (y k) 5 r tandard equation of a circle ( ()) 1 (y 4) 5 4 ubstitute. ( 1 ) 1 (y 4) 5 16 implify. X 1,, and 3 on pp. 699 700 for s. 4 3 XI Write an equation of the circle shown. 4. 1 y 5. y 6. y Write the standard equation of the circle with the given center and radius. 7. enter (0, 0), radius 9 8. enter (5, ), radius 1.3 9. enter (6, 1), radius 4 30. enter (3, ), radius 16 31. enter (10, 7), radius 3.5 3. enter (0, 0), radius 5. hapter eview 711
In H (, and are points of tangency. ind the value of the variable. 1.. 3. 5 4 6 r 1 8 17 3 1 6 1 r 8 1 15 ell whether the red arcs are congruent. plain why or why not. 4. 608 608 5. 5 1368 H 48 J 5 6. 1198 etermine whether } is a diameter of the circle. plain your reasoning. 7. 10 8.9 8.9 8 8. 908 9. X 0 14 Y 5 ind the indicated measure. 10. m 11. m 1. m HJ J 1068 438 88 H 13. m 1 14. m 15. m 1 388 J 58 118 1688 48 ind the value of. ound decimal answers to the nearest tenth. 16. 14 8 4 17. 9 1 18. 0 8 3 19. ind the center and radius of a circle that has the standard equation ( 1 ) 1 (y 5) 5 169. 71 hapter 10 roperties of ircles
VIW O IOI IOI lgebra classzone.com X 1 actor using greatest common factor actor 3 1 6. Identify the greatest common factor of the terms. he greatest common factor () is the product of all the common factors. irst, factor each term. 3 5 ppp and 6 5 p 3pp hen, write the product of the common terms. 5 pp5 inally, use the distributive property with the. 3 1 6 5 ( 1 3) X actor binomials and trinomials actor. a. 5 1 3 b. 9 olution a. ake a table of possible factorizations. ecause the middle term, 5, is negative, both factors of the third term, 3, must be negative. actors of actors of 3 ossible factorization iddle term when multiplied 1, 3, 1 ( 3)( 1) 6 5 7 1, 1, 3 ( 1)( 3) 3 5 5 orrect b. Use the special factoring pattern a b 5 (a 1 b)(a b). 95 3 Write in the form a b. 5 ( 1 3)( 3) actor using the pattern. X 1 for s. 1 9 X for s. 10 4 XI actor. 1. 6 1 18 4. 16a 4b 3. 9r 15rs 4. 14 5 1 7 3 5. 8t 4 1 6t 10t 6. 9z 3 1 3z 1 1z 7. 5y 6 4y 5 1 y 3 8. 30v 7 5v 5 10v 4 9. 6 3 y 1 15 y 3 10. 1 6 1 8 11. y y 6 1. a 64 13. z 8z 1 16 14. 3s 1 s 1 15. 5b 16b 1 3 16. 4 4 49 17. 5r 81 18. 4 1 1 1 9 19. 1 10 1 1 0. z 11 1. y 1 y 6. z 1 1z 1 36 3. 49 4. 1 14 lgebra eview 713
tandardized IO UI HOI QUIO If you have difficulty solving a multiple choice question directly, you may be able to use another approach to eliminate incorrect answer choices and obtain the correct answer. O 1 In the diagram, nq is inscribed in a circle. he ratio of the angle measures of nq is 4 : 7 : 7. What is m Q? 08 408 808 1408 HO 1 OV IY Use the Interior ngles heorem to find m Q. hen use the fact that Q intercepts Q to find m Q. 1 Use the ratio of the angle measures to write an equation. ecause n is isosceles, its base angles are congruent. et 48 5 m Q. hen m Q 5 m 5 78. You can write: m Q 1 m Q 1 m 5 1808 48 1 78 1 78 5 1808 olve the equation to find the value of. 48 1 78 1 78 5 1808 188 5 1808 5 10 3 ind m Q. rom tep 1, m Q 5 48, so m Q 5 4 p 108 5 408. 4 ind m Q. ecause Q intercepts Q, m Q 5 p m Q. o, m Q 5 p 408 5 808. he correct answer is. HO II HOI ecause Q intercepts Q, m Q 5 1 } p m Q. lso, because nq is isosceles, its base angles, Q and, are congruent. or each choice, find m Q, m Q, and m. etermine whether the ratio of the angle measures is 4 : 7 : 7. hoice : If m Q 5 08, m Q 5 108. o, m Q 1 m 5 1808 108 5 1708, and m Q 5 m 5 170 } 5 858. he angle measures 108, 858, and 858 are not in the ratio 4 : 7 : 7, so hoice is not correct. hoice : If m Q 5 408, m Q 5 08. o, m Q 1 m 5 1808 08 5 1608, and m Q 5 m 5 808. he angle measures 08, 808, and 808 are not in the ratio 4 : 7 : 7, so hoice is not correct. hoice : If m Q 5 808, m Q 5 408. o, m Q 1 m 5 1808 408 5 1408, and m Q 5 m 5 708. he angle measures 408, 708, and 708 are in the ratio 4 : 7 : 7. o, m Q 5 808. he correct answer is. 714 hapter 10 roperties of ircles
O In the circle shown, } J intersects } at point. What is the value of? 1 7 10 J 6 4 7 HO 1 OV IY Write and solve an equation. 1 Write an equation. y the egments of a hord heorem, Jp 5 p. You can write ( )( 7) 5 6 p 4 5 4. olve the equation. ( )( 7) 5 4 9 1 14 5 4 9 10 5 0 ( 10)( 1 1) 5 0 o, 5 10 or 5 1. 3 ecide which value makes sense. If 5 1, then J 5 1 5 3. ut a distance cannot be negative. If 5 10, then J 5 10 5 8, and 5 10 7 5 3. o, 5 10. he correct answer is. HO II HOI heck to see if any choices do not make sense. 1 heck to see if any choices give impossible values for J and. Use the fact that J 5 and 5 7. hoice : If 5 1, then J 5 3 and 5 8. distance cannot be negative, so you can eliminate hoice. hoice : If 5, then J 5 0 and 5 5. distance cannot be negative or 0, so you can eliminate hoice. hoice : If 5 7, then J 5 5 and 5 0. distance cannot be 0, so you can eliminate hoice. Verify that hoice is correct. y the egments of a hord heorem, ( 7)( ) 5 6(4). his equation is true when 5 10. he correct answer is. XI plain why you can eliminate the highlighted answer choice. 1. In the diagram, what is m Q? 08 08 68 78 408 58. Isosceles trapezoid H is inscribed in a circle, m 5 ( 1 8)8, and m 5 (3 1 1)8. What is the value of? 17 10 40 7 tandardized est reparation 715
tandardized I UI HOI 1. In (, } > } Q. Which statement is not necessarily true? > Q > Q Q > Q Q >. In (, V 5 5 and 5 4 1 14. What is the value of? 6. In the design for a jewelry store sign, UV is inscribed inside a circle, 5 U 5 1 inches, and V 5 UV 5 18 inches. What is the approimate diameter of the circle? 17 in. 5 in. V U in. 30 in. V 7. In the diagram shown, ] Q is tangent to ( at. What is m? 10 3 1 16 3. What are the coordinates of the center of a circle with equation ( 1 ) 1 (y 4) 5 9? (, 4) (, 4) (, 4) (, 4) 4. In the circle shown below, what is m Q? 1058 78 48 78 488 968 5. egular heagon HJ is inscribed in a circle. What is m? 68 608 108 408 68 68 1188 148 368 8. wo distinct circles intersect. What is the maimum number of common tangents? 1 3 4 9. In the circle shown, m 5 1468 and m H 5 178. What is the value of? 38 H 8 10.5 1 4 336 716 hapter 10 roperties of ircles
I classzone.com I W 10. } is tangent to ( at. } is tangent to ( at. ind the value of. 1 1 5 1 11. In (H, find m H in degrees. 1118 1. ind the value of. 6 H 0 HO O 13. plain why n is similar to nq. 14. et 8 be the measure of an inscribed angle, and let y8 be the measure of its intercepted arc. raph y as a function of for all possible values of. ive the slope of the graph. 15. In (J, } J > } JH. Write two true statements about congruent arcs and two true statements about congruent segments in (J. Justify each statement. J H X O 16. he diagram shows a piece of broken pottery found by an archaeologist. he archaeologist thinks that the pottery is part of a circular plate and wants to estimate the diameter of the plate. a. race the outermost arc of the diagram on a piece of paper. raw any two chords whose endpoints lie on the arc. b. onstruct the perpendicular bisector of each chord. ark the point of intersection of the perpendiculars bisectors. How is this point related to the circular plate? c. ased on your results, describe a method the archaeologist could use to estimate the diameter of the actual plate. plain your reasoning. 17. he point (3, 8) lies on a circle with center (, 4). a. Write an equation for (. b. Write an equation for the line that contains radius }. plain. c. Write an equation for the line that is tangent to ( at point. plain. tandardized est ractice 717