Properties of Circles

Transcription

1 roperties of ircles Use roperties of Tangents 10.2 ind rc Measures 10.3 pply roperties of hords 10.4 Use Inscribed ngles and olygons 10.5 pply Other ngle elationships in ircles 10.6 ind egment Lengths in ircles 10.7 Write and Graph quations of ircles efore In previous chapters, you learned the following skills, which you ll use in hapter 10: classifying triangles, finding angle measures, and solving equations. rerequisite kills VOULY HK opy and complete the statement. 1. Two similar triangles have congruent corresponding angles and? corresponding sides. 2. Two angles whose sides form two pairs of opposite rays are called?. 3. The? of an angle is all of the points between the sides of the angle. KILL N LG HK Use the onverse of the ythagorean Theorem to classify the triangle. (eview p. 441 for 10.1.) , 0.8, , 12, , 2, 2.5 ind the value of the variable. (eview pp. 24, 35 for 10.2, 10.4.) (5 1 40)8 5 8 (6 2 8)8 (8 2 2)8 (2 1 2)

2 Now In hapter 10, you will apply the big ideas listed below and reviewed in the hapter ummary on page 707. You will also use the key vocabulary listed below. ig Ideas 1 Using properties of segments that intersect circles 2 pplying angle relationships in circles 3 Using circles in the coordinate plane KY VOULY 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 congruent circles, p. 660 congruent arcs, p. 660 inscribed angle, p. 672 intercepted arc, p. 672 standard equation of a circle, p. 699 Why? ircles can be used to model a wide variety of natural phenomena. You can use properties of circles to investigate the Northern Lights. Geometry The animation illustrated below for ample 4 on page 682 helps you answer this question: rom what part of arth are the Northern Lights visible? Your goal is to determine from what part of arth you can see the Northern Lights. To begin, complete a justification of the statement that >. Geometry at classzone.com Other animations for hapte r 10: page s 655, 661, 671, 691, and

3 Investigating g Geometry 10.1 plore Tangent egments MTIL compass ruler TIVITY Use before Lesson 10.1 QUTION How are the lengths of tangent segments related? line can intersect a circle at 0, 1, or 2 points. If a line is in the plane of a circle and intersects the circle at 1 point, the line is a tangent. XLO raw tangents to a circle T 1 T 2 T 3 raw a circle Use a compass to draw a circle. Label the center. raw tangents raw lines ] and ] so that they intersect ( only at and, respectively. These lines are called tangents. Measure segments } and } are called tangent segments. Measure and compare the lengths of the tangent segments. W ONLUION Use your observations to complete these eercises 1. epeat teps 1 3 with three different circles. 2. Use your results from ercise 1 to make a conjecture about the lengths of tangent segments that have a common endpoint. 3. In the diagram, L, Q, N, and are points of tangency. Use your conjecture from ercise 2 to find LQ and N if LM 5 7 and M L N M 4. In the diagram below,,,, and are points of tangency. Use your conjecture from ercise 2 to eplain why } > }. 650 hapter 10 roperties of ircles

4 Use roperties 10.1 of Tangents efore You found the circumference and area of circles. Now You will use properties of a tangent to a circle. Why? o you can find the range of a G satellite, as in. 37. Key Vocabulary circle center, radius, diameter chord secant tangent circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. circle with center is called circle and can be written (. segment whose endpoints are the center and any point on the circle is a radius. chord is a segment whose endpoints are on a circle. diameter is a chord that contains the center of the circle. chord radius diameter center secant is a line that intersects a circle in two points. tangent is a line in the plane of a circle that intersects the circle in eactly one point, the point of tangency. The tangent ray ] and the tangent segment } are also called tangents. secant point of tangency tangent XML 1 Identify special segments and lines Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of (. a. } b. } c. ] d. ] G olution a. } is a radius because is the center and is a point on the circle. b. } is a diameter because it is a chord that contains the center. ] c. is a tangent ray because it is contained in a line that intersects the circle at only one point. ] d. is a secant because it is a line that intersects the circle in two points. GUI TI for ample 1 1. In ample 1, what word best describes } G? }? 2. In ample 1, name a tangent and a tangent segment Use roperties of Tangents 651

5 VOULY The plural of radius is radii. ll radii of a circle are congruent. IU N IMT The words radius and diameter are used for lengths as well as segments. or a given circle, think of a radius and a diameter as segments and the radius and the diameter as lengths. XML 2 ind lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. adius of ( y b. iameter of ( c. adius of ( d. iameter of ( 1 olution 1 a. The radius of ( is 3 units. b. The diameter of ( is 6 units. c. The radius of ( is 2 units. d. The diameter of ( is 4 units. GUI TI for ample 2 3. Use the diagram in ample 2 to find the radius and diameter of ( and (. OLN IL Two circles can intersect in two points, one point, or no points. oplanar circles that intersect in one point are called tangent circles. oplanar circles that have a common center are called concentric. concentric circles 2 points of intersection 1 point of intersection (tangent circles) no points of intersection VOULY line that intersects a circle in eactly one point is said to be tangent to the circle. OMMON TNGNT line, ray, or segment that is tangent to two coplanar circles is called a common tangent. common tangents 652 hapter 10 roperties of ircles

6 XML 3 raw common tangents Tell how many common tangents the circles have and draw them. a. b. c. olution a. 4 common tangents b. 3 common tangents c. 2 common tangents GUI TI for ample 3 Tell how many common tangents the circles have and draw them THOM or Your Notebook THOM 10.1 In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. m roof: s , p. 658 Line m is tangent to (Q if and only if m } Q. XML 4 Verify a tangent to a circle In the diagram, } T is a radius of (. Is } T tangent to (? 35 T olution Use the onverse of the ythagorean Theorem. ecause , nt is a right triangle and } T } T. o, } T is perpendicular to a radius of ( at its endpoint on (. y Theorem 10.1, } T is tangent to ( Use roperties of Tangents 653

7 XML 5 ind the radius of a circle In the diagram, is a point of tangency. ind the radius r of (. olution You know from Theorem 10.1 that } },son is a right triangle. You can use the ythagorean Theorem (r 1 50) 2 5 r ubstitute. r r r Multiply. 100r ythagorean Theorem ubtract from each side. r 5 39 ft ivide each side by ft 80 ft r r THOM THOM 10.2 Tangent segments from a common eternal point are congruent. roof:. 41, p. 658 or Your Notebook T If } and } T are tangent segments, then } > } T. XML 6 ind the radius of a circle } is tangent to ( at and } T is tangent to ( at T. ind the value of. olution T 5 T Tangent segments from the same point are > ubstitute. 85 olve for. GUI TI for amples 4, 5, and 6 7. Is } tangent to (? 8. } T is tangent to (Q. 9. ind the value(s) ind the value of r. of r r T hapter 10 roperties of ircles

8 10.1 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 7, 19, and 37 5 TNIZ TT TI s. 2, 29, 33, and VOULY opy and complete: The points and are on (. If is a point on }, then } is a?. 2. WITING plain how you can determine from the contet whether the words radius and diameter are referring to a segment or a length. XML 1 on p. 651 for s MTHING TM Match the notation with the term that best describes it. 3.. enter ] 4. H. adius 5. }. hord ] 6.. iameter ] 7.. ecant 8. G. Tangent 9. } G. oint of tangency 10. } H. ommon tangent G H at classzone.com 11. O NLYI escribe and correct the error in the statement about the diagram. 6 9 The length of secant } is 6. XML 2 and 3 on pp for s OOINT GOMTY Use the diagram at the right. 12. What are the radius and diameter of (? 13. What are the radius and diameter of (? 14. opy the circles. Then draw all the common tangents of the two circles y WING TNGNT opy the diagram. Tell how many common tangents the circles have and draw them Use roperties of Tangents 655

9 XML 4 on p. 653 for s XML 5 and 6 on p. 654 for s TMINING TNGNY etermine whether } is tangent to (. plain LG ind the value(s) of the variable. In ercises 24 26, and are points of tangency. 21. r r r 18 r r r OMMON TNGNT common internal tangent intersects the segment that joins the centers of two circles. common eternal tangent does not intersect the segment that joins the centers of the two circles. etermine whether the common tangents shown are internal or eternal MULTIL HOI In the diagram, ( and (Q are tangent circles. } is a common tangent. ind. 22Ï } Ï } ONING In the diagram, ] is tangent to (Q and (. plain why } > } > } even though the radius of (Q is not equal to the radius of (. 31. TNGNT LIN When will two lines tangent to the same circle not intersect? Use Theorem 10.1 to eplain your answer WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

10 32. NGL ITO In the diagram at right, and are points of tangency on (. plain how you know that ] bisects. (Hint: Use Theorem 5.6, page 310.) 33. HOT ON or any point outside of a circle, is there ever only one tangent to the circle that passes through the point? re there ever more than two such tangents? plain your reasoning. 34. HLLNG In the diagram at the right, , 5 8, and all three segments are tangent to (. What is the radius of (? OLM OLVING IYL On modern bicycles, rear wheels usually have tangential spokes. Occasionally, front wheels have radial spokes. Use the definitions of tangent and radius to determine if the wheel shown has tangential spokes or radial spokes XML 4 on p. 653 for GLOL OITIONING YTM (G) G satellites orbit about 11,000 miles above arth. The mean radius of arth is about 3959 miles. ecause G signals cannot travel through arth, a satellite can transmit signals only as far as points and from point, as shown. ind and to the nearest mile. 38. HOT ON In the diagram, } is a common internal tangent (see ercises 27 28) to ( and (. Use similar triangles to eplain why } 5 } Use roperties of Tangents 657

11 39. OVING THOM 10.1 Use parts (a) (c) to prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. GIVN c Line m is tangent to (Q at. OV c m } Q m a. ssume m is not perpendicular to } Q. Then the perpendicular segment from Q to m intersects m at some other point. ecause m is a tangent, cannot be inside (Q. ompare the length Q to Q. b. ecause } Q is the perpendicular segment from Q to m, } Q is the shortest segment from Q to m. Now compare Q to Q. c. Use your results from parts (a) and (b) to complete the indirect proof. 40. OVING THOM 10.1 Write an indirect proof that if a line is perpendicular to a radius at its endpoint, the line is a tangent. GIVN c m } Q OV c Line m is tangent to (Q. m 41. OVING THOM 10.2 Write a proof that tangent segments from a common eternal point are congruent. GIVN c } and } T are tangent to (. OV c } > } T lan for roof Use the Hypotenuse Leg ongruence Theorem to show that n > nt. T 42. HLLNG oint is located at the origin. Line l is tangent to ( at (24, 3). Use the diagram at the right to complete the problem. a. ind the slope of line l. (24, 3) y l b. Write the equation for l. c. ind the radius of (. d. ind the distance from l to ( along the y-ais. MIX VIW VIW repare for Lesson 10.2 in is in the interior of. If m and m 5 708, find m. (p. 24) ind the values of and y. (p. 154) y y (4y 2 7)8 (2 1 3) triangle has sides of lengths 8 and 13. Use an inequality to describe the possible length of the third side. What if two sides have lengths 4 and 11? (p. 328) 658 XT TI for Lesson 10.1, p. 914 ONLIN QUIZ at classzone.com

12 10.2 ind rc Measures efore You found angle measures. Now You will use angle measures to find arc measures. Why? o you can describe the arc made by a bridge, as in. 22. Key Vocabulary central angle minor arc major arc semicircle measure minor arc, major arc congruent circles congruent arcs central angle of a circle is an angle whose verte is the center of the circle. In the diagram, is a central angle of (. If m is less than 1808, then the points on ( that lie in the interior of form a minor arc with endpoints and. The points on ( that do not lie on minor arc form a major arc with endpoints and. semicircle is an arc with endpoints that are the endpoints of a diameter. minor NMING Minor arcs are named by their endpoints. The minor arc associated with is named. Major arcs and semicircles are named by their endpoints and a point on the arc. The major arc associated with can be named. major arc $ KY ONT or Your Notebook Measuring rcs The measure of a minor arc is the measure of its central angle. The epression m is read as the measure of arc. The measure of the entire circle is The measure of a major arc is the difference between 3608 and the measure of the related minor arc. The measure of a semicircle is m m XML 1 ind measures of arcs ind the measure of each arc of (, where } T is a diameter. a. b. T c. T olution T 1108 a. is a minor arc, so m 5 m b. T is a major arc, so m T c. } T is a diameter, so T is a semicircle, and m T ind rc Measures 659

13 JNT Two arcs of the same circle are adjacent if they have a common endpoint. You can add the measures of two adjacent arcs. OTULT OTULT 23 rc ddition ostulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. or Your Notebook m 5 m 1 m XML 2 ind measures of arcs UVY recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. ind the indicated arc measures. a. m b. m c. m d. m Whom Would You ather Meet? Musician thlete Inventor 798 Other ctor MU The measure of a minor arc is less than The measure of a major arc is greater than olution a. m 5 m 1 m b. m 5 m 1 m c. m m d. m m GUI TI for amples 1 and 2 Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. 1. TQ 2. QT 3. TQ 4. Q 5. T 6. T T ONGUNT IL N Two circles are congruent circles if they have the same radius. Two arcs are congruent arcs if they have the same measure and they are arcs of the same circle or of congruent circles. If ( is congruent to (, then you can write ( > (. 660 hapter 10 roperties of ircles

14 XML 3 Identify congruent arcs Tell whether the red arcs are congruent. plain why or why not. a. b. T c U V 958 X Y 958 Z olution a. > because they are in the same circle and m 5 m. b. and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent. c. VX > YZ because they are in congruent circles and mvx 5 myz. at classzone.com GUI TI for ample 3 Tell whether the red arcs are congruent. plain why or why not M N XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 5, 13, and 23 5 TNIZ TT TI s. 2, 11, 17, 18, and VOULY opy and complete: If and are congruent central angles of (, then and are?. 2. WITING What do you need to know about two circles to show that they are congruent? plain. XML 1 and 2 on pp for s MUING } and } are diameters of (. etermine whether the arc is a minor arc, a major arc, or a semicircle of (. Then find the measure of the arc ind rc Measures 661

15 11. MULTIL HOI In the diagram, } Q is a diameter of (. Which arc represents a semicircle? Q Q QT QT T XML 3 on p. 661 for s ONGUNT Tell whether the red arcs are congruent. plain why or why not L 858 M 14. V W X N Y Z 15. O NLYI plain what is wrong with the statement. You cannot tell if ( > ( because the radii are not given. 16. Two diameters of ( are } and }.Ifm 5 208, find m and m. 17. MULTIL HOI ( has a radius of 3 and has a measure of 908. What is the length of }? 3Ï } 2 3Ï } HOT ON On (, m , m G , and m G If H is on ( so that m GH , eplain why H must be on. 19. ONING In (, m 5 608, m 5 258, m 5 708, and m ind two possible values for m. 20. HLLNG In the diagram shown, } Q }, }Q is tangent to (, and mv What is mu? U V 21. HLLNG In the coordinate plane shown, is at the origin. ind the following arc measures on (. a. m y (3, 4) (4, 3) b. m c. m (5, 0) WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

16 OLM OLVING XML 1 on p. 659 for IG The deck of a bascule bridge creates an arc when it is moved from the closed position to the open position. ind the measure of the arc. 23. T On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle? 24. XTN ON surveillance camera is mounted on a corner of a building. It rotates clockwise and counterclockwise continuously between Wall and Wall at a rate of 108 per minute. a. What is the measure of the arc surveyed by the camera? b. How long does it take the camera to survey the entire area once? c. If the camera is at an angle of 858 from Wall while rotating counterclockwise, how long will it take for the camera to return to that same position? d. The camera is rotating counterclockwise and is 508 from Wall. ind the location of the camera after 15 minutes. 25. HLLNG clock with hour and minute hands is set to 1:00.M. a. fter 20 minutes, what will be the measure of the minor arc formed by the hour and minute hands? b. t what time before 2:00.M., to the nearest minute, will the hour and minute hands form a diameter? MIX VIW VIW repare for Lesson 10.3 in s etermine if the lines with the given equations are parallel. (p. 180) 26. y , y 5 5(1 2 ) 27. 2y , y Trace nxyz and point. raw a counterclockwise rotation of nxyz 1458 about. (p. 598) X Z Y ind the product. (p. 641) 29. ( 1 2)( 1 3) 30. (2y 2 5)(y 1 7) 31. ( 1 6)( 2 6) 32. (z 2 3) (3 1 7)(5 1 4) 34. (z 2 1)(z 2 4) XT TI for Lesson 10.2, p. 914 ONLIN QUIZ at classzone.com 663

17 10.3 pply roperties of hords efore You used relationships of central angles and arcs in a circle. Now You will use relationships of arcs and chords in a circle. Why? o you can design a logo for a company, as in. 25. Key Vocabulary chord, p. 651 arc, p. 659 semicircle, p. 659 ecall that a chord is a segment with endpoints on a circle. ecause its endpoints lie on the circle, any chord divides the circle into two arcs. diameter divides a circle into two semicircles. ny other chord divides a circle into a minor arc and a major arc. semicircle diameter semicircle major arc chord minor arc THOM THOM 10.3 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. roof: s , p. 669 or Your Notebook > if and only if } > }. XML 1 Use congruent chords to find an arc measure In the diagram, ( > (Q, } G > } JK, and mjk ind m G. olution ecause } G and } JK are congruent chords in congruent circles, the corresponding minor arcs G and JK are congruent. c o, m G 5 m JK G 808 J K GUI TI for ample 1 Use the diagram of (. 1. If m , find m. 2. If m , find m hapter 10 roperties of ircles

18 ITING If XY > YZ, then the point Y, and any line, segment, or ray that contains Y, bisects XYZ. X } Y bisects XYZ. Y Z THOM or Your Notebook THOM 10.4 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If } Q is a perpendicular bisector of } T, then } Q is a diameter of the circle. roof:. 31, p. 670 T THOM 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If } G is a diameter and } G }, then } H > } H and G > G. roof:. 32, p. 670 H G XML 2 Use perpendicular bisectors GNING Three bushes are arranged in a garden as shown. Where should you place a sprinkler so that it is the same distance from each bush? olution T 1 T 2 T 3 sprinkler Label the bushes,, and, as shown. raw segments } and }. raw the perpendicular bisectors of } and }. y Theorem 10.4, these are diameters of the circle containing,, and. ind the point where these bisectors intersect. This is the center of the circle through,, and, and so it is equidistant from each point pply roperties of hords 665

19 XML 3 Use a diameter Use the diagram of ( to find the length of }. Tell what theorem you use. olution iameter } is perpendicular to }. o, by Theorem 10.5, } bisects },and 5. Therefore, 5 2( ) 5 2(7) GUI TI for amples 2 and 3 ind the measure of the indicated arc in the diagram (80 2 )8 THOM THOM 10.6 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. roof:. 33, p. 670 or Your Notebook G } > } if and only if 5 G. XML 4 Use Theorem 10.6 In the diagram of (, Q 5 T ind U. olution hords } Q and } T are congruent, so by Theorem 10.6 they are equidisant from. Therefore, U 5 V. U 5 V Use Theorem ubstitute. 5 3 olve for. c o, U (3) U V T 16 GUI TI for ample 4 In the diagram in ample 4, suppose T 5 32, and U 5 V ind the given length. 6. Q 7. QU 8. The radius of ( 666 hapter 10 roperties of ircles

20 10.3 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 7, 9, and 25 5 TNIZ TT TI s. 2, 15, 22, and VOULY escribe what it means to bisect an arc. 2. WITING Two chords of a circle are perpendicular and congruent. oes one of them have to be a diameter? plain your reasoning. XML 1 and 3 on pp. 664, 666 for s. 3 5 INING MU ind the measure of the red arc or chord in ( G 8 J H XML 3 and 4 on p. 666 for s LG ind the value of in (Q. plain your reasoning M L N U T H G ONING In ercises 12 14, what can you conclude about the diagram shown? tate a theorem that justifies your answer H G J 14. N L M 15. MULTIL HOI In the diagram of (, which congruence relation is not necessarily true? } Q > } QN MN > M } NL > } L } N > } L M L N 10.3 pply roperties of hords 667

21 16. O NLYI plain what is 17. O NLYI plain why the wrong with the diagram of (. congruence statement is wrong. 6 6 G 7 7 H > INTIYING IMT etermine whether } is a diameter of the circle. plain your reasoning ONING In the diagram of semicircle Q, } ù } and m plain how you can conclude that n ù n. 22. WITING Theorem 10.4 is nearly the converse of Theorem a. Write the converse of Theorem plain how it is different from Theorem b. opy the diagram of ( and draw auiliary segments } and }. Use congruent triangles to prove the converse of Theorem c. Use the converse of Theorem 10.5 to show that Q 5 Q in the diagram of (. 23. LG In ( below, }, }, 24. HLLNG In ( below, the and all arcs have integer measures. how that must be even. T lengths of the parallel chords are 20, 16, and 12. ind m. Q WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

22 OLM OLVING 25. LOGO IGN The owner of a new company would like the company logo to be a picture of an arrow inscribed in a circle, as shown. or symmetry, she wants to be congruent to. How should } and } be related in order for the logo to be eactly as desired? XML 2 on p. 665 for ON-N MTH In the cross section of the submarine shown, the control panels are parallel and the same length. plain two ways you can find the center of the cross section. OVING THOM 10.3 In ercises 27 and 28, prove Theorem GIVN c } and } are congruent chords. OV c > 28. GIVN c } and } are chords and >. OV c } > } 29. HO LNGTH Make and prove a conjecture about chord lengths. a. ketch a circle with two noncongruent chords. Is the longer chord or the shorter chord closer to the center of the circle? epeat this eperiment several times. b. orm a conjecture related to your eperiment in part (a). c. Use the ythagorean Theorem to prove your conjecture. 30. MULTI-T OLM If a car goes around a turn too quickly, it can leave tracks that form an arc of a circle. y finding the radius of the circle, accident investigators can estimate the speed of the car. a. To find the radius, choose points and on the tire marks. Then find the midpoint of }. Measure }, as shown. ind the radius r of the circle. b. The formula Ï } fr can be used to estimate a car s speed in miles per hours, where f is the coefficient of friction and r is the radius of the circle in feet. The coefficient of friction measures how slippery a road is. If f 5 0.7, estimate the car s speed in part (a) pply roperties of hords 669

23 OVING THOM 10.4 N 10.5 Write proofs. 31. GIVN c } Q is the perpendicular 32. GIVN c } G is a diameter of (L. bisector of } T. G } } OV c } Q is a diameter of (L. OV c } > }, G > G lan for roof Use indirect reasoning. lan for roof raw } L and } L. ssume center L is not on } Q.rove Use congruent triangles to show that nl > ntl, so } L } T. Then } > } and LG > LG. use the erpendicular ostulate. Then show G > G. L T G L 33. OVING THOM 10.6 or Theorem 10.6, prove both cases of the biconditional. Use the diagram shown for the theorem on page HLLNG car is designed so that the rear wheel is only partially visible below the body of the car, as shown. The bottom panel is parallel to the ground. rove that the point where the tire touches the ground bisects. MIX VIW VIW repare for Lesson 10.4 in s The measures of the interior angles of a quadrilateral are 1008, 1408, ( 1 20)8, and (2 1 10)8. ind the value of. (p. 507) Quadrilateral JKLM is a parallelogram. Graph ~JKLM. ecide whether it is best described as a rectangle, a rhombus, or a square. (p. 552) 36. J(23, 5), K(2, 5), L(2, 21), M(23, 21) 37. J(25, 2), K(1, 1), L(2, 25), M(24, 24) QUIZ for Lessons etermine whether } is tangent to (. plain your reasoning. (p. 651) If m G , and m 5 808, find m G and m G. (p. 659) 4. The points,, and are on (, } > },andm What is the measure of? (p. 664) 670 XT TI for Lesson 10.3, p. 914 ONLIN QUIZ at classzone.com

24 Investigating g Geometry TIVITY Use before Lesson plore Inscribed ngles MTIL compass straightedge protractor QUTION How are inscribed angles related to central angles? The verte of a central angle is at the center of the circle. The verte of an inscribed angle is on the circle, and its sides form chords of the circle. XLO onstruct inscribed angles of a circle T 1 T 2 T 3 U T U T V V raw a central angle Use a compass to draw a circle. Label the center. Use a straightedge to draw a central angle. Label it. raw points Locate three points on ( in the eterior of and label them T, U, and V. Measure angles raw T, U, and V. These are called inscribed angles. Measure each angle. at classzone.com W ONLUION Use your observations to complete these eercises 1. opy and complete the table. entral angle Inscribed angle 1 Inscribed angle 2 Inscribed angle 3 Name T U V Measure???? 2. raw two more circles. epeat teps 1 3 using different central angles. ecord the measures in a table similar to the one above. 3. Use your results to make a conjecture about how the measure of an inscribed angle is related to the measure of the corresponding central angle Use Inscribed ngles and olygons 671

25 10.4 Use Inscribed ngles and olygons efore You used central angles of circles. Now You will use inscribed angles of circles. Why? o you can take a picture from multiple angles, as in ample 4. Key Vocabulary inscribed angle intercepted arc inscribed polygon circumscribed circle n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. inscribed angle intercepted arc THOM or Your Notebook THOM 10.7 Measure of an Inscribed ngle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc. roof: s , p. 678 m 5 1 } 2 m The proof of Theorem 10.7 in ercises involves three cases. ase 1 enter is on a side of the inscribed angle. ase 2 enter is inside the inscribed angle. ase 3 enter is outside the inscribed angle. XML 1 Use inscribed angles ind the indicated measure in (. a. m T b. m Q olution a. m T 5 } 1 2 m 5 1 }2 (488) T b. m TQ 5 2m 5 2 p ecause TQ is a semicircle, m Q m TQ o, m Q hapter 10 roperties of ircles

26 XML 2 ind the measure of an intercepted arc ind m and m T. What do you notice about T and U? T 318 U olution rom Theorem 10.7, you know that m 5 2m U 5 2(318) lso, m T 5 } 1 2 m 5 1 }2 (628) o, T > U. INTTING TH M ample 2 suggests Theorem THOM or Your Notebook THOM 10.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. roof:. 34, p. 678 > XML 3 tandardized Test ractice Name two pairs of congruent angles in the figure. JKM > KJL, JLM > KML JLM > KJL, JKM > KML J K JKM > JLM, KJL > KML JLM > KJL, JLM > JKM M L LIMINT HOI You can eliminate choices and, because they do not include the pair JKM > JLM. olution Notice that JKM and JLM intercept the same arc, and so JKM > JLM by Theorem lso, KJL and KML intercept the same arc, so they must also be congruent. Only choice contains both pairs of angles. c o, by Theorem 10.8, the correct answer is. GUI TI for amples 1, 2, and 3 ind the measure of the red arc or angle. 1. G H T V 388 U 3. Z Y 728 X W 10.4 Use Inscribed ngles and olygons 673

27 OLYGON polygon is an inscribed polygon if all of its vertices lie on a circle. The circle that contains the vertices is a circumscribed circle. inscribed triangle circumscribed circles inscribed quadrilateral THOM THOM 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. roof:. 35, p. 678 or Your Notebook m if and only if } is a diameter of the circle. XML 4 Use a circumscribed circle HOTOGHY Your camera has a 908 field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way? olution rom Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. o, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera s 908 field of vision from any point on the semicircle in front of the statue. GUI TI for ample 4 4. WHT I? In ample 4, eplain how to find locations if you want to frame the front and left side of the statue in your picture. 674 hapter 10 roperties of ircles

28 INI QUILTL Only certain quadrilaterals can be inscribed in a circle. Theorem describes these quadrilaterals. THOM or Your Notebook THOM quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.,,, and G lie on ( if and only if m 1 m 5 m 1 m G roof:. 30, p. 678; p. 938 G XML 5 Use Theorem ind the value of each variable. a y8 8 b. K L 2a8 4b8 2a8 J 2b8 M olution a. Q is inscribed in a circle, so opposite angles are supplementary. m 1 m m Q 1 m y y b. JKLM is inscribed in a circle, so opposite angles are supplementary. m J 1 m L m K 1 m M a8 1 2a b812b a b a 5 45 b 5 30 GUI TI for ample 5 ind the value of each variable y c8 T U V (2c 2 6) Use Inscribed ngles and olygons 675

29 10.4 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 11, 13, and 29 5 TNIZ TT TI s. 2, 16, 18, 29, and VOULY opy and complete: If a circle is circumscribed about a polygon, then the polygon is? in the circle. 2. WITING plain why the diagonals of a rectangle inscribed in a circle are diameters of the circle. XML 1 and 2 on pp for s. 3 9 INI NGL ind the indicated measure. 3. m 4. m G 5. m N G 1208 N 1608 L M 6. m 7. m VU 8. mwx Y 678 T 308 U W V X 9. O NLYI escribe the error in the diagram of (. ind two ways to correct the error. Q 45º 100º XML 3 ONGUNT NGL Name two pairs of congruent angles. on p. 673 for s K 12. W J M L X Y Z XML 5 LG ind the values of the variables. on p. 675 for s y m k8 608 G 15. J 548 M 4b8 K 3a T 1308 L 676 hapter 10 roperties of ircles

30 16. MULTIL HOI In the diagram, is a central angle and m What is m? INI NGL In each star below, all of the inscribed angles are congruent. ind the measure of an inscribed angle for each star. Then find the sum of all the inscribed angles for each star. a. b. c. 18. MULTIL HOI What is the value of? ( )8 (8 1 10)8 G 19. LLLOGM arallelogram QT is inscribed in (. ind m. ONING etermine whether the quadrilateral can always be inscribed in a circle. plain your reasoning. 20. quare 21. ectangle 22. arallelogram 23. Kite 24. hombus 25. Isosceles trapezoid 26. HLLNG In the diagram, is a right angle. If you draw the smallest possible circle through and tangent to }, the circle will intersect } at J and } at K. ind the eact length of } JK OLM OLVING 27. TONOMY uppose three moons,, and orbit 100,000 kilometers above the surface of a planet. uppose m 5 908, and the planet is 20,000 kilometers in diameter. raw a diagram of the situation. How far is moon from moon? XML 4 on p. 674 for NT 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 a carpenter s square to draw a diameter on the circular piece of wood? 10.4 Use Inscribed ngles and olygons 677

31 29. WITING right triangle is inscribed in a circle and the radius of the circle is given. plain how to find the length of the hypotenuse. 30. OVING THOM opy and complete the proof that opposite angles of an inscribed quadrilateral are supplementary. GIVN c ( with inscribed quadrilateral G OV c m 1 m , m 1 m G y the rc ddition ostulate, m G 1? and m G 1 m Using the? Theorem, m G 5 2m, m G 5 2m, m 5 2m G, and m G 5 2m. y the ubstitution roperty, 2m 1? , so?. imilarly,?. G OVING THOM 10.7 If an angle is inscribed in (Q, the center Q can be on a side of the angle, in the interior of the angle, or in the eterior of the angle. In ercises 31 33, you will prove Theorem 10.7 for each of these cases. 31. ase 1 rove ase 1 of Theorem GIVN c is inscribed in (Q. Let m 5 8. oint Q lies on }. OV c m 5 1 } 2 m lan for roof how that n Q is isosceles. Use the ase ngles Theorem and the terior ngles Theorem to show that m Q Then, show that m olve for, and show that m 5 1 } 2 m ase 2 Use the diagram and auiliary line to write GIVN and OV statements for ase 2 of Theorem Then write a plan for proof. 33. ase 3 Use the diagram and auiliary line to write GIVN and OV statements for ase 3 of Theorem Then write a plan for proof. 34. OVING THOM 10.8 Write a paragraph proof of Theorem irst draw a diagram and write GIVN and OV statements. 35. OVING THOM 10.9 Theorem 10.9 is written as a conditional statement and its converse. Write a plan for proof of each statement. 36. XTN ON In the diagram, ( and (M intersect at, and } is a diameter of (M. plain why ] is tangent to (. M WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

32 HLLNG In ercises 37 and 38, use the following information. You are making a circular cutting board. To begin, you glue eight 1 inch by 2 inch boards together, as shown at the right. Then you draw and cut a circle with an 8 inch diameter from the boards. 37. } H is a diameter of the circular cutting board. Write a proportion relating GJ and JH. tate a theorem to justify your answer. 38. ind J, JH, and JG. What is the length of the cutting board seam labeled } GK? L M J G K H 39. HUTTL To maimize thrust on a N space shuttle, engineers drill an 11-point star out of the solid fuel that fills each booster. They begin by drilling a hole with radius 2 feet, and they would like each side of the star to be 1.5 feet. Is this possible if the fuel cannot have angles greater than 458 at its points? 1.5 ft 2 ft MIX VIW VIW repare for Lesson 10.5 in s ind the approimate length of the hypotenuse. ound your answer to the nearest tenth. (p. 433) Graph the reflection of the polygon in the given line. (p. 589) 43. y-ais y 5 2 y y y G ΠH 1 1 ketch the image of (3, 24) after the described glide reflection. (p. 608) 46. Translation: (, y) (, y 2 2) 47. Translation: (, y) ( 1 1, y 1 4) eflection: in the y-ais eflection: in y 5 4 XT TI for Lesson 10.4, p. 915 ONLIN QUIZ at classzone.com 679

33 10.5 pply Other ngle elationships in ircles efore You found the measures of angles formed on a circle. Now You will find the measures of angles inside or outside a circle. Why o you can determine the part of arth seen from a hot air balloon, as in. 25. Key Vocabulary chord, p. 651 secant, p. 651 tangent, p. 651 You know that the measure of an inscribed angle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. THOM or Your Notebook THOM If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. roof:. 27, p. 685 m } 2 m m }2 m 2 1 XML 1 ind angle and arc measures Line m is tangent to the circle. ind the measure of the red angle or arc. a. b. K m m J 1258 L olution a. m } 2 (1308) b. mkjl 5 2(1258) GUI TI for ample 1 ind the indicated measure. 1. m 1 2. m T 3. m XY T 988 Y 808 X 680 hapter 10 roperties of ircles

34 INTTING LIN N IL If two lines intersect a circle, there are three places where the lines can intersect. on the circle inside the circle outside the circle You can use Theorems and to find measures when the lines intersect inside or outside the circle. THOM THOM ngles Inside the ircle Theorem If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. roof:. 28, p. 685 or Your Notebook 1 2 m 1 5 } 1 1 m 2 1 m 2, m 2 5 } 1 1 m 2 1 m 2 THOM ngles Outside the ircle Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 2 m } 2 1 m 2 m 2 m } 2 1m Q 2 m 2 m }2 1 m XY 2 m WZ 2 roof:. 29, p. 685 W 3 Z X Y XML 2 ind an angle measure inside a circle ind the value of M olution J 8 L The chords } JL and } KM intersect inside the circle. K } 2 1mJM 1 mlk 2 Use Theorem } 2 ( ) ubstitute. implify pply Other ngle elationships in ircles 681

35 XML 3 ind an angle measure outside a circle ind the value of. 8 olution The tangent ] and the secant ] intersect outside the circle. m 5 } 1 1 m 2 m 2 Use Theorem } 2 ( ) 5 51 ubstitute. implify. XML 4 olve a real-world problem IN The Northern Lights are bright flashes of colored light between 50 and 200 miles above arth. uppose a flash occurs 150 miles above arth. What is the measure of arc, the portion of arth from which the flash is visible? (arth s radius is approimately 4000 miles.) 4150 mi 4000 mi olution ecause } and } are tangents, } } and } }. lso, } > } and } > }.o,n > n by the Hypotenuse-Leg ongruence Theorem, and >. olve right n to find that m ø o, m ø 2(74.58) ø Let m 5 8. m 5 1 } 2 1 m 2 m 2 Use Theorem Not drawn to scale VOI O ecause the value for m is an approimation, use the symbol ø instead of ø 1 } 2 [(360828) 2 8] ubstitute. ø 31 olve for. c The measure of the arc from which the flash is visible is about 318. at classzone.com GUI TI for amples 2, 3, and 4 ind the value of the variable. 4. y J K 448 H 308 G a T 3 U hapter 10 roperties of ircles

36 10.5 XI KILL TI HOMWOK KY 5 WOK-OUT OLUTION on p. W1 for s. 3, 9, and 23 5 TNIZ TT TI s. 2, 6, 13, 15, 19, and VOULY opy and complete: The points,,, and are on a circle and ] intersects ] at. If m 5 1 } 2 1m 2 m 2, then is? (inside, on, or outside) the circle. 2. WITING What does it mean in Theorem if m 5 08? Is this consistent with what you learned in Lesson 10.4? plain your answer. XML 1 on p. 680 for s. 3 6 INING MU Line t is tangent to the circle. ind the indicated measure. 3. m 4. m 5. m 1 t t 658 t MULTIL HOI The diagram at the right is not drawn to scale. } is any chord that is not a diameter of the circle. Line m is tangent to the circle at point. Which statement must be true? m 5 90 Þ 90 INING MU ind the value of. XML 2 on p. 681 for s H G 9. J 308 M 8 K L (2 2 30)8 XML 3 on p. 682 for s G U 348 T ( 1 6)8 (3 2 2)8 W V 13. MULTIL HOI In the diagram, l is tangent to the circle at. Which relationship is not true? m m m m T l 10.5 pply Other ngle elationships in ircles 683

37 14. O NLYI escribe the error in the diagram below HOT ON In the diagram at the right, ] L is tangent to the circle and } KJ is a diameter. What is the range of possible angle measures of LJ? plain. K L 16. ONNTI IL The circles below are concentric. a. ind the value of. b. press c in terms of a and b. J a8 b8 c8 17. INI IL In the diagram, the circle is inscribed in nq. ind m, m G,andm G. 18. LG In the diagram, ] is tangent to (. ind m. 408 G WITING oints and are on a circle and t is a tangent line containing and another point. a. raw two different diagrams that illustrate this situation. b. Write an equation for m in terms of m for each diagram. c. When will these equations give the same value for m? HLLNG ind the indicated measure(s). 20. ind m if mwzy ind m and m. Z W Y X J G 1158 H WOK-OUT OLUTION on p. W1 5 TNIZ TT TI

38 OLM OLVING VIO OING In the diagram at the right, television cameras are positioned at,, and to record what happens on stage. The stage is an arc of (. Use the diagram for ercises ind m, m, and m. 23. The wall is tangent to the circle. ind without using the measure of. 24. You would like amera to have a 308 view of the stage. hould you move the camera closer or further away from the stage? plain. XML 4 on p. 682 for HOT I LLOON You are flying in a hot air balloon about 1.2 miles above the ground. Use the method from ample 4 to find the measure of the arc that represents the part of arth that you can see. The radius of arth is about 4000 miles. 26. XTN ON cart is resting on its handle. The angle between the handle and the ground is 148 and the handle connects to the center of the wheel. What are the measures of the arcs of the wheel between the ground and the cart? plain. 27. OVING THOM The proof of Theorem can be split into three cases. The diagram at the right shows the case where } contains the center of the circle. Use Theorem 10.1 to write a paragraph proof for this case. What are the other two cases? (Hint: ee ercises on page 678.) raw a diagram and write plans for proof for the other cases. 28. OVING THOM Write a proof of Theorem GIVN c hords } and } intersect. OV c m } 2 1 m 1 m OVING THOM Use the diagram at the right to prove Theorem for the case of a tangent and a secant. raw }. plain how to use the terior ngle Theorem in the proof of this case. Then copy the diagrams for the other two cases from page 681, draw appropriate auiliary segments, and write plans for proof for these cases pply Other ngle elationships in ircles 685

39 30. OO Q and are points on a circle. is a point outside the circle. } Q and } are tangents to the circle. rove that } Q is not a diameter. 31. HLLNG block and tackle system composed of two pulleys and a rope is shown at the right. The distance between the centers of the pulleys is 113 centimeters and the pulleys each have a radius of 15 centimeters. What percent of the circumference of the bottom pulley is not touching the rope? MIX VIW lassify the dilation and find its scale factor. (p. 626) VIW repare for Lesson 10.6 in s Use the quadratic formula to solve the equation. ound decimal answers to the nearest hundredth. (pp. 641, 883) QUIZ for Lessons ind the value(s) of the variable(s). 1. m 5 z8 (p. 672) 2. m GH 5 z8 (p. 672) 3. m JKL 5 z8 (p. 672) K G 8 y8 J (11 1 y)8 H M L (p. 680) (p. 680) (p. 680) 7. MOUNTIN You are on top of a mountain about 1.37 miles above sea level. ind the measure of the arc that represents the part of arth that you can see. arth s radius is approimately 4000 miles. (p. 680) 686 XT TI for Lesson 10.5, p. 915 ONLIN QUIZ at classzone.com

40 MIX VIW of roblem olving Lessons MULTI-T OLM n official stands 2 meters from the edge of a discus circle and 3 meters from a point of tangency. TT TT TI classzone.com 4. XTN ON The Navy ier erris Wheel in hicago is 150 feet tall and has 40 spokes. 2 m 3 m a. ind the radius of the discus circle. b. How far is the official from the center of the discus circle? 2. GI NW In the diagram, } XY > } YZ and m XQZ ind m YZ in degrees. Y X Z 3. MULTI-T OLM wind turbine has three equally spaced blades that are each 131 feet long. a. ind the measure of the angle between any two spokes. b. Two spokes form a central angle of 728. How many spokes are between the two spokes? c. The bottom of the wheel is 10 feet from the ground. ind the diameter and radius of the wheel. plain your reasoning. 5. ON-N raw a quadrilateral inscribed in a circle. Measure two consecutive angles. Then find the measures of the other two angles algebraically. 6. MULTI-T OLM Use the diagram. y 938 L 8 M 358 N a. What is the measure of the arc between any two blades? b. The highest point reached by a blade is 361 feet above the ground. ind the distance between the lowest point reached by the blades and the ground. c. What is the distance y from the tip of one blade to the tip of another blade? ound your answer to the nearest tenth. a. ind the value of. K b. ind the measures of the other three angles formed by the intersecting chords. 7. HOT ON Use the diagram to show that m 5 y y8 Mied eview of roblem olving 687

41 Investigating g Geometry TIVITY 10.6 Investigate egment Lengths MTIL graphing calculator or computer Use before Lesson 10.6 classzone.com Keystrokes QUTION What is the relationship between the lengths of segments in a circle? You can use geometry drawing software to find a relationship between the segments formed by two intersecting chords. XLO raw a circle with two chords =6.93 = T 1 raw a circle raw a circle and choose four points on the circle. Label them,,, and. T 2 raw secants raw secants ] and ] and label the intersection point. T 3 Measure segments Note that } and } are chords. Measure }, }, }, and } in your diagram. T 4 erform calculations alculate the products p and p. W ONLUION Use your observations to complete these eercises 1. What do you notice about the products you found in tep 4? 2. rag points,,, and, keeping point inside the circle. What do you notice about the new products from tep 4? 3. Make a conjecture about the relationship between the four chord segments. 4. Let } Q and } be two chords of a circle that intersect at the point T. If T 5 9, QT 5 5, and T 5 15, use your conjecture from ercise 3 to find T. 688 hapter 10 roperties of ircles

42 10.6 ind egment Lengths in ircles efore You found angle and arc measures in circles. Now You will find segment lengths in circles. Why? o you can find distances in astronomy, as in ample 4. Key Vocabulary segments of a chord secant segment eternal segment When two chords intersect in the interior of a circle, each chord is divided into two segments that are called segments of the chord. THOM or Your Notebook THOM egments of hords Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. roof:. 21, p. 694 p 5 p lan for roof To prove Theorem 10.14, construct two similar triangles. The lengths of the corresponding sides are proportional, so } 5 }. y the ross roducts roperty, p 5 p. XML 1 ind lengths using Theorem LG ind ML and JK. M olution NK p NJ 5 NL p NM Use Theorem p ( 1 4) 5 ( 1 1) p ( 1 2) ubstitute implify. K 1 2 N L J ubtract 2 from each side. 5 2 olve for. ind ML and JK by substitution. ML 5 ( 1 2) 1 ( 1 1) JK 5 1 ( 1 4) ind egment Lengths in ircles 689