roperties of ircles 1010.1 Use roperties of Tangents 10.2 ind rc Measures 10.3 ppl roperties of hords 10.4 Use Inscribed ngles and olgons 10.5 ppl Other ngle elationships in ircles 10.6 ind egment Lengths in ircles 10.7 Write and Graph Equations of ircles efore In previous chapters, ou learned the following skills, which ou ll use in hapter 10: classifing triangles, finding angle measures, and solving equations. rerequisite kills VOULY HEK op 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 ras are called?. 3. The? of an angle is all of the points between the sides of the angle. KILL N LGE HEK Use the onverse of the thagorean Theorem to classif the triangle. (eview p. 441 for 10.1.) 4. 0.6, 0.8, 0.9 5. 11, 12, 17 6. 1.5, 2, 2.5 ind the value of the variable. (eview pp. 24, 35 for 10.2, 10.4.) 7. 8. 9. 5 8 (6 2 8)8 (8 2 2)8 (2 1 2)8 (5 1 40)8 78 648
Now In hapter 10, ou will appl the big ideas listed below and reviewed in the hapter ummar on page 707. You will also use the ke vocabular listed below. ig Ideas 1 Using properties of segments that intersect circles 2 ppling angle relationships in circles 3 Using circles in the coordinate plane KEY 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 Wh? ircles can be used to model a wide variet of natural phenomena. You can use properties of circles to investigate the Northern Lights. Geometr The animation illustrated below for Eample 4 on page 682 helps ou answer this question: rom what part of Earth are the Northern Lights visible? Your goal is to determine from what part of Earth ou can see the Northern Lights. To begin, complete a justification of the statement that >. Geometr at classzone.com Other animations for hapte r 10: page s 655, 661, 671, 691, and 701 649
Investigating g Geometr 10.1 Eplore Tangent egments MTEIL compass ruler TIVITY Use before Lesson 10.1 Q U E T I O N 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. E X L O E raw tangents to a circle TE 1 TE 2 TE 3 raw a circle Use a compass to draw a circle. Label the center. raw tangents raw lines ] and ] so that the intersect ( onl at and, respectivel. These lines are called tangents. Measure segments } and } are called tangent segments. Measure and compare the lengths of the tangent segments. W O N L U I O N Use our observations to complete these eercises 1. epeat teps 1 3 with three different circles. 2. Use our results from Eercise 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 tangenc. Use our conjecture from Eercise 2 to find LQ and N if LM 5 7 and M 5 5.5. L N 7 5.5 M 4. In the diagram below,,,, and E are points of tangenc. Use our conjecture from Eercise 2 to eplain wh } > } E. E 650 hapter 10 roperties of ircles
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. Wh? o ou can find the range of a G satellite, as in E. 37. Ke Vocabular 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 an 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 eactl one point, the point of tangenc. The tangent ra ] and the tangent segment } are also called tangents. secant point of tangenc tangent E X M L E 1 Identif special segments and lines Tell whether the line, ra, or segment is best described as a radius, chord, diameter, secant, or tangent of (. a. } b. } c. ] E d. ] E G E 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. E is a tangent ra because it is contained in a line that intersects the circle at onl one point. ] d. E is a secant because it is a line that intersects the circle in two points. GUIE TIE for Eample 1 1. In Eample 1, what word best describes } G? }? 2. In Eample 1, name a tangent and a tangent segment. 10.1 Use roperties of Tangents 651
E VOULY The plural of radius is radii. ll radii of a circle are congruent. IU N IMETE 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. E X M L E 2 ind lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. adius of ( 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. GUIE TIE for Eample 2 3. Use the diagram in Eample 2 to find the radius and diameter of ( and (. OLN ILE 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 E VOULY line that intersects a circle in eactl one point is said to be tangent to the circle. OMMON TNGENT line, ra, or segment that is tangent to two coplanar circles is called a common tangent. common tangents 652 hapter 10 roperties of ircles
E X M L E 3 raw common tangents Tell how man common tangents the circles have and draw them. a. b. c. olution a. 4 common tangents b. 3 common tangents c. 2 common tangents GUIE TIE for Eample 3 Tell how man common tangents the circles have and draw them. 4. 5. 6. THEOEM or Your Notebook THEOEM 10.1 In a plane, a line is tangent to a circle if and onl if the line is perpendicular to a radius of the circle at its endpoint on the circle. m roof: Es. 39 40, p. 658 Line m is tangent to (Q if and onl if m } Q. E X M L E 4 Verif a tangent to a circle In the diagram, } T is a radius of (. Is } T tangent to (? 35 T 12 37 olution Use the onverse of the thagorean Theorem. ecause 12 2 1 35 2 5 37 2, nt is a right triangle and } T } T. o, } T is perpendicular to a radius of ( at its endpoint on (. Theorem 10.1, } T is tangent to (. 10.1 Use roperties of Tangents 653
E X M L E 5 ind the radius of a circle In the diagram, is a point of tangenc. ind the radius r of (. olution You know from Theorem 10.1 that } }, so n is a right triangle. You can use the thagorean Theorem. 2 5 2 1 2 (r 1 50) 2 5 r 2 1 80 2 ubstitute. r 2 1 100r 1 2500 5 r 2 1 6400 100r 5 3900 thagorean Theorem Multipl. ubtract from each side. r 5 39 ft ivide each side b 100. 50 ft 80 ft r r THEOEM THEOEM 10.2 Tangent segments from a common eternal point are congruent. roof: E. 41, p. 658 or Your Notebook T If } and } T are tangent segments, then } > } T. E X M L E 6 ind the radius of a circle } is tangent to ( at and } T is tangent to ( at T. ind the value of. olution 28 3 1 4 T 5 T Tangent segments from the same point are >. 28 5 3 1 4 ubstitute. 8 5 olve for. GUIE TIE for Eamples 4, 5, and 6 7. Is } E tangent to (? 8. } T is tangent to (Q. 9. ind the value(s) ind the value of r. of. 3 2 4 E r r 18 24 T 9 2 654 hapter 10 roperties of ircles
10.1 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 for Es. 7, 19, and 37 5 TNIZE TET TIE Es. 2, 29, 33, and 38 1. VOULY op and complete: The points and are on (. If is a point on }, then } is a?. 2. WITING Eplain how ou can determine from the contet whether the words radius and diameter are referring to a segment or a length. EXMLE 1 on p. 651 for Es. 3 11 MTHING TEM Match the notation with the term that best describes it. 3.. enter ] 4. H. adius 5. }. hord ] 6.. iameter ] 7. E E. ecant 8. G. Tangent 9. } G. oint of tangenc 10. } H. ommon tangent E G H at classzone.com 11. EO NLYI escribe and correct the error in the statement about the diagram. 6 9 E The length of secant } is 6. EXMLE 2 and 3 on pp. 652 653 for Es. 12 17 OOINTE GEOMETY Use the diagram at the right. 12. What are the radius and diameter of (? 13. What are the radius and diameter of (? 14. op the circles. Then draw all the common tangents of the two circles. 9 6 3 3 6 9 WING TNGENT op the diagram. Tell how man common tangents the circles have and draw them. 15. 16. 17. 10.1 Use roperties of Tangents 655
EXMLE 4 on p. 653 for Es. 18 20 EXMLE 5 and 6 on p. 654 for Es. 21 26 ETEMINING TNGENY etermine whether } is tangent to (. Eplain. 18. 3 4 5 19. LGE ind the value(s) of the variable. In Eercises 24 26, and are points of tangenc. 21. r r 16 24 22. 9 6 9 15 r 18 r 20. 23. 14 r r 7 52 48 10 24. 3 1 10 7 2 6 25. 2 2 1 5 13 26. 4 2 1 3 2 1 4 2 4 OMMON TNGENT 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. 27. 28. 29. MULTILE HOIE In the diagram, ( and (Q are tangent circles. } is a common tangent. ind. 22Ï } 15 4 2Ï } 15 8 5 3 30. EONING In the diagram, ] is tangent to (Q and (. Eplain wh } > } > } even though the radius of (Q is not equal to the radius of (. 31. TNGENT LINE When will two lines tangent to the same circle not intersect? Use Theorem 10.1 to eplain our answer. 656 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
32. NGLE IETO In the diagram at right, and are points of tangenc on (. Eplain how ou know that ] bisects. (Hint: Use Theorem 5.6, page 310.) 33. HOT EONE or an point outside of a circle, is there ever onl one tangent to the circle that passes through the point? re there ever more than two such tangents? Eplain our reasoning. 34. HLLENGE In the diagram at the right, 5 5 12, 5 8, and all three segments are tangent to (. What is the radius of (? E OLEM OLVING IYLE On modern biccles, rear wheels usuall have tangential spokes. Occasionall, front wheels have radial spokes. Use the definitions of tangent and radius to determine if the wheel shown has tangential spokes or radial spokes. 35. 36. EXMLE 4 on p. 653 for E. 37 37. GLOL OITIONING YTEM (G) G satellites orbit about 11,000 miles above Earth. The mean radius of Earth is about 3959 miles. ecause G signals cannot travel through Earth, a satellite can transmit signals onl as far as points and from point, as shown. ind and to the nearest mile. 38. HOT EONE In the diagram, } is a common internal tangent (see Eercises 27 28) to ( and (. Use similar triangles to eplain wh } 5 }. 10.1 Use roperties of Tangents 657
39. OVING THEOEM 10.1 Use parts (a) (c) to prove indirectl that if a line is tangent to a circle, then it is perpendicular to a radius. GIVEN c Line m is tangent to (Q at. OVE 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 our results from parts (a) and (b) to complete the indirect proof. 40. OVING THEOEM 10.1 Write an indirect proof that if a line is perpendicular to a radius at its endpoint, the line is a tangent. GIVEN c m } Q OVE c Line m is tangent to (Q. m 41. OVING THEOEM 10.2 Write a proof that tangent segments from a common eternal point are congruent. GIVEN c } and } T are tangent to (. OVE c } > } T lan for roof Use the Hpotenuse Leg ongruence Theorem to show that n > nt. T 42. HLLENGE 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 linel. (24, 3) l b. Write the equation for l. c. ind the radius of (. d. ind the distance fromlto ( along the -ais. MIXE EVIEW EVIEW repare for Lesson 10.2 in E. 43. 43. is in the interior of. If m 5 258 and m 5 708, find m. (p. 24) ind the values of and. (p. 154) 44. 8 508 8 45. 1028 8 38 46. (4 2 7)8 (2 1 3)8 1378 47. triangle has sides of lengths 8 and 13. Use an inequalit to describe the possible length of the third side. What if two sides have lengths 4 and 11? (p. 328) 658 EXT TIE for Lesson 10.1, p. 914 ONLINE QUIZ at classzone.com
10.2 ind rc Measures efore You found angle measures. Now You will use angle measures to find arc measures. Wh? o ou can describe the arc made b a bridge, as in E. 22. Ke Vocabular 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. NMING Minor arcs are named b their endpoints. The minor arc associated with is named. Major arcs and semicircles are named b their endpoints and a point on the arc. The major arc associated with can be named. major arc $ minor arc @ KEY ONET 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 3608. 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 1808. m 5 508 508 m 5 3608 2 508 5 3108 E X M L E 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 5 1108. b. T is a major arc, so m T 5 3608 2 1108 5 2508. c. } T is a diameter, so T is a semicircle, and m T 5 1808. 10.2 ind rc Measures 659
JENT Two arcs of the same circle are adjacent if the have a common endpoint. You can add the measures of two adjacent arcs. OTULTE OTULTE 23 rc ddition ostulate The measure of an arc formed b two adjacent arcs is the sum of the measures of the two arcs. or Your Notebook m 5 m 1 m E X M L E 2 ind measures of arcs UVEY recent surve asked teenagers if the 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 E Whom Would You ather Meet? Musician thlete 1088 838 618 298 Inventor 798 Other E ctor MEUE The measure of a minor arc is less than 1808. The measure of a major arc is greater than 1808. olution a. m 5 m 1 m b. m 5 m 1 m 5 298 1 1088 5 1378 1 838 5 1378 5 2208 c. m 5 3608 2 m d. m E 5 3608 2 me 5 3608 2 1378 5 3608 2 618 5 2238 5 2998 GUIE TIE for Eamples 1 and 2 Identif 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 1208 808 608 ONGUENT ILE N Two circles are congruent circles if the have the same radius. Two arcs are congruent arcs if the have the same measure and the are arcs of the same circle or of congruent circles. If ( is congruent to (, then ou can write ( > (. 660 hapter 10 roperties of ircles
E X M L E 3 Identif congruent arcs Tell whether the red arcs are congruent. Eplain wh or wh not. a. E b. T c. 808 808 U V 958 X Y 958 Z olution a. > E because the are in the same circle and m 5 me. b. and TU have the same measure, but are not congruent because the are arcs of circles that are not congruent. c. VX > YZ because the are in congruent circles and mvx 5 myz. at classzone.com GUIE TIE for Eample 3 Tell whether the red arcs are congruent. Eplain wh or wh not. 7. 1458 1458 8. M N 1208 1208 5 4 10.2 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 for Es. 5, 13, and 23 5 TNIZE TET TIE Es. 2, 11, 17, 18, and 24 1. VOULY op and complete: If and E are congruent central angles of (, then and E are?. 2. WITING What do ou need to know about two circles to show that the are congruent? Eplain. EXMLE 1 and 2 on pp. 659 660 for Es. 3 11 MEUING } and } E 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. 3. 4. 5. 6. E 7. 8. 9. 10. E E 458 708 10.2 ind rc Measures 661
11. MULTILE HOIE In the diagram, } Q is a diameter of (. Which arc represents a semicircle? Q Q QT QT T EXMLE 3 on p. 661 for Es. 12 14 ONGUENT Tell whether the red arcs are congruent. Eplain wh or wh not. 12. 1808 708 408 13. L 858 M 14. V 928 8 W X 928 16 N Y Z 15. EO NLYI Eplain what is wrong with the statement. You cannot tell if ( > ( because the radii are not given. 16. Two diameters of ( are } and }. If m 5 208, find m and m. 17. MULTILE HOIE ( has a radius of 3 and has a measure of 908. What is the length of }? 3Ï } 2 3Ï } 3 6 9 18. HOT EONE On (, me 5 1008, m G 5 1208, and m EG 5 2208. If H is on ( so that m GH 5 1508, eplain wh H must be on E. 19. EONING In (, m 5 608, m 5 258, m 5 708, and me 5 208. ind two possible values for me. 20. HLLENGE In the diagram shown, } Q }, }Q is tangent to (, and mv 5 608. What is mu? U V 21. HLLENGE In the coordinate plane shown, is at the origin. ind the following arc measures on (. a. m (3, 4) (4, 3) b. m c. m (5, 0) 662 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
OLEM OLVING EXMLE 1 on p. 659 for E. 22 22. IGE 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 twent congruent sections. What is the measure of each arc in this circle? 24. EXTENE EONE surveillance camera is mounted on a corner of a building. It rotates clockwise and counterclockwise continuousl between Wall and Wall at a rate of 108 per minute. a. What is the measure of the arc surveed b the camera? b. How long does it take the camera to surve 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. HLLENGE 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 b 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? MIXE EVIEW EVIEW repare for Lesson 10.3 in Es. 26 27. etermine if the lines with the given equations are parallel. (p. 180) 26. 5 5 1 2, 5 5(1 2 ) 27. 2 1 2 5 5, 5 4 2 28. 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. (2 2 5)( 1 7) 31. ( 1 6)( 2 6) 32. (z 2 3) 2 33. (3 1 7)(5 1 4) 34. (z 2 1)(z 2 4) EXT TIE for Lesson 10.2, p. 914 ONLINE QUIZ at classzone.com 663
10.3 ppl 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. Wh? o ou can design a logo for a compan, as in E. 25. Ke Vocabular 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, an chord divides the circle into two arcs. diameter divides a circle into two semicircles. n other chord divides a circle into a minor arc and a major arc. semicircle diameter semicircle major arc chord minor arc THEOEM THEOEM 10.3 In the same circle, or in congruent circles, two minor arcs are congruent if and onl if their corresponding chords are congruent. roof: Es. 27 28, p. 669 or Your Notebook > if and onl if } > }. E X M L E 1 Use congruent chords to find an arc measure In the diagram, ( > (Q, } G > } JK, and mjk 5 808. 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 5 808. G 808 J K GUIE TIE for Eample 1 Use the diagram of (. 1. If m 5 1108, find m. 2. If m 5 1508, find m. 9 9 664 hapter 10 roperties of ircles
IETING If XY > YZ, then the point Y, and an line, segment, or ra that contains Y, bisects XYZ. X } Y bisects XYZ. Y Z THEOEM or Your Notebook THEOEM 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: E. 31, p. 670 T THEOEM 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If } EG is a diameter and } EG }, then } H > } H and G > G. roof: E. 32, p. 670 E H G E X M L E 2 Use perpendicular bisectors GENING Three bushes are arranged in a garden as shown. Where should ou place a sprinkler so that it is the same distance from each bush? olution TE 1 TE 2 TE 3 sprinkler Label the bushes,, and, as shown. raw segments } and }. raw the perpendicular bisectors of } and }. 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. 10.3 ppl roperties of hords 665
E X M L E 3 Use a diameter Use the diagram of (E to find the length of }. Tell what theorem ou use. olution iameter } is perpendicular to }. o, b Theorem 10.5, } bisects }, and 5. Therefore, 5 2( ) 5 2(7) 5 14. 7 E GUIE TIE for Eamples 2 and 3 ind the measure of the indicated arc in the diagram. 3. 4. E 5. E 9 8 E (80 2 )8 THEOEM THEOEM 10.6 In the same circle, or in congruent circles, two chords are congruent if and onl if the are equidistant from the center. roof: E. 33, p. 670 or Your Notebook G E } > } if and onl if E 5 EG. E X M L E 4 Use Theorem 10.6 In the diagram of (, Q 5 T 5 16. ind U. olution hords } Q and } T are congruent, so b Theorem 10.6 the are equidisant from. Therefore, U 5 V. U 5 V Use Theorem 10.6. 2 5 5 2 9 ubstitute. 16 U 2 5 2 9 V 16 T 5 3 olve for. c o, U 5 2 5 2(3) 5 6. GUIE TIE for Eample 4 In the diagram in Eample 4, suppose T 5 32, and U 5 V 5 12. ind the given length. 6. Q 7. QU 8. The radius of ( 666 hapter 10 roperties of ircles
10.3 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 for Es. 7, 9, and 25 5 TNIZE TET TIE Es. 2, 15, 22, and 26 1. 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? Eplain our reasoning. EXMLE 1 and 3 on pp. 664, 666 for Es. 3 5 INING MEUE ind the measure of the red arc or chord in (. 3. 758 E 4. 1288 34 34 5. G 8 E J H EXMLE 3 and 4 on p. 666 for Es. 6 11 6. LGE ind the value of in (Q. Eplain our reasoning. 7. 3 1 7 4 M 8. 5 2 6 L 908 2 1 9 N U T 6 1 9 8 2 13 9. 18 5 27 10. 3 1 2 6 6 22 11. E H 15 15 4 1 1 1 8 G EONING In Eercises 12 14, what can ou conclude about the diagram shown? tate a theorem that justifies our answer. 12. E 13. H G J 14. N L M 15. MULTILE HOIE In the diagram of (, which congruence relation is not necessaril true? } Q > } QN MN > M } NL > } L } N > } L M L N 10.3 ppl roperties of hords 667
16. EO NLYI Eplain what is 17. EO NLYI Eplain wh the wrong with the diagram of (. congruence statement is wrong. E 6 6 G 7 7 H E > IENTIYING IMETE etermine whether } is a diameter of the circle. Eplain our reasoning. 18. 4 6 6 9 19. 20. 3 3 E 5 21. EONING In the diagram of semicircle Q, } ù } and m 5 308. Eplain how ou can conclude that n ù n. 22. WITING Theorem 10.4 is nearl the converse of Theorem 10.5. a. Write the converse of Theorem 10.5. Eplain how it is different from Theorem 10.4. b. op the diagram of ( and draw auiliar segments } and }. Use congruent triangles to prove the converse of Theorem 10.5. c. Use the converse of Theorem 10.5 to show that Q 5 Q in the diagram of (. 23. LGE In ( below, }, }, 24. HLLENGE 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 8 668 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
OLEM OLVING 25. LOGO EIGN The owner of a new compan would like the compan logo to be a picture of an arrow inscribed in a circle, as shown. or smmetr, she wants to be congruent to. How should } and } be related in order for the logo to be eactl as desired? EXMLE 2 on p. 665 for E. 26 26. OEN-ENE MTH In the cross section of the submarine shown, the control panels are parallel and the same length. Eplain two was ou can find the center of the cross section. OVING THEOEM 10.3 In Eercises 27 and 28, prove Theorem 10.3. 27. GIVEN c } and } are congruent chords. OVE c > 28. GIVEN c } and } are chords and >. OVE c } > } 29. HO LENGTH 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 our eperiment in part (a). c. Use the thagorean Theorem to prove our conjecture. 30. MULTI-TE OLEM If a car goes around a turn too quickl, it can leave tracks that form an arc of a circle. 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 5 3.86Ï } 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 slipper a road is. If f 5 0.7, estimate the car s speed in part (a). 10.3 ppl roperties of hords 669
OVING THEOEM 10.4 N 10.5 Write proofs. 31. GIVEN c } Q is the perpendicular 32. GIVEN c } EG is a diameter of (L. bisector of } T. EG } } OVE c } Q is a diameter of (L. OVE 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 E 33. OVING THEOEM 10.6 or Theorem 10.6, prove both cases of the biconditional. Use the diagram shown for the theorem on page 666. 34. HLLENGE car is designed so that the rear wheel is onl partiall visible below the bod of the car, as shown. The bottom panel is parallel to the ground. rove that the point where the tire touches the ground bisects. MIXE EVIEW EVIEW repare for Lesson 10.4 in Es. 35 37. 35. 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 10.1 10.3 etermine whether } is tangent to (. Eplain our reasoning. (p. 651) 1. 12 15 9 2. 12 5 9 3. If m EG 5 1958, and m E 5 808, find m G and m EG. (p. 659) 4. The points,, and are on (, } > }, and m 5 1948. What is the measure of? (p. 664) 670 EXT TIE for Lesson 10.3, p. 914 ONLINE QUIZ at classzone.com
Investigating g Geometr TIVITY Use before Lesson 10.4 10.4 Eplore Inscribed ngles M T E I L compass straightedge protractor Q U E T I O N 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. E X L O E onstruct inscribed angles of a circle TE 1 TE 2 TE 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 O N L U I O N Use our observations to complete these eercises 1. op 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 our results to make a conjecture about how the measure of an inscribed angle is related to the measure of the corresponding central angle. 10.4 Use Inscribed ngles and olgons 671
10.4 Use Inscribed ngles and olgons efore You used central angles of circles. Now You will use inscribed angles of circles. Wh? o ou can take a picture from multiple angles, as in Eample 4. Ke Vocabular inscribed angle intercepted arc inscribed polgon 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 THEOEM or Your Notebook THEOEM 10.7 Measure of an Inscribed ngle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc. roof: Es. 31 33, p. 678 m 5 1 } 2 m The proof of Theorem 10.7 in Eercises 31 33 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. E X M L E 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) 5 248 T 508 488 b. m TQ 5 2m 5 2p 508 5 1008. ecause TQ is a semicircle, m Q 5 1808 2 m TQ 5 1808 2 1008 5 808. o, m Q 5 808. 672 hapter 10 roperties of ircles
E X M L E 2 ind the measure of an intercepted arc ind m and m T. What do ou notice about T and U? T 318 U olution rom Theorem 10.7, ou know that m 5 2m U 5 2(318) 5 628. lso, m T 5 } 1 2 m 5 1 }2 (628) 5 318. o, T > U. INTEETING THE ME Eample 2 suggests Theorem 10.8. THEOEM or Your Notebook THEOEM 10.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. roof: E. 34, p. 678 > E X M L E 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 ELIMINTE HOIE You can eliminate choices and, because the do not include the pair JKM > JLM. olution Notice that JKM and JLM intercept the same arc, and so JKM > JLM b Theorem 10.8. lso, KJL and KML intercept the same arc, so the must also be congruent. Onl choice contains both pairs of angles. c o, b Theorem 10.8, the correct answer is. GUIE TIE for Eamples 1, 2, and 3 ind the measure of the red arc or angle. 1. G H 908 2. T V 388 U 3. Z Y 728 X W 10.4 Use Inscribed ngles and olgons 673
OLYGON polgon is an inscribed polgon 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 THEOEM THEOEM 10.9 If a right triangle is inscribed in a circle, then the hpotenuse is a diameter of the circle. onversel, 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: E. 35, p. 678 or Your Notebook m 5 908 if and onl if } is a diameter of the circle. E X M L E 4 Use a circumscribed circle HOTOGHY Your camera has a 908 field of vision and ou want to photograph the front of a statue. You move to a spot where the statue is the onl thing captured in our picture, as shown. You want to change our position. Where else can ou stand so that the statue is perfectl framed in this wa? olution rom Theorem 10.9, ou know that if a right triangle is inscribed in a circle, then the hpotenuse 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 perfectl within our camera s 908 field of vision from an point on the semicircle in front of the statue. GUIE TIE for Eample 4 4. WHT I? In Eample 4, eplain how to find locations if ou want to frame the front and left side of the statue in our picture. 674 hapter 10 roperties of ircles
INIE QUILTEL Onl certain quadrilaterals can be inscribed in a circle. Theorem 10.10 describes these quadrilaterals. THEOEM or Your Notebook THEOEM 10.10 quadrilateral can be inscribed in a circle if and onl if its opposite angles are supplementar., E,, and G lie on ( if and onl if m 1 m 5 m E 1 m G 5 1808. roof: E. 30, p. 678; p. 938 E G E X M L E 5 Use Theorem 10.10 ind the value of each variable. a. 808 758 8 8 b. K L 2a8 4b8 2a8 J 2b8 M olution a. Q is inscribed in a circle, so opposite angles are supplementar. m 1 m 5 1808 m Q 1 m 5 1808 758 1 8 5 1808 808 1 8 5 1808 5 105 5 100 b. JKLM is inscribed in a circle, so opposite angles are supplementar. m J 1 m L 5 1808 m K 1 m M 5 1808 2a8 1 2a8 5 1808 4b8 1 2b8 5 1808 4a 5 180 6b 5 180 a 5 45 b 5 30 GUIE TIE for Eample 5 ind the value of each variable. 5. 8 8 688 828 6. c8 T 108 88 U V (2c 2 6)8 10.4 Use Inscribed ngles and olgons 675
10.4 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 for Es. 11, 13, and 29 5 TNIZE TET TIE Es. 2, 16, 18, 29, and 36 1. VOULY op and complete: If a circle is circumscribed about a polgon, then the polgon is? in the circle. 2. WITING Eplain wh the diagonals of a rectangle inscribed in a circle are diameters of the circle. EXMLE 1 and 2 on pp. 672 673 for Es. 3 9 INIE NGLE ind the indicated measure. 3. m 4. m G 5. m N 708 848 G 1208 N 1608 L M 6. m 7. m VU 8. mwx Y 678 T 308 U W 758 1108 V X 9. EO NLYI escribe the error in the diagram of (. ind two was to correct the error. Q 45º 100º EXMLE 3 ONGUENT NGLE Name two pairs of congruent angles. on p. 673 for Es. 10 12 10. 11. K 12. W J M L X Y Z EXMLE 5 LGE ind the values of the variables. on p. 675 for Es. 13 15 13. 8 958 8 808 14. E m8 608 2k8 608 G 15. J 548 M 4b8 K 3a8 1108 T 1308 L 676 hapter 10 roperties of ircles
16. MULTILE HOIE In the diagram, is a central angle and m 5 608. What is m? 158 308 608 1208 17. INIE NGLE 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. MULTILE HOIE What is the value of? 5 10 13 15 E (12 1 40)8 (8 1 10)8 G 19. LLELOGM arallelogram QT is inscribed in (. ind m. EONING etermine whether the quadrilateral can alwas be inscribed in a circle. Eplain our reasoning. 20. quare 21. ectangle 22. arallelogram 23. Kite 24. hombus 25. Isosceles trapezoid 26. HLLENGE In the diagram, is a right angle. If ou draw the smallest possible circle through and tangent to }, the circle will intersect } at J and } at K. ind the eact length of } JK. 3 5 4 OLEM 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? EXMLE 4 on p. 674 for E. 28 28. ENTE 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 ou use a carpenter s square to draw a diameter on the circular piece of wood? 10.4 Use Inscribed ngles and olgons 677
29. WITING right triangle is inscribed in a circle and the radius of the circle is given. Eplain how to find the length of the hpotenuse. 30. OVING THEOEM 10.10 op and complete the proof that opposite angles of an inscribed quadrilateral are supplementar. GIVEN c ( with inscribed quadrilateral EG OVE c m 1 m 5 1808, m E 1 m G 5 1808. the rc ddition ostulate, m EG 1? 5 3608 and m G 1 m E 5 3608. Using the? Theorem, m EG 5 2m, m EG 5 2m, m E 5 2m G, and m G 5 2m E. the ubstitution ropert, 2m 1? 5 3608, so?. imilarl,?. E G OVING THEOEM 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 Eercises 31 33, ou will prove Theorem 10.7 for each of these cases. 31. ase 1 rove ase 1 of Theorem 10.7. GIVEN c is inscribed in (Q. Let m 5 8. oint Q lies on }. OVE c m 5 1 } 2 m lan for roof how that n Q is isosceles. Use the ase ngles Theorem and the Eterior ngles Theorem to show that m Q 5 28. Then, show that m 5 28. olve for, and show that m 5 1 } 2 m. 8 32. ase 2 Use the diagram and auiliar line to write GIVEN and OVE statements for ase 2 of Theorem 10.7. Then write a plan for proof. 33. ase 3 Use the diagram and auiliar line to write GIVEN and OVE statements for ase 3 of Theorem 10.7. Then write a plan for proof. 34. OVING THEOEM 10.8 Write a paragraph proof of Theorem 10.8. irst draw a diagram and write GIVEN and OVE statements. 35. OVING THEOEM 10.9 Theorem 10.9 is written as a conditional statement and its converse. Write a plan for proof of each statement. 36. EXTENE EONE In the diagram, ( and (M intersect at, and } is a diameter of (M. Eplain wh ] is tangent to (. M 678 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
HLLENGE In Eercises 37 and 38, use the following information. You are making a circular cutting board. To begin, ou glue eight 1 inch b 2 inch boards together, as shown at the right. Then ou 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 justif our 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. E HUTTLE To maimize thrust on a N space shuttle, engineers drill an 11-point star out of the solid fuel that fills each booster. The begin b drilling a hole with radius 2 feet, and the 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 MIXE EVIEW EVIEW repare for Lesson 10.5 in Es. 40 42. ind the approimate length of the hpotenuse. ound our answer to the nearest tenth. (p. 433) 40. 55 41. 38 42. 26 16 60 82 Graph the reflection of the polgon in the given line. (p. 589) 43. -ais 44. 5 3 45. 5 2 E G Π1 1 1 1 H 1 1 ketch the image of (3, 24) after the described glide reflection. (p. 608) 46. Translation: (, ) (, 2 2) 47. Translation: (, ) ( 1 1, 1 4) eflection: in the -ais eflection: in 5 4 EXT TIE for Lesson 10.4, p. 915 ONLINE QUIZ at classzone.com 679
10.7 Write and Graph Equations of ircles efore You wrote equations of lines in the coordinate plane. Now You will write equations of circles in the coordinate plane. Wh? o ou can determine zones of a commuter sstem, as in E. 36. Ke Vocabular standard equation of a circle Let (, ) represent an point on a circle with center at the origin and radius r. the thagorean Theorem, 2 1 2 5 r 2. This is the equation of a circle with radius r and center at the origin. r (, ) E X M L E 1 Write an equation of a circle Write the equation of the circle shown. olution The radius is 3 and the center is at the origin. 2 1 2 5 r 2 Equation of circle 1 1 2 1 2 5 3 2 ubstitute. 2 1 2 5 9 implif. c The equation of the circle is 2 1 2 5 9. ILE ENTEE T (h, k) You can write the equation of an circle if ou know its radius and the coordinates of its center. uppose a circle has radius r and center (h, k). Let (, ) be a point on the circle. The distance between (, ) and (h, k) is r, so b the istance ormula Ï }} ( 2 h) 2 1 ( 2 k) 2 5 r. quare both sides to find the standard equation of a circle. r (h, k) (, ) KEY ONET or Your Notebook tandard Equation of a ircle The standard equation of a circle with center (h, k) and radius r is: ( 2 h) 2 1 ( 2 k) 2 5 r 2 10.7 Write and Graph Equations of ircles 699
E X M L E 2 Write the standard equation of a circle Write the standard equation of a circle with center (0, 29) and radius 4.2. olution ( 2 h) 2 1 ( 2 k) 2 5 r 2 tandard equation of a circle ( 2 0) 2 1 ( 2 (29)) 2 5 4.2 2 ubstitute. 2 1 ( 1 9) 2 5 17.64 implif. GUIE TIE for Eamples 1 and 2 Write the standard equation of the circle with the given center and radius. 1. enter (0, 0), radius 2.5 2. enter (22, 5), radius 7 E X M L E 3 Write the standard equation of a circle The point (25, 6) is on a circle with center (21, 3). Write the standard equation of the circle. (25, 6) olution To write the standard equation, ou need to know the values of h, k, and r. To find r, find the distance between the center and the point (25, 6) on the circle. (21, 3) 1 1 r 5 Ï }}} [25 2 (21)] 2 1 (6 2 3) 2 istance ormula 5 Ï } (24) 2 1 3 2 implif. 5 5 implif. ubstitute (h, k) 5 (21, 3) and r 5 5 into the standard equation of a circle. ( 2 h) 2 1 ( 2 k) 2 5 r 2 tandard equation of a circle [ 2 (21)] 2 1 ( 2 3) 2 5 5 2 ubstitute. ( 1 1) 2 1 ( 2 3) 2 5 25 implif. c The standard equation of the circle is ( 1 1) 2 1 ( 2 3) 2 5 25. GUIE TIE for Eample 3 3. The point (3, 4) is on a circle whose center is (1, 4). Write the standard equation of the circle. 4. The point (21, 2) is on a circle whose center is (2, 6). Write the standard equation of the circle. 700 hapter 10 roperties of ircles
E X M L E 4 Graph a circle UE EQUTION If ou know the equation of a circle, ou can graph the circle b identifing its center and radius. The equation of a circle is ( 2 4) 2 1 ( 1 2) 2 5 36. Graph the circle. olution ewrite the equation to find the center and radius. ( 2 4) 2 1 ( 1 2) 2 5 36 ( 2 4) 2 1 [ 2 (22)] 2 5 6 2 The center is (4, 22) and the radius is 6. Use a compass to graph the circle. 4 2 (4, 22) E X M L E 5 Use graphs of circles ETHQUKE The epicenter of an earthquake is the point on Earth s surface directl above the earthquake s origin. seismograph can be used to determine the distance to the epicenter of an earthquake. eismographs are needed in three different places to locate an earthquake s epicenter. Use the seismograph readings from locations,, and to find the epicenter of an earthquake. The epicenter is 7 miles awa from (22, 2.5). The epicenter is 4 miles awa from (4, 6). The epicenter is 5 miles awa from (3, 22.5). olution The set of all points equidistant from a given point is a circle, so the epicenter is located on each of the following circles. ( with center (22, 2.5) and radius 7 ( with center (4, 6) and radius 4 8 4 ( with center (3, 22.5) and radius 5 To find the epicenter, graph the circles on a graph where units are measured in miles. ind the point of intersection of all three circles. 24 28 2 c The epicenter is at about (5, 2). at classzone.com GUIE TIE for Eamples 4 and 5 5. The equation of a circle is ( 2 4) 2 1 ( 1 3) 2 5 16. Graph the circle. 6. The equation of a circle is ( 1 8) 2 1 ( 1 5) 2 5 121. Graph the circle. 7. Wh are three seismographs needed to locate an earthquake s epicenter? 10.7 Write and Graph Equations of ircles 701
10.7 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 for Es. 7, 17, and 37 5 TNIZE TET TIE Es. 2, 16, 26, and 42 1. VOULY op and complete: The standard equation of a circle can be written for an circle with known? and?. 2. WITING Eplain wh the location of the center and one point on a circle is enough information to draw the rest of the circle. EXMLE 1 and 2 on pp. 699 700 for Es. 3 16 WITING EQUTION Write the standard equation of the circle. 3. 4. 5. 1 10 1 1 10 1 6. 15 7. 8. 3 5 3 10 10 WITING EQUTION Write the standard equation of the circle with the given center and radius. 9. enter (0, 0), radius 7 10. enter (24, 1), radius 1 11. enter (7, 26), radius 8 12. enter (4, 1), radius 5 13. enter (3, 25), radius 7 14. enter (23, 4), radius 5 15. EO NLYI escribe and correct the error in writing the equation of a circle. n equation of a circle with center (23, 25) and radius 3 is ( 2 3) 2 1 ( 2 5) 2 5 9. 16. MULTILE HOIE The standard equation of a circle is ( 2 2) 2 1 ( 1 1) 2 5 16. What is the diameter of the circle? 2 4 8 16 EXMLE 3 on p. 700 for Es. 17 19 WITING EQUTION Use the given information to write the standard equation of the circle. 17. The center is (0, 0), and a point on the circle is (0, 6). 18. The center is (1, 2), and a point on the circle is (4, 2). 19. The center is (23, 5), and a point on the circle is (1, 8). 702 hapter 10 roperties of ircles
EXMLE 4 on p. 701 for Es. 20 25 GHING ILE Graph the equation. 20. 2 1 2 5 49 21. ( 2 3) 2 1 2 5 16 22. 2 1 ( 1 2) 2 5 36 23. ( 2 4) 2 1 ( 2 1) 2 5 1 24. ( 1 5) 2 1 ( 2 3) 2 5 9 25. ( 1 2) 2 1 ( 1 6) 2 5 25 26. MULTILE HOIE Which of the points does not lie on the circle described b the equation ( 1 2) 2 1 ( 2 4) 2 5 25? (22, 21) (1, 8) (3, 4) (0, 5) LGE etermine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form. 27. 2 1 2 2 6 1 9 5 4 28. 2 2 8 1 16 1 2 1 2 1 4 5 25 29. 2 1 2 1 4 1 3 5 16 30. 2 2 2 1 5 1 2 5 81 IENTIYING TYE O LINE Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. 31. ircle: ( 2 4) 2 1 ( 2 3) 2 5 9 32. ircle: ( 1 2) 2 1 ( 2 2) 2 5 16 Line: 5 23 1 6 Line: 5 2 2 4 33. ircle: ( 2 5) 2 1 ( 1 1) 2 5 4 34. ircle: ( 1 3) 2 1 ( 2 6) 2 5 25 Line: 5 1 } 5 2 3 Line: 5 2 4 } 3 1 2 35. HLLENGE our tangent circles are centered on the -ais. The radius of ( is twice the radius of (O. The radius of ( is three times the radius of (O. The radius of ( is four times the radius of (O. ll circles have integer radii and the point (63, 16) is on (. What is the equation of (? O OLEM OLVING EXMLE 5 on p. 701 for E. 36 36. OMMUTE TIN cit s commuter sstem has three zones covering the regions described. Zone 1 covers people living within three miles of the cit center. Zone 2 covers those between three and seven miles from the center, and Zone 3 covers those over seven miles from the center. a. Graph this situation with the cit center at the origin, where units are measured in miles. b. ind which zone covers people living at (3, 4), (6, 5), (1, 2), (0, 3), and (1, 6). Zone 3 Zone 1 87 0 4 mi 40 Zone 2 10.7 Write and Graph Equations of ircles 703
37. OMT I The diameter of a is about 4.8 inches. The diameter of the hole in the center is about 0.6 inches. You place a on the coordinate plane with center at (0, 0). Write the equations for the outside edge of the disc and the edge of the hole in the center. 4.8 in. 0.6 in. EULEUX OLYGON In Eercises 38 41, use the following information. The figure at the right is called a euleau polgon. It is not a true polgon because its sides are not straight. n is equilateral. 38. J lies on a circle with center and radius. Write an equation of this circle. 39. E lies on a circle with center and radius. Write an equation of this circle. E 40. ONTUTION The remaining arcs of the polgon are constructed in the same wa as J and E in Eercises 38 and 39. onstruct a euleau polgon on a piece of cardboard. 41. ut out the euleau polgon from Eercise 40. oll it on its edge like a wheel and measure its height when it is in different orientations. Eplain wh a euleau polgon is said to have constant width. 42. EXTENE EONE Telecommunication towers can be used to transmit cellular phone calls. Towers have a range of about 3 km. graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). a. raw the graph and locate the towers. re there an areas that ma receive calls from more than one tower? b. uppose our home is located at (2, 6) and our school is at (2.5, 3). an ou use our cell phone at either or both of these locations? c. it is located at (22, 2.5) and it is at (5, 4). Each cit has a radius of 1.5 km. Which cit seems to have better cell phone coverage? Eplain. J 1 1 H G 43. EONING The lines 5 } 3 1 2 and 5 2} 3 4 4 1 16 are tangent to ( at the points (4, 5) and (4, 13), respectivel. a. ind the coordinates of and the radius of (. Eplain our steps. b. Write the standard equation of ( and draw its graph. 44. OO Write a proof. GIVEN c circle passing through the points (21, 0) and (1, 0) OVE c The equation of the circle is 2 2 2k 1 2 5 1 with center at (0, k). (21, 0) (1, 0) 704 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
45. HLLENGE The intersecting lines m and n are tangent to ( at the points (8, 6) and (10, 8), respectivel. a. What is the intersection point of m and n if the radius r of ( is 2? What is their intersection point if r is 10? What do ou notice about the two intersection points and the center? b. Write the equation that describes the locus of intersection points of m and n for all possible values of r. MIXE EVIEW EVIEW repare for Lesson 11.1 in Es. 46 48. ind the perimeter of the figure. 46. (p. 49) 47. (p. 49) 48. (p. 433) 9 in. 18 ft 40 m 22 in. 57 m ind the circumference of the circle with given radius r or diameter d. Use p 5 3.14. (p. 49) 49. r 5 7 cm 50. d 5 160 in. 51. d 5 48 d ind the radius r of (. (p. 651) 52. 15 53. 54. 28 r r 9 r r 15 20 r r 21 QUIZ for Lessons 10.6 10.7 ind the value of. (p. 689) 1. 8 6 9 2. 7 6 5 3. 16 12 In Eercises 4 and 5, use the given information to write the standard equation of the circle. (p. 699) 4. The center is (1, 4), and the radius is 6. 5. The center is (5, 27), and a point on the circle is (5, 23). 6. TIE The diameter of a certain tire is 24.2 inches. The diameter of the rim in the center is 14 inches. raw the tire in a coordinate plane with center at (24, 3). Write the equations for the outer edge of the tire and for the rim where units are measured in inches. (p. 699) EXT TIE for Lesson 10.7, p. 915 ONLINE QUIZ at classzone.com 705
MIXE EVIEW of roblem olving Lessons 10.6 10.7 1. HOT EONE local radio station can broadcast its signal 20 miles. The station is located at the point (20, 30) where units are measured in miles. a. Write an inequalit that represents the area covered b the radio station. b. etermine whether ou can receive the radio station s signal when ou are located at each of the following points: E(25, 25), (10, 10), G(20, 16), and H(35, 30). 4. HOT EONE You are at point, about 6 feet from a circular aquarium tank. The distance from ou to a point of tangenc on the tank is 17 feet. 17 ft 6 ft r ft r ft TTE TET TIE classzone.com 2. EXTENE EONE ell phone towers are used to transmit calls. n area has cell phone towers at points (2, 3), (4, 5), and (5, 3) where units are measured in miles. Each tower has a transmission radius of 2 miles. a. raw the area on a graph and locate the three cell phone towers. re there an areas that can transmit calls using more than one tower? b. uppose ou live at (3, 5) and our friend lives at (1, 7). an ou use our cell phone at either or both of our homes? c. it is located at (21, 1) and it is located at (4, 7). Each cit has a radius of 5 miles. Which cit has better coverage from the cell phone towers? 3. HOT EONE You are standing at point inside a go-kart track. To determine if the track is a circle, ou measure the distance to four points on the track, as shown in the diagram. What can ou conclude about the shape of the track? Eplain. a. What is the radius of the tank? b. uppose ou are standing 4 feet from another aquarium tank that has a diameter of 12 feet. How far, in feet, are ou from a point of tangenc? 5. EXTENE EONE You are given seismograph readings from three locations. t (22, 3), the epicenter is 4 miles awa. t (5, 21), the epicenter is 5 miles awa. t (2, 5), the epicenter is 2 miles awa. a. Graph circles centered at,, and with radii of 4, 5, and 2 miles, respectivel. b. Locate the epicenter. c. The earthquake could be felt up to 12 miles awa. If ou live at (14, 16), could ou feel the earthquake? Eplain. 6. MULTI-TE OLEM Use the diagram. 15 8 a. Use Theorem 10.16 and the quadratic formula to write an equation for in terms of. b. ind the value of. c. ind the value of. 706 hapter 10 roperties of ircles