opeties of icles 1010.1 Use opeties of Tangents 10.2 Find c Measues 10.3 pply opeties of hods 10.4 Use Inscibed ngles and olygons 10.5 pply Othe ngle elationships in icles 10.6 Find egment Lengths in icles 10.7 Wite and Gaph Equations of icles efoe In pevious chaptes, you leaned the following skills, which you ll use in hapte 10: classifying tiangles, finding angle measues, and solving equations. eequisite kills VOULY HEK opy and complete the statement. 1. Two simila tiangles have conguent coesponding angles and? coesponding sides. 2. Two angles whose sides fom two pais of opposite ays ae called?. 3. The? of an angle is all of the points between the sides of the angle. KILL N LGE HEK Use the onvese of the ythagoean Theoem to classify the tiangle. (eview p. 441 fo 10.1.) 4. 0.6, 0.8, 0.9 5. 11, 12, 17 6. 1.5, 2, 2.5 Find the value of the vaiable. (eview pp. 24, 35 fo 10.2, 10.4.) 7. 8. 9. 5x 8 (6x 2 8)8 (8x 2 2)8 (2x 1 2)8 (5x 1 40)8 7x8 648
Now In hapte 10, you will apply the big ideas listed below and eviewed in the hapte ummay on page 707. You will also use the key vocabulay listed below. ig Ideas 1 Using popeties of segments that intesect cicles 2 pplying angle elationships in cicles 3 Using cicles in the coodinate plane KEY VOULY cicle, p. 651 cente, adius, diamete chod, p. 651 secant, p. 651 tangent, p. 651 cental angle, p. 659 mino ac, p. 659 majo ac, p. 659 semicicle, p. 659 conguent cicles, p. 660 conguent acs, p. 660 inscibed angle, p. 672 intecepted ac, p. 672 standad equation of a cicle, p. 699 Why? icles can be used to model a wide vaiety of natual phenomena. You can use popeties of cicles to investigate the Nothen Lights. Geomety The animation illustated below fo Example 4 on page 682 helps you answe this question: Fom what pat of Eath ae the Nothen Lights visible? You goal is to detemine fom what pat of Eath you can see the Nothen Lights. To begin, complete a justification of the statement that >. Geomety at classzone.com Othe animations fo hapte 10: page s 655, 661, 671, 691, and 701 649
Investigating g Geomety 10.1 Exploe Tangent egments MTEIL compass ule TIVITY Use befoe Lesson 10.1 Q U E T I O N How ae the lengths of tangent segments elated? line can intesect a cicle at 0, 1, o 2 points. If a line is in the plane of a cicle and intesects the cicle at 1 point, the line is a tangent. E X L O E aw tangents to a cicle TE 1 TE 2 TE 3 aw a cicle Use a compass to daw a cicle. Label the cente. aw tangents aw lines and ] so that they intesect ( only at and, espectively. These lines ae called tangents. Measue segments } and } ae called tangent segments. Measue and compae the lengths of the tangent segments. W O N L U I O N Use you obsevations to complete these execises 1. epeat teps 1 3 with thee diffeent cicles. 2. Use you esults fom Execise 1 to make a conjectue about the lengths of tangent segments that have a common endpoint. 3. In the diagam, L, Q, N, and ae points of tangency. Use you conjectue fom Execise 2 to find LQ and N if LM 5 7 and M 5 5.5. L N 7 5.5 M 4. In the diagam below,,,, and E ae points of tangency. Use you conjectue fom Execise 2 to explain why } > } E. E 650 hapte 10 opeties of icles
Use opeties 10.1 of Tangents efoe You found the cicumfeence and aea of cicles. Now You will use popeties of a tangent to a cicle. Why? o you can find the ange of a G satellite, as in Ex. 37. Key Vocabulay cicle cente, adius, diamete chod secant tangent cicle is the set of all points in a plane that ae equidistant fom a given point called the cente of the cicle. cicle with cente is called cicle and can be witten (. segment whose endpoints ae the cente and any point on the cicle is a adius. chod is a segment whose endpoints ae on a cicle. diamete is a chod that contains the cente of the cicle. chod adius diamete cente secant is a line that intesects a cicle in two points. tangent is a line in the plane of a cicle that intesects the cicle in exactly one point, the point of tangency. The tangent ay ] and the tangent segment } ae also called tangents. secant point of tangency tangent E X M L E 1 Identify special segments and lines Tell whethe the line, ay, o segment is best descibed as a adius, chod, diamete, secant, o tangent of (. a. } b. } c. ] E d. E G E olution a. } is a adius because is the cente and is a point on the cicle. b. } is a diamete because it is a chod that contains the cente. ] c. E is a tangent ay because it is contained in a line that intesects the cicle at only one point. d. E is a secant because it is a line that intesects the cicle in two points. GUIE TIE fo Example 1 1. In Example 1, what wod best descibes } G? }? 2. In Example 1, name a tangent and a tangent segment. 10.1 Use opeties of Tangents 651
E VOULY The plual of adius is adii. ll adii of a cicle ae conguent. IU N IMETE The wods adius and diamete ae used fo lengths as well as segments. Fo a given cicle, think of a adius and a diamete as segments and the adius and the diamete as lengths. E X M L E 2 Find lengths in cicles in a coodinate plane Use the diagam to find the given lengths. a. adius of ( y b. iamete of ( c. adius of ( d. iamete of ( 1 olution 1 x a. The adius of ( is 3 units. b. The diamete of ( is 6 units. c. The adius of ( is 2 units. d. The diamete of ( is 4 units. GUIE TIE fo Example 2 3. Use the diagam in Example 2 to find the adius and diamete of ( and (. OLN ILE Two cicles can intesect in two points, one point, o no points. oplana cicles that intesect in one point ae called tangent cicles. oplana cicles that have a common cente ae called concentic. concentic cicles 2 points of intesection 1 point of intesection (tangent cicles) no points of intesection E VOULY line that intesects a cicle in exactly one point is said to be tangent to the cicle. OMMON TNGENT line, ay, o segment that is tangent to two coplana cicles is called a common tangent. common tangents 652 hapte 10 opeties of icles
E X M L E 3 aw common tangents Tell how many common tangents the cicles have and daw them. a. b. c. olution a. 4 common tangents b. 3 common tangents c. 2 common tangents GUIE TIE fo Example 3 Tell how many common tangents the cicles have and daw them. 4. 5. 6. THEOEM Fo You Notebook THEOEM 10.1 In a plane, a line is tangent to a cicle if and only if the line is pependicula to a adius of the cicle at its endpoint on the cicle. m oof: Exs. 39 40, p. 658 Line m is tangent to (Q if and only if m } Q. E X M L E 4 Veify a tangent to a cicle In the diagam, } T is a adius of (. Is } T tangent to (? 35 T 12 37 olution Use the onvese of the ythagoean Theoem. ecause 12 2 1 35 2 5 37 2, nt is a ight tiangle and } T } T. o, } T is pependicula to a adius of ( at its endpoint on (. y Theoem 10.1, } T is tangent to (. 10.1 Use opeties of Tangents 653
E X M L E 5 Find the adius of a cicle In the diagam, is a point of tangency. Find the adius of (. olution You know fom Theoem 10.1 that } }, so n is a ight tiangle. You can use the ythagoean Theoem. 2 5 2 1 2 ( 1 50) 2 5 2 1 80 2 ubstitute. 2 1 100 1 2500 5 2 1 6400 100 5 3900 ythagoean Theoem Multiply. ubtact fom each side. 5 39 ft ivide each side by 100. 50 ft 80 ft THEOEM THEOEM 10.2 Tangent segments fom a common extenal point ae conguent. oof: Ex. 41, p. 658 Fo You Notebook T If } and } T ae tangent segments, then } > } T. E X M L E 6 Find the adius of a cicle } is tangent to ( at and } T is tangent to ( at T. Find the value of x. olution 28 3x 1 4 T 5 T Tangent segments fom the same point ae >. 28 5 3x 1 4 ubstitute. 8 5 x olve fo x. GUIE TIE fo Examples 4, 5, and 6 7. Is } E tangent to (? 8. } T is tangent to (Q. 9. Find the value(s) Find the value of. of x. 3 2 4 E 18 24 T 9 x 2 654 hapte 10 opeties of icles
10.1 EXEIE KILL TIE HOMEWOK KEY 5 WOKE-OUT OLUTION on p. W1 fo Exs. 7, 19, and 37 5 TNIZE TET TIE Exs. 2, 29, 33, and 38 1. VOULY opy and complete: The points and ae on (. If is a point on }, then } is a?. 2. WITING Explain how you can detemine fom the context whethe the wods adius and diamete ae efeing to a segment o a length. EXMLE 1 on p. 651 fo Exs. 3 11 MTHING TEM Match the notation with the tem that best descibes it. 3.. ente 4. H. adius 5. }. hod 6.. iamete 7. E E. ecant 8. G F. Tangent 9. } G. oint of tangency 10. } H. ommon tangent E G H at classzone.com 11. EO NLYI escibe and coect the eo in the statement about the diagam. 6 9 E The length of secant } is 6. EXMLE 2 and 3 on pp. 652 653 fo Exs. 12 17 OOINTE GEOMETY Use the diagam at the ight. 12. What ae the adius and diamete of (? 13. What ae the adius and diamete of (? 14. opy the cicles. Then daw all the common tangents of the two cicles. 9 6 3 y 3 6 9 x WING TNGENT opy the diagam. Tell how many common tangents the cicles have and daw them. 15. 16. 17. 10.1 Use opeties of Tangents 655
EXMLE 4 on p. 653 fo Exs. 18 20 EXMLE 5 and 6 on p. 654 fo Exs. 21 26 ETEMINING TNGENY etemine whethe } is tangent to (. Explain. 18. 3 4 5 19. LGE Find the value(s) of the vaiable. In Execises 24 26, and ae points of tangency. 21. 16 24 22. 9 6 9 15 18 20. 23. 14 7 52 48 10 24. 3x 1 10 7x 2 6 25. 2x 2 1 5 13 26. 4x 2 1 3x 2 1 4x 2 4 OMMON TNGENT common intenal tangent intesects the segment that joins the centes of two cicles. common extenal tangent does not intesect the segment that joins the centes of the two cicles. etemine whethe the common tangents shown ae intenal o extenal. 27. 28. 29. MULTILE HOIE In the diagam, ( and (Q ae tangent cicles. } is a common tangent. Find. 22Ï } 15 4 2Ï } 15 8 5 3 30. EONING In the diagam, ] is tangent to (Q and (. Explain why } > } > } even though the adius of (Q is not equal to the adius of (. 31. TNGENT LINE When will two lines tangent to the same cicle not intesect? Use Theoem 10.1 to explain you answe. 656 5 WOKE-OUT OLUTION on p. W1 5 TNIZE TET TIE
32. NGLE IETO In the diagam at ight, and ae points of tangency on (. Explain how you know that ] bisects. (Hint: Use Theoem 5.6, page 310.) 33. HOT EONE Fo any point outside of a cicle, is thee eve only one tangent to the cicle that passes though the point? e thee eve moe than two such tangents? Explain you easoning. 34. HLLENGE In the diagam at the ight, 5 5 12, 5 8, and all thee segments ae tangent to (. What is the adius of (? E F OLEM OLVING IYLE On moden bicycles, ea wheels usually have tangential spokes. Occasionally, font wheels have adial spokes. Use the definitions of tangent and adius to detemine if the wheel shown has tangential spokes o adial spokes. 35. 36. EXMLE 4 on p. 653 fo Ex. 37 37. GLOL OITIONING YTEM (G) G satellites obit about 11,000 miles above Eath. The mean adius of Eath is about 3959 miles. ecause G signals cannot tavel though Eath, a satellite can tansmit signals only as fa as points and fom point, as shown. Find and to the neaest mile. 38. HOT EONE In the diagam, } is a common intenal tangent (see Execises 27 28) to ( and (. Use simila tiangles to explain why } 5 }. 10.1 Use opeties of Tangents 657
39. OVING THEOEM 10.1 Use pats (a) (c) to pove indiectly that if a line is tangent to a cicle, then it is pependicula to a adius. GIVEN c Line m is tangent to (Q at. OVE c m } Q m a. ssume m is not pependicula to } Q. Then the pependicula segment fom Q to m intesects m at some othe point. ecause m is a tangent, cannot be inside (Q. ompae the length Q to Q. b. ecause } Q is the pependicula segment fom Q to m, } Q is the shotest segment fom Q to m. Now compae Q to Q. c. Use you esults fom pats (a) and (b) to complete the indiect poof. 40. OVING THEOEM 10.1 Wite an indiect poof that if a line is pependicula to a adius 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 Wite a poof that tangent segments fom a common extenal point ae conguent. GIVEN c } and } T ae tangent to (. OVE c } > } T lan fo oof Use the Hypotenuse Leg onguence Theoem to show that n > nt. T 42. HLLENGE oint is located at the oigin. Line l is tangent to ( at (24, 3). Use the diagam at the ight to complete the poblem. a. Find the slope of linel. (24, 3) y l b. Wite the equation fo l. c. Find the adius of (. x d. Find the distance fomlto ( along the y-axis. MIXE EVIEW EVIEW epae fo Lesson 10.2 in Ex. 43. 43. is in the inteio of. If m 5 258 and m 5 708, find m. (p. 24) Find the values of x and y. (p. 154) 44. x 8 508 y8 45. 1028 x 8 3y8 46. (4y 2 7)8 (2x 1 3)8 1378 47. tiangle has sides of lengths 8 and 13. Use an inequality to descibe the possible length of the thid side. What if two sides have lengths 4 and 11? (p. 328) 658 EXT TIE fo Lesson 10.1, p. 914 ONLINE QUIZ at classzone.com