O x 40 O. O x. Determine whether a tangent line is shown in each diagram. Explain

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1 -1 Pactice Fom G Tangent Lines lgeba ssume that lines that appea to be tangent ae tangent. is the cente of each cicle. What is the value of? The cicle at the ight epesents Eath. The adius of the Eath is about 6400 km. Find the distance d that a peson can see on a clea day fom each of the following heights h above Eath. Round you answe to the neaest tenth of a kilomete. h d 4. km 32.1 km km km km km In each cicle, what is the value of to the neaest tenth? etemine whethe a tangent line is shown in each diagam. Eplain no; u 2 yes; (3"3) yes; TY and ZW ae diametes of (S. TU and U ae tangents of (S. What is m/syz? T 32 U W S Z Y Pentice Hall Gold Geomety Teaching Resouces 3

2 -1 Pactice (continued) Fom G Tangent Lines Each polygon cicumscibes a cicle. What is the peimete of each polygon? mm 70 mm in. 8 mm 17 mm 10 in. 7 mm 6 in. 5 in in. 28 in ft 3 in. 15 ft 5 in. 21 ft 4 in. 24 ft 18. Eo nalysis classmate states that is tangent to (. Eplain how to show that you classmate is wong. If is tangent to (, then ' and ml 5 0; this cannot be tue because the sum of the thee angles would be geate than The peak of Mt. Eveest is about 8850 m above sea level. bout how many kilometes is it fom the peak of Mt. Eveest to the hoizon if the Eath s adius is about 6400 km? aw a diagam to help you solve the poblem. 337 km d 20. The design of the banne at the ight includes a cicle with a -in. diamete. Using the measuements given in the diagam, eplain whethe the lines shown ae tangents to the cicle. no; u 21 2 in. 6 in. 6 in. 6 in. 6 in. 16 in. in. in. 16 in. Pentice Hall Gold Geomety Teaching Resouces 4

3 -1 Pactice Fom K Tangent Lines Lines that appea to be tangent ae tangent. is the cente of each cicle. What is the value of? To stat, identify the type of geometic figue fomed by the tangent lines and adii. The figue fomed is a. quadilateal The cicle at the ight epesents Eath. The adius of Eath is about 6400 km. Find the distance d to the hoizon that a peson can see on a clea day fom each of the following heights h above Eath. Round you answe to the neaest tenth of a kilomete km km m 2.4 km km km h d lgeba In each cicle, what is the value of to the neaest tenth? To stat, use the Pythagoean Theoem () in...7 m 16 m 5 in. m 8 in. 10. Q and UR ae diametes of (P. R RS and TS ae tangents of (P. Find m/upt and m/uqp. 32; 53 Q P 16 S U T Pentice Hall Foundations Geomety Teaching Resouces 5

4 -1 Pactice (continued) Fom K Tangent Lines etemine whethe a tangent is shown in each diagam. Eplain. 11. To stat, use the onvese of the Pythagoean Theoem to elate the side lengths of the tiangle. G yes; no; u 16 2 yes; Each polygon cicumscibes a cicle. What is the peimete of each polygon? in. To stat, find the length of each 15 in. unknown segment. 16 in. 72 in. 3 in. P u u u u cm 2 cm 16. ft 140 ft 3 ft 15 cm 22 cm 1 ft 17. ( is inscibed in a tiangle, which has a peimete of 76 in. What is the value of? 8 in. 5 in. 25 in. H 2 I 18. Reasoning GHI is a tiangle. How can you pove that HI is tangent to (G? mlg 1 mli 5 0. y the Tiangle ngle-sum Thm., mlh 5 0, so HI is tangent to (G by Thm. -2. G 61 Pentice Hall Foundations Geomety Teaching Resouces 6

5 -1 Reteaching Tangent Lines tangent is a line that touches a cicle at eactly one point. In the diagam, is tangent to (Q. You can apply theoems about tangents to solve poblems. Theoem -1 If a line is tangent to a cicle, then that line foms a ight angle with the adius at the point whee the line touches the cicle. Q R Theoem -2 If a line in the same plane as a cicle is pependicula to a adius at its endpoint on the cicle, then the line is tangent to the cicle. Poblem Use the diagam at the ight to solve the poblems below. GH is tangent to (K. H 68 K G What is the measue of /G? ecause GH is tangent to (K, it foms a ight angle with the adius. What is the length of the adius? You can use the Pythagoean Theoem to find missing lengths. The sum of the angles of a tiangle HK 2 1 HG 2 5 GK 2 is always 180. Wite an equation ( 1 ) 2 to find m/g ( 1 )( 1 ) m/g 1 m/h 1 m/k m/g m/g m/g 5 22 So, the measue of /G is 22 and the length of the adius is 3.5 units. Eecises In each cicle, what is the value of? Pentice Hall Geomety Teaching Resouces

6 -1 Reteaching (continued) Tangent Lines In each cicle, what is the value of? Theoem -3 If two segments ae tangent to a cicle fom the same point outside the cicle, then the two segments ae equal in length. In the diagam, and ae both tangent to (. So, they ae also conguent. When cicles ae dawn inside a polygon so that the sides of the polygon ae tangents, the cicle is inscibed in the figue. You can apply Theoem -3 to find the peimete, o distance aound the polygon. Poblem (M is inscibed in quadilateal. What is the peimete of? Z 5 W 5 W Y Y 5 Z 5 3 Now add to find the length of each side: 5 W 1 W Y 1 Y Z 1 Z The peimete is 38 in. in. Z W M Y 3 in. 2 in. 5 in. Eecises Each polygon cicumscibes a cicle. What is the peimete of each polygon? 7. 6 cm 34 cm ft 4 ft 35 ft. 7 cm 4 cm 3.5 cm 3 cm 8 cm 6 ft 5.25 cm 8.75 cm Pentice Hall Geomety Teaching Resouces 10

7 -2 Pactice Fom G hods and cs In Eecises 1 and 2, the ( (E. What can you conclude? 1. Q 2. P E R S lqp lrs le le; all adii ae conguent; all chods dawn ae conguent. Find the value of. Y W lwy lef; WY F; all adii ae conguent. E F In (, 0 is a 0 diamete and E > E. What can you conclude about and? Eplain. 0 0 ; because E E and, must be a pependicula bisecto of by the onvese of the Pependicula isecto Theoem. This means, so 0 0 by Theoem -6,. E 7. In (, Z is the diamete of the cicle and Z ' WY. What conclusions can you make? Justify you answe. 0W 0Y because Z is a pependicula bisecto, and W Y because of Theoem -8. Z W Find the value of to the neaest tenth. Y In the figue at the ight, sphee with adius 15 mm is intesected by a plane 3 mm fom the cente. To the neaest tenth, find the adius of the coss section (Y mm 3 mm Y 15 mm Pentice Hall Gold Geomety Teaching Resouces 13

8 -2 Pactice (continued) Fom G hods and cs 0. Given: (J with diamete HK ; KL 0 0 > LM > MK Pove: nkil > nkim 0 0 Statements: 1) KI KI; 2) KL KM; 3) KM KL; 4) JM JL; 5) KH is the ' bis. of ML; 6) IM IL; 7) kkil kklm; Reasons: 1) Refl. Pop. of ; 2) Given; 3) onvese Thm. -6; 4) ll adii in a cicle ae ; 5) onvese of ' is. Thm.; 6) ef. of a bis.; 7) SSS 13. Given: and ae diametes of (E. Pove: ne > ne Statements: 1) and ae diametes of (E; 2) E E and E E; 3) le le; 4) ke ke; Reasons: 1) Given; 2) ef. of adius; 3) Vet. ngles ae ; 4) SS M K I J H E L (N and ( ae conguent. PQ is a chod of both cicles. 14. If N 5 in. and PQ 5 8 in., how long is the adius to the neaest tenth of an inch? 7.2 in. N P 15. If N 5 30 mm and adius 5 16 mm, how long is PQ to the neaest tenth of a millimete? 11.1 mm Q 16. If adius 5 m and PQ 5 m, how long is N to the neaest tenth? 22.2 m 17. aw a iagam student daws ( with a diamete of cm. Inside the cicle she inscibes equilateal n so that,, and ae all chods of the cicle. The diamete of ( bisects. The section of the diamete fom the cente of the cicle to whee it bisects is 3 cm. To the neaest whole numbe, what is the peimete of the equilateal tiangle inscibed in (? 31 cm 18. Two concentic cicles have adii of 6 mm and mm. segment tangent to the smalle cicle is a chod of the lage cicle. What is the length of the segment to the neaest tenth mm Pentice Hall Gold Geomety Teaching Resouces 14

9 -2 Reteaching hods and cs Seveal elationships between chods, acs, and the cental angles of a cicle ae listed below. The conveses of these theoems ae also tue. Theoem -4 onguent cental angles have conguent acs. Theoem -5 onguent cental angles have conguent chods. Theoem -6 onguent chods have conguent acs. Theoem -7 hods equidistant fom the cente ae conguent. Poblem What is the value of? EF 5 FG Given > hods equidistant fom the cente of a cicle ae conguent. E 3.2 F 5 G 1 G 5 1 G G 5 G Segment ddition Postulate Substitution Given G Substitution The values of is 7. Eecises In Eecises 1 and 2, the cicles ae conguent. What can you conclude? 1. I 2. G H W Z nswes may vay. Samples below: M 0 0 GI L J and GI LJ Find the value of. L K J Y 0 0 W ZY and W ZY Pentice Hall Geomety Teaching Resouces 1

10 -2 Reteaching (continued) hods and cs Useful elationships between diametes, chods, and acs ae listed below. To bisect a figue means to divide it eactly in half. Theoem -8 Theoem - In a cicle, if a diamete is pependicula to a chod, it bisects that chod and its ac. In a cicle, if a diamete bisects a chod that is not a diamete of the cicle, it is pependicula to that chod. Theoem -10 If a point is an equal distance fom the endpoints of a line segment, then that point lies on the pependicula bisecto of the segment. Poblem What is the value of to the neaest tenth? In this poblem, is the adius. To find its value daw adius, which becomes the hypotenuse of ight ne. Then use the Pythagoean Theoem to solve. E 5 E diamete pependicula to a chod bisects the chod. Use the Pythagoean Theoem. 4 6 E Solve fo Find the positive squae oot of each side. The value of is 5. Eecises Find the value of to the neaest tenth Find the measue of each segment to the neaest tenth.. Find c when 5 6 cm and d 5 1 cm cm 10. Find c when 5 cm and d 5 8 cm. 8.2 cm 11. Find d when 5 10 in. and c 5 10 in. 8.7 in.. Find d when 5 8 in. and c 5 15 in. 2.8 in. c d Pentice Hall Geomety Teaching Resouces 20

11 -3 Pactice Fom G Inscibed ngles Find the value of each vaiable. Fo each cicle, the dot epesents the cente ; d 21; 42; 117 3; 0; ; 88; 102; ; 38 58; 0; 61 78; 0; 65 Find the value of each vaiable. Lines that appea to be tangent ae tangent ; 136 Find each indicated measue fo (M. 13. a. m/ 86 b. m/ c. m 102 d. m M 108; 216 Pentice Hall Gold Geomety Teaching Resouces 23

12 -3 Pactice (continued) Fom G Inscibed ngles Find the value of each vaiable. Fo each cicle, the dot epesents the cente d 38 e ; 28; 62 1; 88; ; 55; 52; 70; Given: Quadilateal is inscibed in (Z. * ) Y is tangent to (Z. Pove: m/ 1 m/y 5 m/ Statements: 1) is inscibed in (Z; 2) l is suppl. to l; 3) ml 1 ml 5 180; 4) ml 1 ml 1 mly 5 180; 5) ml 1 ml 1 mly 5 ml 1 ml; 6) ml 1 mly 5 ml; Reasons: 1) Given; 2) oollay 3 to Thm. -11; 3) ef. of suppl.; 4) l dd. Post.; 5) Subst. Pop.; 6) Subt. Pop. 18. Eo nalysis classmate says that m/e 5 0. Eplain why this is E incoect. le is not an inscibed angle because its vete is not a point on the cicle. nly an inscibed angle that intecepts a semicicle has a measue of student inscibes quadilateal inside a cicle. The measues of angles,, and ae given below. Find the measue of each angle of quadilateal. ml 5 2; ml 5 64; ml 5 88; ml m/ m/ m/ Z Y 20. Reasoning Quadilateal WYZ is inscibed in a cicle. If /W and /Y ae each inscibed in a semicicle, does this mean the quadilateal is a ectangle? Eplain. No; lw and ly ae ight angles, but the othes do not have to be. 21. Witing student inscibes an angle inside a semicicle to fom a tiangle. The measues of the angles that ae not the vete of the inscibed angle ae and 2 2. Find the measues of all thee angles of the tiangle. Eplain how you got you answe. 33; 57; 0; if the angle is inscibed in a semicicle it must measue 0. To find the measues of the othe angles, set thei sum equal to 0: Pentice Hall Gold Geomety Teaching Resouces 24

13 -3 Reteaching (continued) Inscibed ngles Eecises In Eecises 1, find the value of each vaiable ; 47; ; 130; ; 70 0; 53; 100 0; 60 Find the value of each vaiable. Lines that appea to be tangent ae tangent ; 4 66; 54; Points,, and lie on (. ml m R m. Find each measue m m/ m/ student inscibes a tiangle inside a cicle. The tiangle divides the cicle into acs with the following measues: 468, 1028, and 28. What ae the measues of the angles of the tiangle? 23; 51; student inscibes NPQ inside (Y. The measue of m/n 5 68 and m/ 5 4. Find the measues of the othe angles of the quadilateal. mlp 5 1; mlq 5 86 Pentice Hall Geomety Teaching Resouces 30

On a piece of graph paper, draw a circle that has a radius of 5 and center at ( 0, 0 ).

On a piece of graph paper, draw a circle that has a radius of 5 and center at ( 0, 0 ). 10.1 Stat Thinking On a piece of gaph pape, daw a cicle that has a adius of 5 and cente at ( 0, 0 ). 1. aw the segment that connects the points ( 3, 4 ) and ( 4, 3) on the cicle. Is this segment a diamete?