A discus thrower spins around in a circle one and a half times, then releases the discus. The discus forms a path tangent to the circle.

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1 Page 1 of Popeties of Tangents Goal Use popeties of a tangent to a cicle. Key Wods point of tangency p. 589 pependicula p. 108 tangent segment discus thowe spins aound in a cicle one and a half times, then eleases the discus. The discus foms a path tangent to the cicle. THEOREMS 11.1 and 11.2 Theoem 11.1 stating point of thow Wods If a line is tangent to a cicle, then it is pependicula to the adius dawn at the point of tangency. elease point path of discus discus l Symbols If l is tangent to at, then l &*. Theoem 11.2 Wods In a plane, if a line is pependicula to a adius of a cicle at its endpoint on the cicle, then the line is tangent to the cicle. l Symbols If l &*, then l is tangent to at. EXMPLE 1 Use Popeties of Tangents VOULRY TIP Tangent is based on a Latin wod meaning to touch. &*( is tangent to at point. Find. 13 &* is a adius of, so you can apply Theoem 11.1 to conclude that &* and &*(ae pependicula. So, a is a ight angle, and T is a ight tiangle. To find, use the Pythagoean Theoem. () 2 () 2 () 2 Pythagoean Theoem 13 2 () Substitute 13 fo and 12 fo. 169 () Multiply. 25 () 2 Subtact 144 fom each side. 5 Find the positive squae oot Popeties of Tangents 595

2 Page 2 of 6 gicultue EXMPLE 2 Find the Radius of a icle You ae standing at, 8 feet fom a silo. The distance to a point of tangency is 16 feet. What is the adius of the silo? SILOS ae used as stoage bins fo feed fo fam animals. Round silos allow fo the feed to be tightly packed, which pevents it fom spoiling. Tangent ^&*( is pependicula to adius &* at, so T is a ight tiangle. So, you can use the Pythagoean Theoem. () 2 () 2 () 2 ( 8) ft 8 ft Pythagoean Theoem Substitute 8 fo, fo, and 16 fo ( 8)( 8) Subtact 2 fom each side Subtact 64 fom each side. 12 ivide each side by 16. NSWER The adius of the silo is 12 feet. You can use the onvese of the Pythagoean Theoem to show that a line is tangent to a cicle. LOOK K To eview the onvese of the Pythagoean Theoem, see p EXMPLE How can you show that EF ^**( must be tangent to? 3 Veify a Tangent to a icle Use the onvese of the Pythagoean Theoem to detemine whethe T EF is a ight tiangle. 9 E 12 F (F) 2 (E) 2 (EF ) 2 ompae (F ) 2 with (E) 2 (EF ) Substitute fo F, 9 fo E, and 12 fo EF Multiply Simplify. T EF is a ight tiangle with ight angle E. So, EF &* is pependicula to E &*. y Theoem 11.2, it follows that EF ^&*( is tangent to. 596 hapte 11 icles

3 Page 3 of 6 VOULRY TIP tangent segment is often simply called a tangent. Tangent Segment tangent segment touches a cicle at one of the segment s endpoints and lies in the line that is tangent to the cicle at that point. tangent segment ctivity 11.2, on page 594, shows that tangent segments fom the same eteio point ae conguent. THEOREM 11.3 Wods If two segments fom the same point outside a cicle ae tangent to the cicle, then they ae conguent. P R S Symbols If SR &* and ST &* ae tangent to P at points R and T, then SR &*cst &*. T SKILLS REVIEW To eview solving equations, see p EXMPLE &* is tangent to at. &** is tangent to at. Find the value of Use Popeties of Tangents Two tangent segments fom the same point ae conguent. Substitute 2 3 fo and 11 fo. Subtact 3 fom each side. 4 ivide each side by Use Popeties of Tangents &* and &* ae tangent to. Find the value of Popeties of Tangents 597

4 Page 4 of Eecises Guided Pactice Vocabulay heck 1. omplete the statement: In the diagam at the ight, ^&*( is? to, and point is the?. Skill heck 2. In the diagam below, ^&*( XY is 3. In the diagam below, tangent to at point P. 5 and 7. What is mapx? Eplain. Is &* tangent to? Eplain. X P Y &* is tangent to at and &* is tangent to at. Find the value of Pactice and pplications Eta Pactice See p Finding Segment Lengths ^&*( is tangent to. Find the value of Homewok Help Eample 1: Es. 7 9, 27 Eample 2: Es Eample 3: Es Eample 4: Es , Finding Segment Lengths &* and &* ae tangent to. Find the value of hapte 11 icles

5 Page 5 of 6 Using lgeba Squae the binomial. 13. ( 2) ( 4) 2. ( 7) ( 12) 2 I I L S S Z O N E. O M HOMEWORK HELP Eta help with poblem solving in Es is at classzone.com Finding the Radius of a icle &* is tangent to. Find the value of Veifying Tangents Tell whethe &* is tangent to. Eplain you easoning Finding onguent Pats In Eecises 22 24, &* and &* ae tangent to. 22. Name all conguent segments. 23. Name all conguent angles. 24. Name two conguent tiangles. Science Visualize It! In Eecises 25 and 26, L has adii LJ &* and LK&* that ae pependicula. KM &** and JM&* ae tangent to L. 25. Sketch L, LJ &*, LK &*, KM &**, and JM&*. 26. Is T JLM conguent to T KLM? Eplain you easoning. 27. Global Positioning System GPS satellites obit 12,500 miles above Eath. ecause GPS signals can t tavel though Eath, a satellite can tansmit signals only as fa as points and fom point. Find and to the neaest mile. GPS Hikes sometimes cay navigation devices which utilize GPS technology. GPS helps hikes calculate whee they ae and how to get to anothe location. pplication Links L S S Z O N E. O M Obit 3,960 mi 12,500 mi GPS satellite 11.2 Popeties of Tangents 599

6 Page 6 of 6 LOOK K Fo moe about fiewoks, see p hallenge You ae cuising away fom a fiewoks show ove a bay at point. The highest point of the fiewoks is point F. When you ship eaches point, you can no longe see the fiewoks ove the hoizon. You ae standing at point E. FE&* is tangent to Eath at. Find FE. Round you answe to the neaest mile. Highest point of fiewoks, 0.2 mi above wate F Not dawn to scale Radius of Eath 3960 mi E You ae hee, 0.01 mi above wate Standadized Test Pactice 29. Multiple hoice In the diagam below, EF &* and EG &* ae tangent to. What is the value of? F G 4 5 E 30. Multiple hoice In the diagam below, SR&* is tangent to P. Find the adius of P. R F 18 G P H 36 J S Mied Review Using the Tiangle Inequality an the side lengths fom a tiangle? Eplain. (Lesson 4.7) 31. 5, 11, , 14, , 3, , 25, , 3, , 6, 29 Using the Midsegment Theoem Find the value of. (Lesson 7.5) lgeba Skills Finding Slope Find the slope of the line that passes though the points. (Skills Review, p. 665) 40. (0, 0) and ( 3, 6) 41. (2, 4) and (8, 0) 42. (1, 5) and ( 2, 1) 43. (0, 3) and (4, 7) 44. ( 1, 6) and (4, 5) 45. ( 7, 2) and ( 1, 4) 600 hapte 11 icles