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Dans un triangle rectangle, la tangente d'un angle est égale au rapport de la longueur du côté opposé à cet angle sur la longueur du côté adjacent à ce même angle.
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Dans un triangle rectangle, la tangente d'un angle est égale au rapport de la longueur du côté opposé à cet angle sur la longueur du côté adjacent à ce même angle.
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11.2Properties of Tangents595

Goal

Use properties of a

tangent to a circle.

Key Words

• point of tangency p. 589 • perpendicular p. 108 • tangent segment

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.

11.211.2Properties of Tangents

VOCABULARYTIP

Tangentis based on

a Latin word meaning "to touch."

Student Help

discusstarting point of throw path of discus release point

Theorem 11.1

WordsIf a line is tangent to a circle, then it is perpendicular to the radius drawn at the point of tangency.

Theorem 11.2

WordsIn a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

THEOREMS 11.1 and 11.2

B C l B C l

AC&*(is tangent to ?B at point C. Find BC.

Solution

BC&*is a radius of ?B, so you can apply Theorem 11.1 to conclude that BC &*and AC&*(are perpendicular. So, aBCAis a right angle, and TBCAis a right triangle. To find BC, use the Pythagorean Theorem.

BA)2?(BC)2?(AC)2Pythagorean Theorem

132?(BC)2?122Substitute 13 for BAand 12 for AC.

169? (BC)2?144Multiply.

25?(BC)2Subtract 144 from each side.

5?BCFind the positive square root.

EXAMPLE1Use Properties of Tangents

B C A13 12

Page 1 of 6

You can use the Converse of the Pythagorean Theorem to show that a line is tangent to a circle.

596Chapter 11CirclesYou are standing at C, 8 feet from a silo. The distance to a

point of tangency is 16 feet. What is the radius of the silo?

Solution

Tangent BC^&*(is perpendicular to radius AB&*at B, so TABCis a right triangle. So, you can use the Pythagorean Theorem.

AC)2?(AB)2?(BC)2Pythagorean Theorem

(r ?8)2?r2?162 r2?16r?64?r2?256(r?8)(r?8) ?r2?16r?64

16r?64?256Subtract r2from each side.

16r?192Subtract 64 from each side.

r?12Divide each side by 16.

ANSWER ?The radius of the silo is 12 feet.

Substitute r?8 for AC, rfor AB,

and 16 for BC.

EXAMPLE2Find the Radius of a Circle

How can you show that EF^***(must be tangent to ?D?

Solution

Use the Converse of the Pythagorean Theorem to determine whether

TDEFis a right triangle.

DF)2? (DE)2?(EF)2Compare (DF)2with (DE)2?(EF)2.

152? 92?122Substitute 15 for DF, 9 for DE, and 12 for EF.

225? 81 ?144Multiply.

225?225Simplify.

TDEF is a right triangle with right angle E. So, EF&*is perpendicular to DE &*. By Theorem 11.2, it follows that EF^&*(is tangent to ?D.

EXAMPLE3Verify a Tangent to a Circle

LOOKBACK

To review the Converse

of the Pythagorean

Theorem, see p. 200.

Student HelpD

E F15 12 9

SILOS are used as storage

bins for feed for farm animals.

Round silos allow for the feed

to be tightly packed, which prevents it from spoiling.

Agriculture

16 ft8 ft

Page 2 of 6

11.2Properties of Tangents597

VOCABULARYTIP

A tangent segment is

often simply called a tangent.

Student Help

AB&*is tangent to ?Cat B.

AD &**is tangent to ?Cat D.

Find the value of x.

Solution

AD?AB

2x?3?11

Substitute 2x?3 for ADand 11 for AB.

2x?8Subtract 3 from each side.

x?4Divide each side by 2.Two tangent segments fromthe same point are congruent.

EXAMPLE4Use Properties of Tangents

CD B A 112
x ? 3

Use Properties of Tangents

CB&*and CD&*are tangent to ?A. Find the value of x. 1.2. AB D C143 x ? 5 AB DC 15 x

Tangent SegmentA touches a circle at one of the

segment"s endpoints and lies in the line that is tangent to the circle at that point. Activity 11.2, on page 594, shows that tangent segments from the same exterior point are congruent.tangent segment

WordsIf two segments from the same point outside a circle are tangent to the circle, then they are congruent.

SymbolsIf SR&*and ST&*are tangent to ?P

at points Rand T, then SR&* cST&*.

THEOREM 11.3

R PS T

SKILLSREVIEW

To review solving

equations, see p. 673.

Student Help

CA

Btangent segment

Page 3 of 6

598Chapter 11Circles1.Complete the statement: In the diagramat the right, AB

^&*(is __?__ to ?C, and point B is the __?__.

2.In the diagram below, XY

^&*(is 3.In the diagram below, tangent to ?Cat point P. AB?BD?5 and AD?7.

What is maCPX? Explain. Is BD

&*tangent to ?C? Explain. AB&*is tangent to ?Cat Aand DB&*is tangent to ?Cat D.

Find the value of

x.

4. 5. 6.

Finding Segment LengthsAB^&*(is tangent to ?C. Find the value of r.

7. 8. 9.

Finding Segment LengthsAB&*and AD&*are tangent to ?C.

Find the value of

x.

10. 11. 12.A

C BD

2x ? 75x ? 8

A CB D 19 x ? 4 AC B D7 x C B A 25r
20 CA B17 r15 CB A 5 r 4

Practice and Applications

2x C 10 A D B2 Cx A DB 4 Cx A D B C DB AX P YC

Skill CheckVocabulary Check

Guided Practice

Exercises11.211.2

Extra Practice

See p. 695.

BA C

Example 1:Exs. 7?9, 27

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