2 Tangent LinesMathematics Florida StandardsMAFS.912.G-C.1.2 Identify and describe relationshipsamong inscribed angles, radii, and chords... the radiusof a circle is perpendicular to the tangent where theradius intersects the circle.

MPl.MP 3,MP4

Ob|ecHve To use properties of a tangent to a circle

Try this again with, a different circle.

MATHEMATICAL

PRACTICES

Getting Ready!

Draw a diagram like the one at the right. Each rayfrom Point A touches the circle in only one placeno matter how far it extends. Measure AB and AC.

Repeat the procedure with a point farther awayfrom the circle. Consider any two rays with acommon endpoint outside the circle. Make aconjecture about the lengths of the two segmentsformed when the rays touch the circle.

<> X C

Lesson

Vocabularytangent to acircle ipoint of tangenc^i

In the Solve It, you drew lines that touch a circle at only one point. These lines are called

tangents. This use of the word tangent is related to, but different from, the tangent ratioin right triangles that you studied in Chapter 8.

A tangent to a circle is a line in the plane of thecircle that intersects the circle in exactly one point.

The point where a circle and a tangent intersectIs the point of tangency.

BA is a tangent ray and BA is a tangent segment.

Essential Understanding A radius of a circle and the tangent that Intersects theendpoint of the radius on the circle have a special relationship.

Theorem 12-1. "Tis-rs JSaSS

Theorem

If a line is tangent to acircle, then the line is

perpendicular to the radius

at the point of tangency.

If^. .AB is tangent to OO at P

A

Then...

AB

762 Chapter 12 Circles

Proof Indirect Proof of Theorem 12-1

Given: n is tangent to OO at P.

Prove: n 1 OP

Step 1 Assume that n is not perpendicular to OP.

Step 2 If line n is not perpendicular to OP, then, for some otherpoint L on n, OL must be perpendicular to n. Also thereis a point iCon n such that LK = LP. /-OLK = LOLPbecause perpendicular lines form congruent adjacentangles. OL = OL. So, AOLK = AOLP by SAS.

Since corresponding parts of congruent triangles arecongruent, OK = OP. So K and P are both on OOby the definition of a circle. For two points on n to also be on 00 contradictsthe given fact that n is tangent to 00 at P. So the assumption that n is notperpendicular to OP must be false.

Step 3 Therefore, n L OP must be true.

Of

Ih'r'-What kind of angleis formed by a radiusand a tangent?The angle formed isa right angle, so themeasure is 90.

Finding Angle Measures

Multiple Choice ML and MN are tangent to OO. What is the value of a:?

® 58

CE:>63 CESll?

Since ML and MN are tangent to OO, Z.L and Z.N are right angles.LMNO is a quadrilateral. So the sum of the angle measures is 360.

mZ-L + mtLM + mAiV + mAO = 360

90 + mZM + 90 + 117 = 360 Substitute.

297 + m/LM = 360 Simplify.

mZ.M= 63 Solve.

The correct answer is B.

Got It? 1. a. ED is tangent to OO. What is the value ofx?" b. Reasoning Consider a quadrilateral like the

one in Problem 1. Write a formula you could

use to find the measure of any angle x formed

by two tangents when you know the measureof the central angle c whose radii intersect thetangents.

C PowerGeome Lesson 12-1 Tangent Lines 763

How does knowingEarth's radius help?The radius forms a rightangle with a tangent linefrom the observation

deck to the horizon. So,you can use two radii,the tower's height, andthe tangent to form aright triangle.

Problem 2 Finding Distance iotdMa

Earth Science The CN Tower in Toronto, Canada, has anobservation deck 447 m above ground level. About how far isit from the observation deck to the horizon? Earth's radius is

about 6400 km.

Step 1 Make a sketch. The length 447 m is about 0.45 km.

0.45 km

Not to scale

Step 2 Use the Pythagorean Theorem.

Ct2 ̂ ■j'^2 +

(6400 + 0.45)2 = j^2 ^ g4Qg2(6400.45)2 = + g4oo2

— tc240,965,760.2025 = TE"- + 40,960,000 Use a calculator.

5760.2025 = TE^

76 ~ TE

Substitute.

Simplify.

Subtract 40,960,000 from each side.

Take the positive square root of each side.

The distance from the CN Tower to the horizon is about 76 km.I

Got It? 2. What is the distance to the horizon that a person can see on a clear day from" ' an airplane 2 mi above Earth? Earth's radius is about 4000 mi.

Theorem 12-2 is the converse oflheorem 12-1. You can use it to prove that a line orsegment is tangent to a circle. You can also use it to construct a tangent to a circle.

Theorem 12-2

Theorem

If a line in the plane of acircle is perpendicular toa radius at its endpoint onthe circle, then the line istangent to the circle.

AB ± OPatP

Then . . .

AB is tangent to OO

You will prove Theorem 12-2 In Exercise 30.

764 Chapter 12 Circles

Why does the value xappear on each sideof the equation?The length of thehypotenuse, is the radiusplus 8, which is on theleft side of the equation.On the right side of theequation, the radius isone side of the triangle.

What information

does the diagramgive you?• LMN\s a triangle.• NM = 25, LM = 24,

Ni = l

• NL IS a radius.

Problem 3 Finding a Radius

What is the radius of 0C?

AC^ = AB^ + BC^

(x + 8)2 = 12^ +

x2 + 16x + 64 = 144 + x2

16a: = 80

a:= 5

Ihe radius is 5.

Pythagorean Theorem

Substitute.

Simplify.

Subtract and 64 from each side.

Divide each side by 16.

Got It? 3. What is the radius of OO?10

0

Problem 4I identifying a Tangent

Is ML tangent to ON at I? Explain.

The lengths of the sides ofALMN

<rjn

MT is a tangent if M 1 Wi. Use the Converse ofthe Pythagorean Theorem to determine whetherALMN is a right triangle.

To determine whether ML is

tangent to OO

NL^ + ML2 ̂ A/m2

72 + 242 ̂ 252 Substitute.

625 = 625 Simplify.

By the Converse of the Pythagorean Theorem, ALMN is a right triangle with ML _L NL.

So ML is tangent to ON at L because it is perpendicular to the radius at the point of

tangency (Theorem 12-2).

Got It? 4. Use the diagram in Problem 4. If NL = 4, ML = 7, and NM = 8, is MLtangent to GN at I? Explain.

In the Solve It, you made a conjecture about the lengths of two tangents from acommon endpolnt outside a circle. Your conjecture may be confirmed by the

following theorem.

c PowerGeometry.com Lesson 12-1 Tangent Lines 765

How can you find thelength of BClFind the segmentscongruent to Bf andfC.Then use segmentaddition.

Theorem 12-3

Theorem

If two tangent segments to

a circle share a common

endpoint outside the circle,

then the two segments are

congruent.

If ... Then ...

BA and EC are tangent to ©O BA = BC

0

Vou will prove Theorem 12-3 in Exercise 23.

In the figure at the right, the sides of the triangle are tangent to the circle. Thecircle is inscribed in the triangle. The triangle is circumscribed aboutthe circle.

Problem 5 Circles Inscribed in Polygons

O O is inscribed in AABC. What is the perimeter of AABC?

AD = AF= 10 cm

BD = BE = 15 cm

CF= CE = 8 cm

Two segments tangent to a circle from apoint outside the circle are congruent, sothey have the same length.

10 cm £) 15 cm

F8 cm^ E

C

p — AB + BC + CA

= AD + DB + BE + EC+CF +FA

= 10+15 + 15 + 8 + 8+10

= 66

The perimeter is 66 cm.

Definition of perimeter p

Segment Addition Postulate

Substitute.

Got It? 5. 00 is inscribed in APQR, which has a perimeter of 88 cm.What is the length of QF?

& Lesson Check

Do you know HOW?1. If mLA = 58, what is m/LACBl

2. If BC = 8 and DC = 4, what is the

radius?

3. If^C= 12 and BC = 9, what is the

radius?

_ MATHEMATICAL

Do you UNDERSTAND? PRACTICES

4. Vocabulary How are the phrases tangent ratio and

tangent ofa circle used differently?

5. Error Analysis A classmate

insists that DF is a tangent

to QE. Explain how to show

that your classmate is wrong.

^24 -1J ;

766 Chapter 12 Circles

MATHEMATICAL

Practice and Problem-Solving Exercises practices

Practice Algebra Lines that appear to be tangent are tangent. O is the center

of each circle. What is the value of x?

^ See Problem 1.

6. 7.

Earth Science The circle at the right represents Earth. The

radius of Earth is about 6400 km. Find the distance d to the

horizon that a person can see on a clear day from each of

the following heights h above Earth. Round your answer to

the nearest tenth of a kilometer.

^ See Problem 2.

9. 5 km 10. 1 km

Algebra In each circle, what is the value ofa;, to the nearest tenth?

12. ^ 13. 10cm

14

11. 2500m

^ See Problem 3.

14. p 15 in.

Determine whether a tangent is shown In each diagram. Explain.

15. —-V 16. 6

^ See Problem 4.

Each polygon circumscribes a circle. What is the perimeter of each polygon? ^ See Problem 5.

19. -r- 1.9 in.

'J3.7 in.3.4 in.

C PowerGeometry.com I Lesson 12-1 Tangent Liner 767

Apply 20. Solar Eclipse Common tangents to two circles maybe inrema/orexrema/. Ifyoudraw a segment joining the centers of the circles, a common internal tangent willintersect the segment. A common external tangent willnot. For this cross-sectional diagram of the sun, moon,

and Earth during a solar eclipse, use the terms above to

describe the types of tangents of each color,a. red b. blue c. greend. Which tangents show the extent on Earth's surface of

total eclipse? Of partial eclipse?

A21. Reasoning A nickel, a dime, and a quarter are touching

as shown. Tangents are drawn from point A to both sides

of each coin. What can you conclude about the four

tangent segments? Explain.

22. Think About a Plan Leonardo da Vinci wrote, "When each

of two squares touch the same circle at four points, one is

double the other." Explain why the statement is true.• How will drawing a sketch help?

• Are both squares inside the circle?

23j Prove Theorem 12-3.Proof _

Given: BA and BC are tangent to OO at A and C,

respectively.

Prove: BA = 5C

Moon

24. Given: BC is tangent to ©A at D.Proof m = w

Prove: AS s AC

25. Given: ©A and ©B with common tangentsProof DFand^

26. a. A belt fits snugly around the two circular pulleys. CE is anauxiliary line from E to BD. CE H BA. What type ofquadrilateral is ABCB? Explain.

b. What is the length of CE?c. What is the distance between the centers of the pulleys

to the nearest tenth?

27. BD and CK at the right are diameters of ©A. BP and QP are tangentsto ©A. What is m/LCDA?

28. Constructions Draw a circle. Label the center T. Locate a point on thecircle and label it R. Construct a tangent to ©T at R.

29. Coordinate Geometry Graph the equation = 9. Then draw a

segment from (0, 5) tangent to the circle. Find the length of the segment.

Prove: A GDC

768 Chapter 12 Circles

Challenge Write an indirect proof of Theorem 12-2.

^ Given: 1 ̂ at P.Prove: AB is tangent to OO.

Proof

31. Two circles that have one point in common are tangent circles. Given

any triangle, explain how to draw three circles that are centered at eachvertex of the triangle and are tangent to each other.

1

Apply What You've Learned MATHEMATICAL

PRACTICES

MR 2, MR 4

Look back at the information given on page 761 about the logo for the

showroom display. The diagram of the logo is shown again below.

^7.2 ft

a. What can you conclude about Z.OJSC? Justify your answer.

b. Write and solve an equation to find the length of OE.

c. Explain howyou knowyour answer to part (b) is reasonable.

d. How can you use the length you found in part (b) to find the length of one side ofthe red sail in the logo for the showroom display? What is that length?

C PowerGe6meTry.com I Lesson 12-1 Tangent Liner 769