762 Chapter 12 Circles

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 ratio in right triangles that you studied in Chapter 8.

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B

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A tangent to a circle is a line in the plane of the circle 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 the endpoint of the radius on the circle have a special relationship.

Theorem 12-1

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

If . . . <AB

> is tangent to } O at P

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A

BO

P

Then . . .<AB

># OP

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P

Tangent Lines12-1

Objective To use properties of a tangent to a circle

Draw a diagram like the one at the right. Each ray from Point A touches the circle in only one place no matter how far it extends. Measure AB and AC. Repeat the procedure with a point farther away from the circle. Consider any two rays with a common endpoint outside the circle. Make a conjecture about the lengths of the two segments formed when the rays touch the circle.

A

C

B

OTry this again with a different circle.

Lesson Vocabulary

•tangenttoacircle

•pointoftangency

LessonVocabulary

MATHEMATICAL PRACTICES

G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords . . . the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

MP 1, MP 3, MP 4

Common Core State Standards

Problem 1

Lesson 12-1 TangentLines 763

Indirect Proof of Theorem 12-1

Given: n is tangent to } O at P.

Prove: n # OP

Step 1 Assume that n is not perpendicular to OP.

Step 2 If line n is not perpendicular to OP, then, for some other point L on n, OL must be perpendicular to n. Also there is a point K on n such that LK ≅ LP. ∠OLK ≅ ∠OLP because perpendicular lines form congruent adjacent angles. OL ≅ OL. So, △OLK ≅ △OLP by SAS.

Since corresponding parts of congruent triangles are congruent, OK ≅ OP. So K and P are both on } O by the definition of a circle. For two points on n to also be on } O contradicts the given fact that n is tangent to } O at P. So the assumption that n is not perpendicular to OP must be false.

Step 3 Therefore, n # OP must be true.

Finding Angle Measures

Multiple Choice ML and MN are tangent to } O. What is the value of x?

58 90

63 117

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

m∠L + m∠M + m∠N + m∠O = 360

90 + m∠M + 90 + 117 = 360 Substitute.

297 + m∠M = 360 Simplify.

m∠M = 63 Solve.

The correct answer is B.

1. a. ED is tangent to } O. What is the value of x? 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 measure of the central angle c whose radii intersect the tangents.

Proof

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O

Pn

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O

Pn KL

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O M

L

N

117� x�

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O

D E

38�

x�Got It?

What kind of angle is formed by a radius and a tangent?The angle formed is a right angle, so the measure is 90.

447 m

Problem 2

764 Chapter 12 Circles

Finding Distance

Earth Science The CN Tower in Toronto, Canada, has an observation deck 447 m above ground level. About how far is it 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.

Step 2 Use the Pythagorean Theorem.

CT 2 = TE2 + CE2

(6400 + 0.45)2 = TE2 + 64002 Substitute.

(6400.45)2 = TE2 + 64002 Simplify.

40,965,760.2025 = TE2 + 40,960,000 Use a calculator.

5760.2025 = TE2 Subtract 40,960,000 from each side.

76 ≈ TE Take the positive square root of each side.

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

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 of Theorem 12-1. You can use it to prove that a line or segment is tangent to a circle. You can also use it to construct a tangent to a circle.

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E

C6400 km

0.45 km

Not to scale

Got It?

Theorem 12-2

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

If . . . <AB

># OP at P

Then . . .<AB

> is tangent to } O

You will prove Theorem 12-2 in Exercise 30.

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P

How does knowing Earth’s radius help?The radius forms a right angle with a tangent line from the observation deck to the horizon. So, you can use two radii, the tower’s height, and the tangent to form a right triangle.

STEM

Problem 3

Problem 4

Lesson 12-1 Tangent Lines 765

Finding a Radius

What is the radius of }C?

AC2 = AB2 + BC2 Pythagorean Theorem

(x + 8)2 = 122 + x2 Substitute.

x2 + 16x + 64 = 144 + x2 Simplify.

16x = 80 Subtract x2 and 64 from each side.

x = 5 Divide each side by 16.

The radius is 5.

3. What is the radius of } O?

Identifying a Tangent

Is ML tangent to } N at L? Explain.

NL2 + ML2 ≟ NM2

72 + 242 ≟ 252 Substitute.

625 = 625 Simplify.

By the Converse of the Pythagorean Theorem, △LMN is a right triangle with ML # NL.So ML is tangent to }N at L because it is perpendicular to the radius at the point of tangency (Theorem 12-2).

4. Use the diagram in Problem 4. If NL = 4, ML = 7, and NM = 8, is ML tangent to }N at L? Explain.

In the Solve It, you made a conjecture about the lengths of two tangents from a common endpoint outside a circle. Your conjecture may be confirmed by the following theorem.

B A

8

12

x

xC

Got It?

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10

6x

O x

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NM

L7

25

24

ML is a tangent if ML # NL. Use the Converse of the Pythagorean Theorem to determine whether△LMN is a right triangle.

To determine whether ML is tangent to } O

The lengths of the sides of △LMN

Got It?

Why does the value x appear on each side of the equation?The length of AC, the hypotenuse, is the radius plus 8, which is on the left side of the equation. On the right side of the equation, the radius is one side of the triangle.

What information does the diagram give you?•LMN is a triangle.•NM = 25, LM = 24,

NL = 7•NL is a radius.

Problem 5

766 Chapter 12 Circles

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

Circles Inscribed in Polygons

O is inscribed in △ABC. What is the perimeter of △ABC?

AD = AF = 10 cmBD = BE = 15 cmCF = CE = 8 cm

p = AB + BC + CA Definition of perimeter p

= AD + DB + BE + EC + CF + FA Segment Addition Postulate

= 10 + 15 + 15 + 8 + 8 + 10 Substitute.

= 66

The perimeter is 66 cm.

5. } O is inscribed in △PQR, which has a perimeter of 88 cm. What is the length of QY ?

Theorem 12-3

TheoremIf two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.

If . . . BA and BC are tangent to }O

Then . . .BA ≅ BC

You will prove Theorem 12-3 in Exercise 23.

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O

A

C

B

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OA

F

CE

BD10 cm 15 cm

8 cmTwo segments tangent to a circle from a point outside the circle are congruent, so they have the same length.

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OP

Z

R

Y

QX15 cm

17 cm

Got It?

Do you know HOW? 1. If m∠A = 58, what is m∠ACB?

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

3. If AC = 12 and BC = 9, what is the radius?

Do you UNDERSTAND? 4. Vocabulary How are the phrases tangent ratio and

tangent of a circle used differently?

5. Error Analysis A classmate insists that DF is a tangent to } E. Explain how to show that your classmate is wrong.

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BA

D

C

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DE

F25

7

24

Lesson Check

How can you find the length of BC?Find the segments congruent to BE and EC. Then use segment addition.

MATHEMATICAL PRACTICES

Lesson 12-1 TangentLines 767

Practice and Problem-Solving Exercises

Algebra Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x?

6. 7. 8.

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.

9. 5 km 10. 1 km 11. 2500 m

Algebra In each circle, what is the value of x, to the nearest tenth?

12. 13. 14.

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

15. 16. 17.

Each polygon circumscribes a circle. What is the perimeter of each polygon?

18. 19.

PracticeA See Problem 1.

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60�

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O x�

43�

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60�

See Problem 2.

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See Problem 3.

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14x

x10

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x

x

O

10 cm

7 cm

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15 in.

9 in.

P

O

Q

See Problem 4.

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5

15

16

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2.56.5

6

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10

8

6

See Problem 5.

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8 cm 16 cm

6 cm9 cm

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3.7 in.

1.9 in.

3.4 in.

3.6 in.

MATHEMATICAL PRACTICES

STEM

768 Chapter 12 Circles

20. Solar Eclipse Common tangents to two circles may be internal or external. If you draw a segment joining the centers of the circles, a common internal tangent will intersect the segment. A common external tangent will not. 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. green d. Which tangents show the extent on Earth’s surface of

total eclipse? Of partial eclipse?

21. 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?

23. Prove Theorem 12-3.

Given: BA and BC are tangent to }O at A and C, respectively.

Prove: BA ≅ BC

24. Given: BC is tangent to }A at D. DB ≅ DC

Prove: AB ≅ AC

ApplyB

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MoonNot to scale

Sun Earth

A

Proof

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O

A

C

B

Proof

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A

B DC

26. a. A belt fits snugly around the two circular pulleys. CE is an auxiliary line from E to BD. CE } BA. What type of quadrilateral is ABCE ? 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 tangents to }A. What is m∠CDA?

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

29. Coordinate Geometry Graph the equation x2 + y2 = 9. Then draw a segment from (0, 5) tangent to the circle. Find the length of the segment.

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B

CA

D E8 in.

35 in.

14 in.

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C A

B

D Q

KP

25�

25. Given: }A and }B with common tangents DF and CE

Prove: △GDC ∼ △GFE

Proof

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A BG

C

D E

F

STEM

Lesson 12-1 Tangent Lines 769

30. Write an indirect proof of Theorem 12-2.

Given: AB # OP at P.

Prove: AB is tangent to } O.

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 each vertex of the triangle and are tangent to each other.

ChallengeC

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Proof

Apply What You’ve Learned

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 ft9 ft

15 ft27 ft

DB

G

C

A

O

E

a. What can you conclude about ∠OBC? Justify your answer.

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

c. Explain how you know your 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 of the red sail in the logo for the showroom display? What is that length?

PERFO

RMANCE TA

SK MATHEMATICAL PRACTICESMP 2, MP 4