GEOMETRY

9.7 Segment Lengths in Circles

Objectives

Find the lengths of segments of chords.

Find the lengths of segments of tangents and secants.

Finding the Lengths of Chords

When two chords of a circle intersect, each chord is divided into two segments which are called segments of a chord. There are several possible cases.

Theorem 9.14

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. EA • EB = EC • ED

E

A

B

C

D

Finding Segment Lengths

Chords ST and PQ intersect inside the circle. Find the value of x. 6

93

XR

T

S

Q P

RQ • RP = RS • RT Use Theorem 9.14

Substitute values.9 • x = 3 • 6

9x = 18

x = 2

Simplify.

Divide each side by 9.

Using Segments of Tangents and Secants

In the figure shown, PS is called a tangent segment because it is tangent to the circle at an end point. Similarly, PR is a secant segment and PQ is the external segment of PR.

Q

S

P

R

Theorem 9.15

If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

C

A

D

E

B

EA • EB = EC • ED

Theorem 9.16

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equal the square of the length of the tangent segment.

(EA)2 = EC • ED

C

D

E

A

Finding Segment Lengths

Find the value of x.

x

10

11

9

S

P

T

R

Q

RP • RQ = RS • RT Use Theorem 10.16

Substitute values.9•(11 + 9)=10•(x + 10)

180 = 10x + 100

80 = 10x

Simplify.

Subtract 100 from each side.

8 = x Divide each side by 10.

Note:

In Lesson 10.1, you learned how to use the Pythagorean Theorem to estimate the radius of a circle. Example 3 shows you another way to estimate the radius of a circular object.

Estimating the radius of a circle

Aquarium Tank. You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency is about 20 feet. Estimate the radius of the tank.

(CB)2 = CE • CD Use Theorem 10.17

Substitute values.

400 16r + 64

336 16r

Simplify.

21 r Divide each side by 16.

(20)2 8 • (2r + 8)

Subtract 64 from each side.

So, the radius of the tank is about 21 feet.

Practice

3 • ( x + 3) = 4 (4 + 7)

3x + 9 = 4 * 11

3x = 44- 9

3 x = 35

11.7

3 • 6 = 9 • x

18 = 9x

2

(x)2 = 9* (4+ 9)

X2 = 117

X= 10.8

Practice

2 x = 25

X = 12.531 2 = 20( 20 + x )

961 = 400 + 20x

561= 20 x

X = 28.1

10 (10 +3x)= 8 (8+ 22)

100+ 30x = 240

30x = 140

x = 4.7

Practice

6 x = 3 * 8

6x = 24

X = 4

8( 8+ 8 ) = 6 * ( 6 + 2x)

8 * 16 = 36 + 12 x

128 = 36 + 12x

92 = 12x

X = 7.7

Take Notes

4 * ( 4 + 6) = 5 ( 5 + x)

4 *10 = 25 + 5x

40 = 25 + 5x

15 = 5x

X= 3

Take a minute and solve …

7 ( 7+5) = 6( 6+x)

84 = 36 + 6x

48 = 6x

X = 8

Take Notes

4 * 9 = 6 x

36 = 6x

X = 6

Take a minute and solve …

18 2 = 12( 12+ x)

324 = 144 +12x

180 = 12x

15

Take Notes

9 ( x+ 7 + 9) = 8 ( 8 + x + 10)

9 ( x + 16) = 8 ( x + 18)

9x + 144 = 8x + 144

X = 0

Take a minute and solve …

9 ( 9 + x + 4) = 8 ( 8 + x + 8)

9 ( x + 13) = 8 ( x + 16)

9x + 117=8x +128

X = 11

Take Notes

4 ( 9 + x ) = 6 ( x + 6 )

36 + 4x = 6x + 36

X = 0

Take a minute and solve …

8( 2x + 1) = 6 (3x)

16x + 8 = 18x

2x = 8

X = 4

Take Notes

40 2 = 32( 32 + x)

1600= 1024 + 32x

576 = 32x

x = 18

Take a minute and solve …

15 2 = 9( 9+x)

225 = 81 + 9x

144 = 9x

16