6.6 Find Segment Lengths in Circles

Theorem 6.16 Segments of ChordsIf 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.

_____ EC_____EA B

A

C

D

EEB ED

6.6 Find Segment Lengths in Circles

Example 1 Find lengths using Theorem 6.16Find lengths using Theorem 6.16

Find ML and JK.

K

M

L

J

N5x

3xx

1x

____ ____ NJNK NL NM

____ ____5 xx 3x 1x_________52 xx 2x x4 3___x 3

Find ML and JK by substitution.

____ ____ ML 3x 1x__ __ __ __ 3 3 3 1

___ 10

____ __ JK x 5x__ __ __ 3 3 5

___ 11

6.6 Find Segment Lengths in Circles

Theorem 6.17 Segments of SecantsIf two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

_____ EDEA _____ EB EC

B

A

CD

E

6.6 Find Segment Lengths in Circles

Example 2 Find lengths using Theorem 6.17Find lengths using Theorem 6.17

RTRSRPRQ __ ________ x6 6

Q

R

P

S

T

Solution

Use Theorem _____6.17

Substitute.

x 5

64Find the value of x.

4 5 5Simplify.___ _____ x60 5 25Solve for x.x__7

6.6 Find Segment Lengths in CirclesCheckpoint. Find the value of Checkpoint. Find the value of xx..

1.

x

5 126

x 6 5 12x6 60

10x

6.6 Find Segment Lengths in CirclesCheckpoint. Find the value of Checkpoint. Find the value of xx..

2.

x

7

8

66 14 7 7x

84 497 x

5x35 x7

6.6 Find Segment Lengths in Circles

Theorem 6.18 Segments of Secants and TangentsIf a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

____ ____EA2 ED EC

E

A

C

D

Q

S

R

T

6.6 Find Segment Lengths in Circles

Example 3 Find lengths using Theorem 6.18Find lengths using Theorem 6.18

___2 RSRQ _____ __ 2 x14

Solution

Use Theorem _____6.18

Substitute.

x

12

14Find RS.

12xSimplify.xx _____ 2 196 12Write in standard form._____0 2 xx 12

RT

196Use quadratic formula.

__2

_____4_____0

2

12 12 1 1961

Q

S

R

T

6.6 Find Segment Lengths in Circles

Example 3 Find lengths using Theorem 6.18Find lengths using Theorem 6.18

________x

Solution

Simplify.

x

12

14Find RS.

5826 Lengths cannot be __________, so use the _________ solution.

negative positive

So, x = ___________5826 ____, 23.9 .____RS and 23.9

M J

K

L

6.6 Find Segment Lengths in CirclesCheckpoint. Complete the following exercise.Checkpoint. Complete the following exercise.

JK. Find 3.

x

8

6

28x 6x64xx 62

06462 xx___64___62 xx

23x 73

x 733

9 9

3x 73733JK

6.6 Find Segment Lengths in Circles

Pg. 231, 6.6 #1-24