10.6 Segment Lengths in Circles

Segments of a chord: The segments resulting when two

chords intersect inside a circle.

C

E

D

B

A

F

Segments of a chord Rule

(Theorem)

The product of the segments of one

chord is equal to the product of the

segments of the other chord.

BC times CD = AC times CF

GUIDED PRACTICE

Find the value(s) of x.

x (3) = (4) (6) Substitute.

Simplify.

x = 8 Solve for x.

SOLUTION

3x = 24

for Examples 1 and 2 EXAMPLE 1

ALGEBRA Find ML and JK.

SOLUTION

NK NJ = NL NM

x (x + 4) = (x + 1) (x + 2)

x2 + 4x = x2 + 3x + 2

4x = 3x + 2

x = 2

ML = ( x + 2 ) + ( x + 1)

= 2 + 2 + 2 + 1

= 7

JK = x + ( x + 4)

= 2 + 2 + 4

= 8

Secant segment: segment containing a chord of a circle

and has exactly one endpoint outside the circle.

B

C

A

D

E A secant segment has an external

segment and an internal segment.

Segments of secants theorem:

Outside times Whole = Outside times Whole

CD ∙ CE = CB ∙ CA

GUIDED PRACTICE

Find the value(s) of x.

SOLUTION

Use Theorem 10.15. 5 (x + 5) = 6 (6 + 9)

90 = 5x + 25 Simplify.

13 = x Solve for x

GUIDED PRACTICE

Find the value(s) of x.

SOLUTION

Use Theorem 10.15. 3 (3 + x + 2) = (x+1)(x + 1 + x – 1)

3x + 15 = 2x2 + 2x Simplify.

0 = 2x2 – x – 15 Combine Like Terms.

0 = (x – 3) (2x + 5) Solve for x

3 = x Use positive solution

CP

The same rule applies for Secants and/or Tangents.---

Except the outside is the whole on a tangent segment.

E

H

G

F

D

FE ∙FD = FG ∙ FG

GUIDED PRACTICE

Find the value of x.

SOLUTION

x2 = 1 (1 + 3)

x = 2

Use Theorem 10.16.

Simplify.

Simplify.

4.

= 4 x2

GUIDED PRACTICE

Find the value of x.

SOLUTION

5.

49 = 25 + 5x

24 = 5x

72 = 5 (x + 5) Use Theorem 10.16.

Simplify.

Write in standard form.

Simplify. x = 24 5

GUIDED PRACTICE

Determine which theorem you would use to find x. Then

find the value of x.

8.

Use Theorem 10.14.

x (18) = (9) (16) Substitute.

18x = 144 Simplify.

SOLUTION

Simplify. x = 8

GUIDED PRACTICE

Find the value of x.

SOLUTION

6.

144 = x2 + 10x

0 = x2 + 10x – 144

122 = x (x + 10)

x –10 + 102 – 4(1) (– 144)

2(1) =

Use Theorem 10.16.

Simplify.

Write in standard form.

Use quadratic formula

Or factor.

Simplify. x = 8

CP

EXAMPLE 4 Solve a real-world problem

SCIENCE

Tethys, Calypso, and Telesto are three of Saturn’s

moons. Each has a nearly circular orbit 295,000

kilometers in radius. The Cassini-Huygens spacecraft

entered Saturn’s orbit in July 2004. Telesto is on a

point of tangency. Find the distance DB from Cassini

to Tethys.

EXAMPLE 4 Solve a real-world problem

SOLUTION

DC DB = AD2 Use Theorem 10.16.

83,000 DB 203,0002 Substitute.

DB 496,494 Solve for DB.

ANSWER

Cassini is about 496,494 kilometers from Tethys.

Geometry

• Page 692 (2-20ev, 30,34-42ev)

Sophomore Math

• worksheet