CorrectionKey=NL-C;CA-C Name Class Date 15.4 Segment Relationships in ...

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Class

Date

15.4 Segment Relationships in Circles

Essential Question: What are the relationships between the segments in circles?

Explore Exploring Segment Length Relationships in Circles

Any segment connecting two points on a circle is a chord. In some cases, two chords drawn inside the same circle will intersect, creating four segments. In the following activity, you will look for a pattern in how these segments are related and form a conjecture.

A Using ge_ometry_software or a compass and straightedge, construct circle A with two chords CD and EF that intersect inside the circle. Label the intersection point G.

F C

G

D

A

B

Circle1

E

B Repeat your construction with two more circles. Vary the size of the circles

and where you put the intersecting chords inside them.

Circle2

Circle3

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Lesson 4

C Fill in the chart with the lengths of the segments measured to the nearest millimeter and

calculate their products.

DG

GC

EG

GF

DG GC EG GF

Circle 1

Circle 2

Circle 3

Look for a pattern among the measurements and calculations of the segments. From the

table, it appears that

will always equal

.

Reflect

1. Discussion Compare your results with those of your classmates. What do you notice?

2. What conjecture can you make about the products of the segments of two chords that intersect inside a circle?

Conjecture:

Explain 1 Applying the Chord-Chord Product Theorem

In the Explore, you discovered a pattern in the relationship between the parts of two chords that intersect inside a circle. In this Example, you will apply the following theorem to solve problems.

Chord-Chord Product Theorem

If two chords intersect inside a circle, then the products

of the lengths of the segments of the chords are equal.

C

A

B

E D AE ? EB = CE ? ED

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Lesson 4

Example 1 Find the value of x and the length of each chord.

A Set up an equation according to the Chord-Chord Product Theorem and

solve for x. CE ED = AE EB

6(2) = 3(x) 12 = 3x 4 = x

Add the segment lengths to find the length of each chord. CD = CE + ED = 6 + 2 = 8 AB = AE + EB = 4 + 3 = 7

B Set up an equation according to the Chord-Chord Product Theorem and

solve for x: HG GJ = KG GI

( )= ( )

= 6x

= x Add the segment lengths together to find the lengths of each chord:

HJ = HG + GJ =

+ 8=

KI =

+ GI = 6 +

=

C

6 3B

x E2

A

D

H I

9 x

G K 68

J

Your Turn

3. Given AD = 12. Find the value of x and the length of each chord.

B

A 3 14 E

x

C

D

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Lesson 4

Explain 2 Proving the Secant-Secant Product Theorem

A secant is any line that intersects a circle at exactly two points. A secant segment is part of a secant line with at least one point on the circle. A secant segment that lies in the exterior of the circle with one point on the circle is called an external secant segment. Secant segments drawn from the same point in the exterior of a circle maintain a certain relationship that can be stated as a theorem.

Secant-Secant Product Theorem

If two secants intersect in the exterior of a circle, then the

product of the lengths of one secant segment and its

A

external segment equals the product of the lengths of the

other secant segment and its external segment.

C

B

D

E AE ? BE = CE ? DE

Example 2 Use similar triangles to prove the Secant-Secant Product Theorem.

Step 1 Identify the segments in the diagram. The whole secant segments in this

A

diagram are

and

.

The external secant segments in this diagram are Step 2

and

.

C

Given the diagram as shown, prove that AE BE = CE DE. A

Prove: AE BE = CE DE

Proof: Draw auxiliary line segments A?D and C?B. EAD and ECB both

C

intercept

, so

. E E by the

Property. Thus, EAD ECB by

the

. Therefore, corresponding sides

B

D

E

B

D

E

are proportional, so _ AE = _ BE . By the

BE(CE)

_A_E_

CE

=

_D_E_

BE

BE(CE),

and

thus

AE

BE

=

CE

DE.

Property of Equality,

Reflect

4. Rewrite the Secant-Secant Theorem in your own words. Use a diagram or shortcut notation to help you remember what it means.

5. Discussion: Suppose that two secants are drawn so that they intersect on the circle. Can you determine anything about the lengths of the segments formed? Explain.

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Lesson 4

Explain 3 Applying the Secant-Secant Product Theorem

You can use the Secant-Secant Product Theorem to find unknown measures of secants and secant segments by setting up an equation.

Example 3 Find the value of x and the length of each secant segment.

A Set up an equation according to the Secant-Secant Product Theorem and

solve for x. AC AB = AE AD

(5 + x)(5) = (12)(6)

5x + 25 = 72 5x = 47 x = 9.4

Add the segments together to find the lengths of each secant segment. AC = 5 + 9.4 = 14.4; AE = 6 + 6 = 12

5B A

6D

x

C

6 E

B Set up an equation according to the Secant-Secant Product Theorem and

solve for x.

UP TP = SP RP

( ) ( ) (7) =

(6)

8R 6

P

S

7

xT

U

x +

=

x =

x =

Add the segments together to find the lengths of each secant segment.

UP = 7 + =

; SP = 8 + 6 = 14

Your Turn

Find the value of x and the length of each secant segment.

6.

5 Q 5.4

R

P

4S

x

T

7.

H

4

J6

N x P x M5 L

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Lesson 4

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