CorrectionKey=NL-C;CA-C Name Class Date 15.4 Segment Relationships in ...
Name
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
Resource Locker
? Houghton Mifflin Harcourt Publishing Company
Module 15
815
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
? Houghton Mifflin Harcourt Publishing Company
Module 15
816
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
? Houghton Mifflin Harcourt Publishing Company
Module 15
817
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.
? Houghton Mifflin Harcourt Publishing Company
Module 15
818
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
? Houghton Mifflin Harcourt Publishing Company
Module 15
819
Lesson 4
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