1. No category

Page 1 of 6
11.6
Goal
Use properties of chords in
a circle.
Properties of Chords
In Lessons 11.3 and 11.5, you saw how to find the measure of an angle
formed by chords that intersect at the center of a circle or on a circle.
The Geo-Activity below explores the angles formed by chords that
intersect inside a circle.
Key Words
• chord p. 589
Geo-Activity
Properties of Angles Formed By Chords
1 Draw a circle with two central
●
angles measuring 708 and 308.
Label as shown.
2 Draw chords AC
&* and BD
&*.
●
Label the intersection E, as
shown. Find maAEB.
B
B
A
A
70°
E
X
30° C
X
D
C
D
3 Compare your angle measures with those of other students.
●
What do you notice?
4 Repeat Steps 1 and 2 for different central angles.
●
5 What can you say about an angle formed by intersecting chords?
●
The result demonstrated by the Geo-Activity is summarized in the
theorem below.
THEOREM 11.10
Words
If two chords intersect inside a circle, then
the measure of each angle formed is one half
the sum of the measures of the arcs
intercepted by the angle and its vertical angle.
Symbols
1
2
ma1 5 }} (mCD
s 1 mAB
r ),
1
2
ma2 5 }} (mBC
s 1 mAD
s)
620
Chapter 11
Circles
A
2
D
C
1
B
Page 2 of 6
EXAMPLE
1
Find the Measure of an Angle
Find the value of x.
1068
P
T
x8
P
Solution
R
1748
1
2
Use Theorem 11.10.
1
2
Substitute 1068 for mPS
r and 1748 for mRQ
r.
x 5 }}(280)
1
2
Add.
x 5 140
Multiply.
x8 5 }}(mPS
r 1 mRQ
r)
x8 5 }}(1068 1 1748)
IStudent Help
S
EXAMPLE
2
Find the Measure of an Arc
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
Find the value of x.
x8
A
B
808
E
D
C
Solution
608
1
2
Use Theorem 11.10.
1
2
Substitute x8 for mAB
r and 608 for mCD
s.
1
2
Use the distributive property.
1
2
Subtract 30 from each side.
808 5 }}(mAB
r 1 mCD
s)
808 5 }}(x8 1 608)
80 5 }}x 1 30
50 5 }}x
100 5 x
Multiply each side by 2.
Find the Measure of an Angle and an Arc
Find the value of x.
1.
2.
1908
A
A
x8
D
708
3.
B
668
x8
C
B
C
C
B
x8
A
728
708
998
D
D
11.6
Properties of Chords
621
Page 3 of 6
Intersecting Chords When two chords intersect in a circle, four
segments are formed. The following theorem shows the relationship
among these segments.
THEOREM 11.11
Words
B
If two chords intersect inside 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.
C
E
EA p EB 5 EC p ED
Symbols
EXAMPLE
3
D
A
Find Segment Lengths
Find the value of x.
S
3
P
R x
6
9
P
T
Solution
&* and QP
&* are chords that intersect at R.
Notice that ST
RS p RT 5 RQ p RP
3p659px
Use Theorem 11.11.
Substitute 3 for RS, 6 for RT, 9 for RQ, and x for RP.
18 5 9x
Simplify.
25x
Divide each side by 9.
Find Segment Lengths
Find the value of x.
4.
A
5.
N
P
12
6 L
9
F
4 H
8
C
10
5
G
x
G
F
Chapter 11
Circles
V
4
U
D
J
622
C
8.
x
E
6
x S
6
8
T
E
M
7.
R
6 B x
x
12
6.
D
K
K
9.
L
16
5
5
12
N
x
E
J
24
M
Page 4 of 6
11.6 Exercises
Guided Practice
Vocabulary Check
1. In the diagram, name the points
inside the circle.
A
E
C
B
Skill Check
D
Find the measure of a1.
2.
B
3.
A
558
1
A
658
888
C
C
888
B
D
1108
4.
928
1
B
A
C
1
D
928
1688
D
Find the value of x.
5.
6.
A
x
B
B
7.
B
x
D
2
4
6
A
A
14
7
x
C
C
10
D
15
18
C
8
D
Practice and Applications
Extra Practice
See p. 696.
Matching Match each angle with the correct expression
you can use to find its measure.
1
2
A. }}(mBF
r 1 mDE
s)
8. ma1
C
B
1
1
2
B. }}(mAB
r 1 mCE
s)
9. ma2
4
1
2
C. }}(mAE
r 1mBC
s)
10. ma3
3
E
A
1
2
D. }}(mBD
s 1 mFE
r)
11. ma4
D
2
F
Finding Angle Measures Find the value of x.
12.
Example 1: Exs. 8–14
Example 2: Exs. 15–18,
Example 3: Exs. 19–26
13.
A B
Homework Help
1348
x8
1628
C
14.
B
258
A
1308
A
C
x8
B
x8
758
D
D
D
968
11.6
C
Properties of Chords
623
Page 5 of 6
Finding Arc Measures Find the value of x.
15.
16.
C
B
x8
A
708
A
898
558
17.
B
A
B
598
728
1298
x8
D
C
D x8 C
18.
D
You be the Judge A student claims if two chords intersect
and the measure of each angle formed is the same as the measure
of the arc intercepted by the angle, then each angle must be a
central angle. Is he correct? Explain.
19. Animation Design You are designing an animated logo for a Web site.
You want sparkles to leave point C and reach the circle at the same
time. To find out how far each sparkle moves between frames, you
need to know the distances from C to the circle. Three distances are
shown below. Find CN.
15 10
C 12
N
15 10
C 12
N
Frame 1
15 10
C 12
Frame 3
N
Frame 2
Chords in a Circle Find the value of x.
20.
21.
B
C
B
4 6
D
B
A
328
D x8
24.
1228
D
x
8
15
23.
C
B
x
9
A
22.
C
12
10
x
A
14
25.
A B
B
908
1508
x8
x8
C
C
C
D
D
A
1288
A
5
D
1008
26. Technology Use geometry drawing software.
1
●
Draw a circle and label points
A, B, C, and D as shown.
^&*( and CD
^&*(. Label the point
2 Draw lines AB
●
of intersection E.
A
D
3 Measure EA, EB, EC, and ED. Then
●
calculate EA p EB and EC p ED.
What do you notice? What theorem does this demonstrate?
624
Chapter 11
Circles
C
E
B
Page 6 of 6
Standardized Test
Practice
27. Multi-Step Problem In the diagram, AC 5 12, CD 5 3, and
EC 5 9.
a. Find BC.
1008
A
b. What is the measure of aACB?
c. What is the measure of AE
r?
B
368
D
C
d. Is T ACB similar to T ECD? Explain
your reasoning.
Mixed Review
808
E
Finding Side Lengths Find the unknown side length. Round your
answer to the nearest tenth if necessary. (Lesson 4.4)
28.
29.
10
3
30.
c
a
16
3
14
b
30
Algebra Skills
Absolute Values Evaluate. (Skills Review, p. 662)
31. 23
32. 1
33. 219
34. 50
35. 2.7
36. 28
37. 210.01
38. 2100
Quiz 2
Find the value of x in (C. (Lesson 11.4)
1.
2.
A
3x 2 1
C
A
(x 1 63)8
C
C
x
E
3.
A
B
2x 2 5
E
B
D
E
B
x15
D
D
(3x 1 1)8
Find the value of each variable. Explain your reasoning. (Lesson 11.5)
4.
5.
A
6.
B
x8
B
B
988
x8
588
x8
518
A
C
418 y8
C
D
D
1058
y8
A
C
Find the value of x. (Lesson 11.6)
7.
A
808
x8
D
448 C
8.
1078
1358
C
9.
A
D
B
x8
A
D
B
7
x
14
12
C
B
11.6
Properties of Chords
625