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Section 10.3 Using Chords 545
Using Chords
Essential QuestionEssential Question What are two ways to determine when a chord
is a diameter of a circle?
Drawing Diameters
Work with a partner. Use dynamic geometry software to construct a circle of radius 5 with center at the origin. Draw a diameter that has the given point as an endpoint. Explain how you know that the chord you drew is a diameter. a. (4, 3) b. (0, 5) c. (−3, 4) d. (−5, 0)
Writing a Conjecture about Chords
Work with a partner. Use dynamic geometry software to construct a chord BC — of a circle A. Construct a chord on the perpendicular bisector of BC —. What do you notice? Change the original chord and the circle several times. Are your results always the same? Use your results to write a conjecture.
A Chord Perpendicular to a Diameter
Work with a partner. Use dynamic geometry software to construct a diameter BC — of a circle A. Then construct a chord DE — perpendicular to BC — at point F. Find the lengths DF and EF. What do you notice? Change the chord perpendicular to BC — and the circle several times. Do you always get the same results? Write a conjecture about a chord that is perpendicular to a diameter of a circle.
A
C
B
E
F
D
Communicate Your AnswerCommunicate Your Answer
4. What are two ways to determine when a chord is a diameter of a circle?
LOOKING FOR
STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.
A
C
B
546 Chapter 10 Circles
10.3 Lesson^ What You Will LearnWhat You Will Learn
Use chords of circles to find lengths and arc measures.
Using Chords of Circles Recall that a chord is a segment with endpoints on a circle. Because its endpoints lie on the circle, any chord divides the circle into two arcs. A diameter divides a circle into two semicircles. Any other chord divides a circle into a minor arc and a major arc.
Using Congruent Chords to Find an Arc Measure
In the diagram, ⊙ P ≅ ⊙ Q , FG — ≅ JK —, and m JK = 80 °. Find m FG.
SOLUTION
Because FG — and JK — are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent by the Congruent Corresponding Chords Theorem.
So, m FG = m JK = 80 °.
Previous chord arc diameter
Core VocabularyCore Vocabullarry
TheoremsTheorems
Theorem 10.6 Congruent Corresponding Chords Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Proof Ex. 19, p. 550
Theorem 10.7 Perpendicular Chord Bisector Theorem
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Proof Ex. 22, p. 550
Theorem 10.8 Perpendicular Chord Bisector Converse
If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter.
Proof Ex. 23, p. 550
READING
If GD ≅ GF , then the point G , and any line, segment, or ray that contains G , bisects FD.
G
D
F
E
EG bisects FD.
diameter
semicircle
semicircle
chord
major arc
minor arc
A D
B C
G
D
F
E H
G F
P
J
K
80 ° Q
Q
R
T
S P
AB ≅ CD if and only if AB ≅ CD.
If EG is a diameter and EG ⊥ DF , then HD ≅ HF and GD ≅ GF.
If QS is a perpendicular bisector of TR , then QS is a diameter of the circle.
548 Chapter 10 Circles
Using Congruent Chords to Find a Circle’s Radius
In the diagram, QR = ST = 16, CU = 2 x , and CV = 5 x − 9. Find the radius of ⊙ C.
SOLUTION
Because CQ — is a segment whose endpoints are the center and a point on the circle, it is a radius of ⊙ C. Because CU — ⊥ QR —, △ QUC is a right triangle. Apply properties of chords to fi nd the lengths of the legs of △ QUC.
T X
S
R
Q
W U
C
Y V
5 x − 9
2 x
radius
Step 1 Find CU. Because QR — and ST — are congruent chords, QR — and ST — are equidistant from C by the Equidistant Chords Theorem. So, CU = CV. CU = CV Equidistant Chords Theorem 2 x = 5 x − 9 Substitute. x = 3 Solve for x. So, CU = 2 x = 2(3) = 6. Step 2 Find QU. Because diameter WX — ⊥ QR —, WX — bisects QR — by the Perpendicular Chord Bisector Theorem. So, QU = —^12 (16) = 8.
Step 3 Find CQ. Because the lengths of the legs are CU = 6 and QU = 8, △ QUC is a right triangle with the Pythagorean triple 6, 8, 10. So, CQ = 10.
So, the radius of ⊙ C is 10 units.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
5. In the diagram, JK = LM = 24, NP = 3 x , and NQ = 7 x − 12. Find the radius of ⊙ N.
TheoremTheorem
Theorem 10.9 Equidistant Chords Theorem
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Proof Ex. 25, p. 550
A E D
G
F
C
B
T X
S
R
Q
W U
C
Y V
5 x − 9
2 x
K
L M
R
S
J P
N
T Q
7 x − 12
3 x
AB ≅ CD if and only if EF = EG.
Section 10.3 Using Chords 549
10.3 Exercises Dynamic Solutions available at BigIdeasMath.com
In Exercises 3– 6, find the measure of the red arc or chord in ⊙ C****. (See Example 1.)
3. E
C B D
A
75 °
4. T
C V
U
34 °
34 °
5
Z
C
Y
W
X
60 °
110 °
M
L
N Q
R S
C
(^7) P
7
11
120 °
120 °
In Exercises 7–10, find the value of x****. (See Example 2.)
7.
H
G
J F C (^) E
8
x
8. U
C
R T x S 40 °
9. M
N
Q
P
L
2 x + 9
5 x − 6 10.^ G^ F
H
E
D
(5 x + 2)° (7 x − 12)°
11. ERROR ANALYSIS Describe and correct the error in reasoning.
Because AC — bisects DB — , BC ≅ CD.
D
C
B
A
E
12. PROBLEM SOLVING In the cross section of the submarine shown, the control panels are parallel and the same length. Describe a method you can use to fi nd the center of the cross section. Justify your method. (See Example 3.)
In Exercises 13 and 14, determine whether AB — is a diameter of the circle. Explain your reasoning.
13.
D
C B
A
D E
C
3 3
5
B
A
In Exercises 15 and 16, find the radius of ⊙ Q****. (See Example 4.)
15.^16
16
4 x + 3
7 x − 6
A F
H G
B
Q
E 16.
5 5
4 x + 4
6 x − 6
A B
C
D
Q
17. PROBLEM SOLVING An archaeologist finds part of a circular plate. What was the diameter of the plate to the nearest tenth of an inch? Justify your answer. 7 in.
7 in.
6 in.6 in.
6 in.6 in.
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. WRITING Describe what it means to bisect a chord. 2. WRITING Two chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
