9.3: Properties of Chords
Learning Objectives
 Find the lengths of chords in a circle.
 Discover properties of chords and arcs.
Review Queue
 Draw a chord in a circle.
 Draw a diameter in the circle from #1. Is a diameter a chord?

△ABC is an equilateral triangle in⨀A . FindmBCˆ andmBDCˆ . 
△ABC and△ADE are equilateral triangles in⨀A . List a pair of congruent arcs and chords.
Know What? To the right is the Gran Teatro Falla, in Cadiz, Andalucía, Spain. Notice the five windows,
Recall from the first section, a chord is a line segment whose endpoints are on a circle. A diameter is the longest chord in a circle.
Congruent Chords & Congruent Arcs
From #4 in the Review Queue above, we noticed that
Theorem 93: In the same circle or congruent circles, minor arcs are congruent if and only if their corresponding chords are congruent.
In both of these pictures,
In the second circle,
Example 1: Use
a) If
b) If
Solution:
a)
b)
Investigation 92: Perpendicular Bisector of a Chord
Tools Needed: paper, pencil, compass, ruler
1. Draw a circle. Label the center
2. Draw a chord. Label it
3. Find the midpoint of
4. Connect
Theorem 94: The perpendicular bisector of a chord is also a diameter.
If
If
Theorem 95: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.
Example 2: Find the value of
Solution: The diameter perpendicular to the chord. From Theorem 95,
Example 3: Is the converse of Theorem 94 true?
Solution: The converse of Theorem 94 would be: A diameter is also the perpendicular bisector of a chord. This is not true, a diameter cannot always be a perpendicular bisector to every chord. See the picture.
Example 4: Algebra Connection Find the value of
Solution: The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up an equation for
Equidistant Congruent Chords
Investigation 93: Properties of Congruent Chords
Tools Needed: pencil, paper, compass, ruler
 Draw a circle with a radius of 2 inches and two chords that are both 3 inches. Label like the picture to the right. This diagram is drawn to scale.
 From the center, draw the perpendicular segment to
AB¯¯¯¯¯¯¯¯ andCD¯¯¯¯¯¯¯¯ . You can use Investigation 32  Erase the arc marks and lines beyond the points of intersection, leaving
FE¯¯¯¯¯¯¯¯ andEG¯¯¯¯¯¯¯¯ . Find the measure of these segments. What do you notice?
Theorem 96: In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.
The shortest distance from any point to a line is the perpendicular line between them.
If
Example 5: Algebra Connection Find the value of
Solution: Because the distance from the center to the chords is congruent and perpendicular to the chords, the chords are equal.
Example 6:
Solution: First find the radius.
Example 7: Find
Solution: First, find the corresponding central angle,
Know What? Revisited In the picture, the chords from
Review Questions
 Questions 13 use the theorems from this section and similar to Example 3.
 Questions 410 use the definitions and theorems from this section.
 Questions 1116 are similar to Example 1 and 2.
 Questions 1725 are similar to Examples 2, 4, 5, and 6.
 Questions 26 and 27 are similar to Example 7.
 Questions 2830 use the theorems from this section.
 Two chords in a circle are perpendicular and congruent. Does one of them have to be a diameter? Why or why not?
 Write the converse of Theorem 95. Is it true? If not, draw a counterexample.
 Write the original and converse statements from Theorem 93 and Theorem 96.
Fill in the blanks.

−−−−−≅DF¯¯¯¯¯¯¯¯ 
ACˆ≅−−−−− 
DJˆ≅−−−−− 
−−−−−≅EJ¯¯¯¯¯¯¯ 
∠AGH≅−−−−− 
∠DGF≅−−−−−  List all the congruent radii in
⨀G .
Find the value of the indicated arc in

mBCˆ 
mBDˆ 
mBCˆ 
mBDˆ 
mBDˆ 
mBDˆ
Algebra Connection Find the value of

AB=32 
AB=20  Find
mABˆ in Question 20. Round your answer to the nearest tenth of a degree.  Find
mABˆ in Question 25. Round your answer to the nearest tenth of a degree.
In problems 2830, what can you conclude about the picture? State a theorem that justifies your answer. You may assume that
Review Queue Answers
1 & 2. Answers will vary
3.
4.
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