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# Chords in Circles

## Line segments whose endpoints are on a circle.

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Chords in Circles

### Chord Theorems

1. Chord Theorem #1: 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, and .

2. Chord Theorem #2: The perpendicular bisector of a chord is also a diameter.

If and then is a diameter.

3. Chord Theorem #3: If a diameter is perpendicular to a chord, then the diameter bisects the chord and its corresponding arc.

If , then and .

4. Chord Theorem #4: 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 and , then and are equidistant to the center and .

What if you were given a circle with two chords drawn through it? How could you determine if these two chords were congruent?

### Examples

#### Example 1

Find the value of and .

The diameter is perpendicular to the chord, which means it bisects the chord and the arc. Set up equations for and .

#### Example 2

and in . Find the radius.

First find the radius. is a radius, so we can use the right triangle with hypotenuse . From Chord Theorem #3, .

#### Example 3

1. If , find .

, which means the arcs are congruent too. .

1. If , find .

because .

#### Example 4

Find the values of and .

The diameter is perpendicular to the chord. From Chord Theorem #3, and .

#### Example 5

Find the value of .

Because the distance from the center to the chords is equal, the chords are congruent.

### Review

Fill in the blanks.

1. List all the congruent radii in .

Find the value of the indicated arc in .

Find the value of and/or .

1. Find in Question 17. Round your answer to the nearest tenth of a degree.
2. Find in Question 22. Round your answer to the nearest tenth of a degree.

In problems 25-27, what can you conclude about the picture? State a theorem that justifies your answer. You may assume that is the center of the circle.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English Spanish

chord

A line segment whose endpoints are on a circle.

circle

The set of all points that are the same distance away from a specific point, called the center.

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.