### Chord Theorems

There are several important theorems about chords that will help you to analyze circles better.

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

Use to answer the following.

- If , find .

, which means the arcs are congruent too. .

- 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.

- List all the congruent radii in .

Find the value of the indicated arc in .

Find the value of and/or .

- Find in Question 17. Round your answer to the nearest tenth of a degree.
- 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.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.4.