Radius bisects in the circle below. How does relate to chord ? Prove your ideas.

### Central Angles and Chords

A **central** **angle** for a circle is an angle with its vertex at the center of the circle.

In the circle above, is the center and is a central angle. Notice that the central angle meets the circle at two points ( and ), dividing the circle into two sections. Each of circle portions is called an **arc**. The smaller arc (blue below) is called the **minor** **arc**, and is considered the arc that is **intercepted** by the central angle. The larger arc (red below) is called the **major** **arc**.

The minor arc above is named . Notice that and are the endpoints of the arc and there is an arc symbol above the letters indicating that you are referring to an arc.

The major arc above is named . When naming a major arc you should use three letters so that it is clear you are referring to the larger portion of the circle.

Arcs can be **measured in degrees** just like angles. In general, *the* *degree measure of a minor arc is equal to the measure of the central angle that intercepts it*. Because there are in a circle, the sum of the measures of a minor arc and its corresponding major arc will be .

A **chord** is a segment that connects two points on a circle. If a chord passes through the center of the circle then it is a **diameter**. In the circle below, both and are chords.

Notice that each chord has a corresponding arc. is a chord and is an arc. In #2 below you will prove that two chords are congruent if and only if their corresponding arcs are congruent.

Let's take a look at a few problems involving central angles and chords.

1. Find and .

The degree measure of a minor arc is equal to the measure of the central angle that intercepts it. Therefore, . A full circle is , so .

2. Prove that two chords are congruent if and only if their corresponding arcs are congruent.

This statement has two parts that you must prove.

- If two arcs are congruent then their corresponding chords are congruent.
- If two chords are congruent then their corresponding arcs are congruent.

Both statements can be proved by finding congruent triangles. Consider the circle below with center . Note that , , , and are all radii of the circle and therefore are all congruent.

Start by proving statement #1. Assume that . This would imply that . Because the measure of an arc is the same as the measure of its corresponding central angle, it must be true that and thus . because they are all radii of the circle. Therefore, by . This means because they are corresponding parts of congruent triangles.

Now prove the converse (statement #2). Assume that . because they are all radii of the circle. Therefore, by . This means because they are corresponding parts of congruent triangles. This implies that the arcs intercepted by these angles are congruent and therefore .

3. is the center of the circle below. Find the shortest distance from to .

The shortest distance from to will be the length of a segment from to that is perpendicular to . Because is the center of the circle, is a radius and thus the length of any radius will be 3 units. Draw two radii from to and to . Also draw a segment from to that is perpendicular to .

is an equilateral triangle because it has three sides of length 3. This means that and is a 30-60-90 triangle. According to the 30-60-90 pattern, .

**Examples**

**Example 1**

Earlier, you were asked how relates to chord .

Radius bisects in the circle below. How does relate to chord ? Prove your ideas.

Two possible conjectures are that bisects and is perpendicular to . Both of these conjectures can be proved by first proving that .

To prove that , first note that and are both radii of the circle, and therefore . Also note that by the reflexive property. Since it was assumed that bisects , . by .

Because , because they are corresponding parts. Therefore, bisects . Also because , because they are corresponding parts. and are supplementary because they form a line. Therefore, and is perpendicular to .

**This means that if a radius bisects a central angle, then it is the perpendicular bisector of the related chord.**** **

**Example 2**

In the circle below, diameters and are perpendicular and .

Find .

and are perpendicular. This means that and therefore

#### Example 3

Find in the circle under Example 2.

because and are vertical angles and are therefore congruent. This means that and therefore .

#### Example 4

Find in the circle under Example 2.

because it is supplementary with . . Therefore, .

### Review

1. Draw an example of a *central angle* and its *intercepted arc*.

2. What's the relationship between a central angle and its intercepted arc?

3. Draw an example of a chord.

4. A chord that passes through the center of the circle is called a _______________.

In the circle below, and are diameters, bisects , and . Use this circle for #5-#9.

5. Find .

6. Find .

7. Find .

8. Find .

9. How is related to ?

10. Prove that when a radius bisects a chord, it is perpendicular to the chord. *Use the picture below and prove that .*

11. Prove that when a radius is perpendicular to a chord it bisects the chord. *Use the picture below and prove* *that* *.*

In the circle below with center , and .

12. Find .

13. Find .

14. Find .

15. Find .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.3.