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Central Angles and Arcs

Arcs determined by angles whose vertex is the center of a circle and segments that connect two points on a circle.

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Central Angles and Chords

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.

  1. If two arcs are congruent then their corresponding chords are congruent.
  2. 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. 

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Vocabulary

Arc

An arc is a section of the circumference of a circle.

Intercepts

The intercepts of a curve are the locations where the curve intersects the x and y axes. An x intercept is a point at which the curve intersects the x-axis. A y intercept is a point at which the curve intersects the y-axis.

Central Angle

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

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