Chapter 6.1, Problem 20E

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7th Edition

Alexander + 2 others

ISBN: 9781337614085

Textbook Problem

Suppose that a circle is divided into three congruent arcs by points

To determine

**To calculate:**

The measure of each arc and the type of figure that results when *A, B *and *C *are joined by line segments.

Explanation

**Given:**

A circle is divided into three congruent arcs by the points *A, B *and *C*.

**Congruent arcs:**

In a circle or concentric circles, congruent arcs are arcs with equal measures.

**Theorem:**

The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

**Central angle postulate:**

In a circle, the degree measure of a central angle is equal to the degree measure of its intercepted arc.

**Calculation:**

Consider a circle with center *O. *Let *A, B* and *C* be the points such that the circle is divided into three congruent arcs as shown below,

Since the arcs are congruent, they are of equal measure.

Let *x* be the measure of each arc.

By arc addition postulate,

Hence the measure of each arc is

When we join the points *A, B* and *C* with line segments, we get a triangle which is shown below

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