   Chapter 6.1, Problem 20E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Suppose that a circle is divided into three congruent arcs by points A , B , and C . What is the measure of each arc? What type of figure results when A , B , and C are joined by line segments?

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,

mAB+mBC+mCA=360x+x+x=3603x=360x=3603x=120

Hence the measure of each arc is 120.

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

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