   Chapter 7.3, Problem 15E 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

Find the lengths of the side and the radius of an equilateral triangle whose apothem has the length 8 ft.

To determine

To find:

The lengths of the side and the radius of an equilateral triangle.

Explanation

Approach:

1) Any radius of a regular polygon bisects the angle at the vertex to which it is drawn.

2) Any apothem of a regular polygon bisects the side of the polygon to which it is drawn.

3) The measure of any central angle of a regular polygon of n sides is given by c=360n

Calculation:

Consider an equilateral triangle DEF with QE as radius and QG as apothem.

It is given that the length of the apothem of the equilateral triangle DEF is 8 ft i.e., QG = 8 ft

Use the formula of central angle, c=360n

Substitute n = 3 in c=360n.

c=3603=120

Therefore, EQD=60

Any radius of a regular polygon bisects the angle at the vertex to which it is drawn.

Therefore, GQD=60

Now, ΔODE is a 30°-60°-90° triangle.

By properties of 30°-60°-90° triangle,

GD=83ft

Any apothem of a regular polygon bisects the side of the polygon to which it is drawn

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