   Chapter 7.3, Problem 16E 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 a regular hexagon whose apothem has the length 10 m.

To determine

To find:

The lengths of the side and the radius of a regular hexagon.

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 a regular hexagon ABCDEF with QE as radius and QG as apothem.

It is given that the length of the apothem of the regular hexagon ABCDEF is 10 m i.e., QG = 10 cm

Use the formula of central angle, c=360n

Substitute n = 6 in c=360n.

c=3606=60

Therefore, EQD=60

Let us suppose the measure of QD be x m.

With EQD=60 and QE=QD, ΔQED is an equiangular and equilateral triangle.

Therefore, QD = ED = x m.

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

Therefore, GD=x2m

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

Therefore, ODG=30

Now, ΔODE is a 30°-60°-90° triangle

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