   Chapter 8.3, Problem 7E 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

For a regular hexagon, the length of the apothem is 10 cm. Find the length of the radius for the circumscribed circle for this hexagon.

To determine

To find:

The length of the radius for the circumscribed circle for the regular hexagon.

Explanation

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

4) The radius of the circumscribed circle about a hexagon is equal to the radius of the hexagon.

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 cm 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 cm.

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

Therefore, QD = ED = x cm.

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

Therefore, GD=x2cm

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

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