Question

a regular hexagon is shown. what is the measure of the radius, c, rounded to the nearest inch? use the appropriate trigonometric ratio to solve. 6 in. 10 in. 14 in. 24 in.

Explanation:

Step1: Recall hexagon properties

A regular hexagon can be divided into six equilateral triangles. The radius of a regular hexagon is equal to the side - length of the hexagon. If we consider the right - triangle formed inside the hexagon, the central angle of each of the six sectors of the hexagon is $\frac{360^{\circ}}{6}=60^{\circ}$. When we draw the apothem (the 12 - inch line), it bisects the central angle and the side of the hexagon.

Step2: Use cosine ratio

In the right - triangle, $\cos30^{\circ}=\frac{12}{c}$. We know that $\cos30^{\circ}=\frac{\sqrt{3}}{2}$. So, $\frac{\sqrt{3}}{2}=\frac{12}{c}$.

Step3: Solve for c

Cross - multiply to get $c\times\sqrt{3}=24$, then $c = \frac{24}{\sqrt{3}}$. Rationalize the denominator: $c=\frac{24\sqrt{3}}{3}=8\sqrt{3}\approx8\times1.732 = 13.856\approx14$ inches.

Answer:

14 in