a regular hexagon is shown. what is the measure of the radius, c, rounded to the nearest inch? use the…

Explanation:

Step1: Analyze the regular - hexagon

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. When we consider the right - triangle formed inside the hexagon, the central angle of each of the six sectors of the hexagon is $\frac{360^{\circ}}{6}=60^{\circ}$. If we consider the right - triangle with height 12 inches, and assume the radius is $c$.

Step2: Use trigonometric ratio

In the right - triangle, if the height (opposite side with respect to the $30^{\circ}$ angle) is 12 inches and the radius $c$ is the hypotenuse. We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. In a 30 - 60 - 90 triangle formed inside the hexagon, if the angle is $30^{\circ}$ and the opposite side to the $30^{\circ}$ angle is 12 inches. Since $\sin30^{\circ}=\frac{12}{c}$, and $\sin30^{\circ}=\frac{1}{2}$. We have the equation $\frac{1}{2}=\frac{12}{c}$.

Step3: Solve for $c$

Cross - multiply the equation $\frac{1}{2}=\frac{12}{c}$ to get $c = 24$ inches.

Answer:

24 in.