a regular pentagon is shown. what is the measure of the radius, c, rounded to the nearest hundredth? use an…

Explanation:

Step1: Find the central - angle of the pentagon

A regular pentagon has a central - angle $\theta=\frac{360^{\circ}}{n}$, where $n = 5$. So $\theta=\frac{360^{\circ}}{5}=72^{\circ}$. When we consider the right - triangle formed by the apothem (10 cm), half of the side length ($b$), and the radius ($c$), the angle at the center of the pentagon for this right - triangle is $\frac{\theta}{2}=\frac{72^{\circ}}{2}=36^{\circ}$.

Step2: Use the cosine function

We know that $\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}$. In our right - triangle, $\cos36^{\circ}=\frac{10}{c}$.

Step3: Solve for $c$

We can rewrite the equation as $c=\frac{10}{\cos36^{\circ}}$. Since $\cos36^{\circ}\approx0.8090$, then $c=\frac{10}{0.8090}\approx12.36$ cm.

Answer:

12.36 cm