Question

a regular pentagon is shown. what is the measure of the radius, c, rounded to the nearest hundredth? use an appropriate trigonometric ratio to solve. 5.88 cm 8.09 cm 12.36 cm 17.01 cm

Explanation:

Step1: Find the central angle

A regular pentagon has \(n = 5\) sides. The central angle \(\theta=\frac{360^{\circ}}{n}=\frac{360^{\circ}}{5} = 72^{\circ}\). The right - angled triangle is formed by the apothem (\(a = 10\) cm), half of the side length (\(s/2=b\)), and the radius (\(c\)). The angle at the center of the pentagon for the right - angled triangle is \(\alpha=\frac{72^{\circ}}{2}=36^{\circ}\).

Step2: Use the cosine ratio

We know that \(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\cos(36^{\circ})=\frac{10}{c}\).

Step3: Solve for \(c\)

$$c=\frac{10}{\cos(36^{\circ})}$$

Since \(\cos(36^{\circ})\approx0.8090\), then \(c=\frac{10}{0.8090}\approx12.36\)

Answer:

\(12.36\) cm (the third option)