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

The central angle of a regular pentagon for one - fifth of the circle. The total central angle around a point is \(360^{\circ}\). So the central angle \(\theta=\frac{360^{\circ}}{5} = 72^{\circ}\). When we draw the apothem (the perpendicular from the center to the side), it bisects the central angle. So the angle in the right - triangle formed is \(\alpha=\frac{72^{\circ}}{2}=36^{\circ}\).

Step2: Use the trigonometric ratio

We know that \(\sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the length of the apothem is \(10\) cm (opposite side with respect to the angle \(\alpha = 36^{\circ}\)), and the radius \(c\) is the hypotenuse of the right - triangle.
We have the formula \(\sin(36^{\circ})=\frac{10}{c}\).

Step3: Solve for \(c\)

Since \(\sin(36^{\circ})\approx0.5878\), then \(c=\frac{10}{\sin(36^{\circ})}\).
Substitute the value of \(\sin(36^{\circ})\) into the formula: \(c=\frac{10}{0.5878}\approx17.01\)

Answer:

\(17.01\) cm