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 central angle

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

Step2: Use sine ratio

We know that $\sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}}$. In the right - triangle, if $\alpha = 36^{\circ}$ and the opposite side to the $36^{\circ}$ angle is 5 cm and the hypotenuse is the radius $c$. So, $\sin36^{\circ}=\frac{5}{c}$.

Step3: Solve for $c$

We can re - arrange the equation $\sin36^{\circ}=\frac{5}{c}$ to get $c=\frac{5}{\sin36^{\circ}}$. Since $\sin36^{\circ}\approx0.5878$, then $c=\frac{5}{0.5878}\approx8.51$ (this is wrong approach, let's use cosine).
Let's consider another right - triangle setup. If we consider the right - triangle formed by the apothem and half of the side - length, and use the cosine ratio. The central angle of a pentagon is $72^{\circ}$, half of it is $36^{\circ}$. If we know that $\cos36^{\circ}=\frac{b}{c}$ and assume the side - length of the pentagon is 10 cm, half of the side - length is 5 cm. We know that $\cos36^{\circ}\approx0.8090$. From $\cos36^{\circ}=\frac{5}{c}$, we can solve for $c$ as $c = \frac{5}{\cos36^{\circ}}$.
$c=\frac{5}{0.8090}\approx8.09$ cm.

Answer:

8.09 cm