a regular pentagon is shown. what is the length of the apothem, rounded to the nearest tenth? 9.4 cm 8 cm…

Explanation:

Step1: Find central angle

A regular pentagon has 5 sides. The 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, half of a side, and the radius, the angle at the center of the pentagon for this right - triangle is $\frac{\theta}{2}=\frac{72^{\circ}}{2} = 36^{\circ}$.

Step2: Use trigonometry

Let the side length of the pentagon be $s = 9.4$ cm, so half of the side length $a=\frac{s}{2}=\frac{9.4}{2}=4.7$ cm. We know that $\tan\alpha=\frac{\text{opposite}}{\text{adjacent}}$, where $\alpha = 36^{\circ}$ and the opposite side is half of the side length of the pentagon and the adjacent side is the apothem $h$. So $\tan36^{\circ}=\frac{4.7}{h}$.

Step3: Solve for apothem

We can rewrite the equation as $h=\frac{4.7}{\tan36^{\circ}}$. Since $\tan36^{\circ}\approx0.7265$, then $h=\frac{4.7}{0.7265}\approx6.5$ cm.

Answer:

6.5 cm