Question

a regular pentagon is shown. what is the length of the apothem, rounded to the nearest tenth? 2.9 cm 3.3 cm 4.9 cm 6.5 cm

Explanation:

Step1: Recall apothem formula for regular polygon

For a regular polygon, we can use right - triangle relationships. A regular pentagon can be divided into 5 isosceles triangles. The central angle of each of these isosceles triangles is $\theta=\frac{360^{\circ}}{n}$, where $n = 5$ (number of sides of the pentagon), so $\theta = 72^{\circ}$. The apothem is the height of the isosceles triangle. If we consider half of the side - length of the pentagon and the apothem and the radius of the circum - circle of the pentagon, we can use trigonometry. Here, we can also use the fact that if we know the side - length $s$ and the number of sides $n$ of a regular polygon, the apothem $a$ can be calculated using the formula $a=\frac{s}{2\tan(\frac{180^{\circ}}{n})}$. But we can also use the right - triangle formed in the pentagon directly.
Let's assume we use the right - triangle formed in the pentagon. If we consider the right - triangle with hypotenuse equal to the radius of the circum - circle of the pentagon and one side as half of the side of the pentagon. The side of the pentagon $s = 9.4$ cm. Half of the side length $x=\frac{s}{2}=\frac{9.4}{2}=4.7$ cm.
We know that the central angle of the pentagon is $72^{\circ}$, and in the right - triangle formed by the apothem, half of the side of the pentagon and the radius of the circum - circle, the angle at the center of the pentagon for this right - triangle is $\frac{72^{\circ}}{2}=36^{\circ}$.

Step2: Use tangent function

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In our right - triangle, if $\theta = 36^{\circ}$ and the opposite side is half of the side of the pentagon ($x = 4.7$ cm) and the adjacent side is the apothem $a$. So $\tan36^{\circ}=\frac{4.7}{a}$. Then $a=\frac{4.7}{\tan36^{\circ}}$.
Since $\tan36^{\circ}\approx0.7265$, $a=\frac{4.7}{0.7265}\approx6.5$ cm.

Answer:

6.5 cm