Question

the graph shows the endpoints of one side of a regular pentagon. determine the perimeter of the pentagon. enter the unknown segment length(s), then calculate the perimeter of the polygon. perimeter of the pentagon: units

Explanation:

Step1: Recall side - length property of regular pentagon

A regular pentagon has 5 equal - length sides.

Step2: Determine the length of one side

Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ for two points $A(-12,16)$ and $O(3,1)$. Here, $x_1=-12,y_1 = 16,x_2 = 3,y_2 = 1$.
\[

$$\begin{align*} d&=\sqrt{(3-(-12))^2+(1 - 16)^2}\\ &=\sqrt{(3 + 12)^2+(1-16)^2}\\ &=\sqrt{(15)^2+(-15)^2}\\ &=\sqrt{225 + 225}\\ &=\sqrt{450}\\ &\approx21.21 \end{align*}$$

\]

Step3: Calculate the perimeter of the pentagon

The perimeter $P$ of a regular pentagon with side - length $s$ is $P = 5s$. Since $s\approx21.21$, then $P=5\times21.21 = 106.05$.

Answer:

$106.05$