Question

a regular heptagon (7 sides) has an area of $540\\ \mathrm{cm}^2$. a second regular heptagon has an area of $60\\ \mathrm{cm}^2$. the perimeter of the larger heptagon is $p$ cm. what is the perimeter of the smaller heptagon? \\(\bigcirc\\) a. $\frac{p}{81}$ \\(\bigcirc\\) b. $\frac{p}{18}$ \\(\bigcirc\\) c. $\frac{p}{9}$ \\(\bigcirc\\) d. $\frac{p}{6}$ \\(\bigcirc\\) e. $\frac{p}{3}$

Explanation:

Step1: Find area scale factor

The ratio of the areas of the larger to smaller heptagon is $\frac{540}{60} = 9$.

Step2: Find linear scale factor

For similar figures, the linear scale factor is the square root of the area scale factor.
$\sqrt{9} = 3$
This means the perimeter of the larger heptagon is 3 times that of the smaller one.

Step3: Solve for smaller perimeter

Let the smaller perimeter be $P$. We know $p = 3P$, so rearrange to solve for $P$:
$P = \frac{p}{3}$

Answer:

E. $\frac{p}{3}$