Question

If the scale factor of the areas is 16/9, then what is the perimeter of mnp?

In the diagram, def mnp
De = 9
Ef = 15
Df = 12

Answer 1

Given that the scale factor of the areas of two similar shapes is \( \frac{16}{9} \), let's denote the scale factor of the lengths (perimeters) as \( k \). The relationship between the scale factor of the areas and the scale factor of the lengths is given by the square of the scale factor of the lengths:

\[
k^2 = \frac{16}{9}
\]

To find \( k \), we take the square root of both sides:

\[
k = \sqrt{\frac{16}{9}} = \frac{4}{3}
\]

Now, we need to find the perimeter of triangle \( mnp \). We first need to calculate the perimeter of triangle \( def \).

The sides of triangle \( def \) are given by:

- \( De = 9 \)
- \( Ef = 15 \)
- \( Df = 12 \)

To find the perimeter of triangle \( def \), we sum the lengths of its sides:

\[
\text{Perimeter of } def = De + Ef + Df = 9 + 15 + 12
\]
\[
= 36
\]

Now, to find the perimeter of triangle \( mnp \), we apply the scale factor \( k \):

\[
\text{Perimeter of } mnp = k \times \text{Perimeter of } def = \frac{4}{3} \times 36
\]

Calculating this:

\[
\frac{4}{3} \times 36 = \frac{144}{3} = 48
\]

Thus, the perimeter of triangle \( mnp \) is

\[
\boxed{48}
\]