Similar Triangles: Finding Perimeter of ΔDEF When ΔABC Perimeter is 64

To solve this problem, follow these steps:

• Step 1: Find the ratio of similarity between the corresponding sides of the triangles $\triangle DEF$ and $\triangle ABC$.
• Step 2: Simplify the ratio of the sides.
• Step 3: Apply the ratio to find the perimeter of $\triangle DEF$.

Let's follow these steps:

Step 1: Given $EF = 18$ and $BC = 24$ are corresponding sides of similar triangles $\triangle DEF \sim \triangle ABC$, we find the ratio:

$\frac{EF}{BC} = \frac{18}{24}$

Step 2: Simplify this ratio:

$\frac{18}{24} = \frac{3}{4}$

Step 3: The ratio of similarity $\frac{DE}{AB} = \frac{EF}{BC} = \frac{3}{4}$ means the perimeter of $\triangle DEF$ is $\frac{3}{4}$ of the perimeter of $\triangle ABC$.

Given the perimeter of $\triangle ABC$ is 64, compute the perimeter of $\triangle DEF$:

$\frac{3}{4} \times 64 = 48$

Therefore, the perimeter of $\triangle DEF$ is 48.