If the perimeters of two similar triangles $ ABC $ and $ DEF $ are $ 75 ext{ cm}$ and $ 40 ext{ cm}$ respe

Question

If the perimeters of two similar triangles $ ABC $ and $ DEF $ are $ 75 	ext{ cm}$ and $ 40 	ext{ cm}$ respectively, and if $\overline{BC}$ is $ 15 	ext{ cm}$, then the length of $\overline{EF}$ is:

Examoogle · Accepted Answer

The correct answer is Option D: 8 cm. Finding Corresponding Side Length from Ratio of Perimeters of Similar Triangles

Step-by-Step Solution & Concepts:
Key Fact: Finding Corresponding Side Length from Ratio of Perimeters of Similar Triangles

Given: $	riangle ABC \sim 	riangle DEF$, Perimeter of $	riangle ABC = 75	ext{ cm}$, Perimeter of $	riangle DEF = 40	ext{ cm}$, $BC = 15	ext{ cm}$

Key Formulas: $\frac{	ext{Perimeter}_1}{	ext{Perimeter}_2} = \frac{	ext{Side}_1}{	ext{Side}_2}$

Key Points: Perimeters simplify as $75/40 = 15/8$.; Corresponding side $EF = 8	ext{ cm}$.

Steps: Perimeter Ratio Property: For two similar triangles, the ratio of their perimeters equals the ratio of any pair of corresponding sides: $$\frac{	ext{Perimeter}(	riangle ABC)}{	ext{Perimeter}(	riangle DEF)} = \frac{BC}{EF}$$. Calculate Side EF: $$\frac{15}{8} = \frac{15}{EF} \implies EF = 8	ext{ cm}$$

Answer: Option D is correct — 8 cm

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