Triangles and are similar with corresponding to and corresponding to . If , , and , what is the length of ?

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SAT Lines, Angles & Triangles Question 32 | Free SAT Prep | Aniko

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Question 32·Medium·Lines, Angles, and Triangles

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Triangles $ABC$ and $DEF$ are similar with $AB$ corresponding to $DE$ and $BC$ corresponding to $EF$ . If $AB = 6$ , $BC = 8$ , and $DE = 9$ , what is the length of $EF$ ?

For similarity questions, first identify which sides correspond, then compute a single scale factor using any known pair of corresponding sides. Use that same factor in a proportion or direct multiplication to find the unknown side, and avoid mixing operations like adding side lengths—stick with consistent ratios between corresponding sides.

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Use the idea of similar triangles

When two triangles are similar, what can you say about the ratios of their corresponding side lengths?

Find the scale factor first

You know $AB=6$ and its corresponding side $DE=9$ . How can you use these to find the scale factor from triangle $ABC$ to triangle $DEF$ ?

Apply the same ratio to EF

Once you know how much larger triangle $DEF$ is compared to triangle $ABC$ , multiply $BC=8$ by that same scale factor to get $EF$ .

Desmos Guide

Compute EF using the scale factor

In Desmos, type 8*(9/6) (this represents $BC \times DE/AB$ ). Look at the numeric result that appears; that value is the length of $EF$ .

Step-by-step Explanation

Use similarity to set up a ratio

The problem states that triangles $ABC$ and $DEF$ are similar and that $AB$ corresponds to $DE$ and $BC$ corresponds to $EF$ . For similar triangles, corresponding sides are proportional, so the ratio $DE/AB$ must be equal to the ratio $EF/BC$ .

Find the scale factor between the triangles

Use the given corresponding sides $AB=6$ and $DE=9$ to find the scale factor from triangle $ABC$ to triangle $DEF$ :

\text{scale factor} = \frac{DE}{AB} = \frac{9}{6} = \frac{3}{2}.

This means every side in triangle $DEF$ is $3/2$ times the length of the matching side in triangle $ABC$ .

Apply the scale factor to find EF

Now apply the same scale factor to side $BC$ to get $EF$ :

EF = BC \times \frac{3}{2} = 8 \times \frac{3}{2} = 12.

So the length of $EF$ is 12, which corresponds to choice B.

Strategy

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation