Given that triangle $ABC$ is similar to triangle $AXY$ with a ratio of similarity $3:2$, and that $bc=24$ in triangle $ABC$, what is the length of the corresponding side $xy$ in triangle $AXY$?

• A. $36$
• B. $16$
• C. $18$
• D. $12$

Answer: (B)

Verified step by step guidance

1. Identify the corresponding sides in the two similar triangles. Since triangle ABC is similar to triangle AXY, side BC in triangle ABC corresponds to side XY in triangle AXY.
2. Note the given ratio of similarity between the triangles, which is 3:2. This means that every length in triangle ABC is \( \frac{3}{2} \) times the corresponding length in triangle AXY.
3. Set up the proportion relating the sides BC and XY using the similarity ratio: \( \frac{BC}{XY} = \frac{3}{2} \).
4. Substitute the known length of side BC, which is 24, into the proportion: \( \frac{24}{XY} = \frac{3}{2} \).
5. Solve the proportion for \( XY \) by cross-multiplying and isolating \( XY \): \( XY = \frac{24 \times 2}{3} \).