Given two similar right triangles, one with sides $3$, $4$, and $5$, and the other with sides $6$, $x$, and $10$, what is the value of $x$?

• A. $8$
• B. $12$
• C. $7$
• D. $9$

Answer: (A)

Verified step by step guidance

1. Identify that the two triangles are similar, which means their corresponding sides are proportional.
2. Write the ratio of the corresponding sides from the first triangle (3, 4, 5) to the second triangle (6, x, 10). For example, set up the ratio \( \frac{3}{6} = \frac{4}{x} = \frac{5}{10} \).
3. Simplify the known ratios: \( \frac{3}{6} = \frac{1}{2} \) and \( \frac{5}{10} = \frac{1}{2} \), confirming the scale factor between the triangles is \( \frac{1}{2} \).
4. Use the scale factor to find \( x \) by setting \( \frac{4}{x} = \frac{1}{2} \) and solve for \( x \).
5. Cross-multiply and solve the equation \( 4 \times 2 = x \times 1 \) to find the value of \( x \).