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. $9$
• B. $7$
• C. $8$
• D. $12$

Answer: (C)

Verified step by step guidance

1. Identify the corresponding sides of the two similar right triangles. Since the triangles are similar, their corresponding sides are proportional. The first triangle has sides 3, 4, and 5, and the second triangle has sides 6, x, and 10.
2. Determine the scale factor between the two triangles by comparing the known corresponding sides. For example, compare the hypotenuses: the first triangle's hypotenuse is 5, and the second triangle's hypotenuse is 10. Calculate the scale factor as \(\frac{10}{5}\).
3. Use the scale factor to find the unknown side \(x\). Since the side corresponding to 3 in the first triangle corresponds to 6 in the second triangle, check if the scale factor matches. Then apply the same scale factor to the side 4 in the first triangle to find \(x\) by setting up the proportion \(\frac{4}{x} = \frac{3}{6}\) or directly using the scale factor.
4. Write the proportion equation explicitly: \(\frac{4}{x} = \frac{5}{10}\) or equivalently \(x = 4 \times \frac{10}{5}\), depending on which sides you are comparing.
5. Solve the proportion for \(x\) by multiplying both sides appropriately to isolate \(x\). This will give you the value of \(x\) without calculating the final numeric answer here.