Given two triangles with sides of lengths $6$, $8$, $x$ and $9$, $12$, $15$, what value of $x$ will make the triangles similar by the SSS similarity theorem?

• A. $12$
• B. $10$
• C. $18$
• D. $15$

Answer: (D)

Verified step by step guidance

1. Recall that two triangles are similar by the SSS (Side-Side-Side) similarity theorem if their corresponding sides are in proportion.
2. Set up the proportion using the known sides of the two triangles. The sides of the first triangle are 6, 8, and x, and the sides of the second triangle are 9, 12, and 15.
3. Write the ratios of the corresponding sides as equal fractions: \(\frac{6}{9} = \frac{8}{12} = \frac{x}{15}\).
4. Simplify the known ratios: \(\frac{6}{9} = \frac{2}{3}\) and \(\frac{8}{12} = \frac{2}{3}\), confirming the first two pairs are proportional.
5. Use the equality \(\frac{x}{15} = \frac{2}{3}\) to solve for \(x\) by cross-multiplying: \$3x = 2 \times 15\(, then isolate \)x$.