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? $x=$

• A. $10$
• B. $20$
• C. $12$
• D. $15$

Answer: (D)

Verified step by step guidance

1. Recall that the SSS (Side-Side-Side) similarity theorem states that two triangles are similar if their corresponding sides are in proportion.
2. Identify the sides of the two triangles: Triangle 1 has sides 6, 8, and x; Triangle 2 has sides 9, 12, and 15.
3. Set up ratios of corresponding sides to find the value of x that makes the triangles similar. For example, compare the ratios \$\frac{6}{9}\$, \$\frac{8}{12}\$, and \$\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 sides are in the ratio 2:3.
5. Since the triangles are similar by SSS, the third ratio must also be \$\frac{2}{3}\$. Set up the equation \$\frac{x}{15} = \frac{2}{3}\$ and solve for \$x\$ by cross-multiplying.