Question

a triangle has sides whose lengths are 5, 12, and 13. a similar triangle could have sides with lengths of 10, 24, and 26 7, 24, and 25 6, 8, and 10 3, 4, and 5

Explanation:

Step1: Recall similarity of triangles

For two triangles to be similar, their corresponding sides must be in proportion (i.e., the ratios of corresponding sides must be equal). First, let's identify the ratios of the sides of the given triangle (5, 12, 13) and check each option.

Step2: Check Option 1 (10, 24, 26)

Calculate the ratios:
$\frac{10}{5} = 2$, $\frac{24}{12} = 2$, $\frac{26}{13} = 2$.
All ratios are equal (2), so this is a valid similar triangle (scaled by a factor of 2).

Step3: Verify other options (optional, but for completeness)

• Option 2 (7, 24, 25): Check ratios $\frac{7}{5}=1.4$, $\frac{24}{12}=2$, $\frac{25}{13}\approx1.923$. Ratios not equal.
• Option 3 (6, 8, 10): Ratios $\frac{6}{5}=1.2$, $\frac{8}{12}=\frac{2}{3}\approx0.666$, $\frac{10}{13}\approx0.769$. Not equal.
• Option 4 (3, 4, 5): Ratios $\frac{3}{5}=0.6$, $\frac{4}{12}=\frac{1}{3}\approx0.333$, $\frac{5}{13}\approx0.385$. Not equal.

Answer:

10, 24, and 26 (the first option: 10, 24, and 26)