1. The problem states that two triangles are similar, meaning their corresponding sides are proportional. 2. We are given the larger triangle with sides 15, 24, and 27, and the smaller triangle with sides 8, 5, and 9. 3. To find the ratio of the lengths of any two corresponding sides, we compare pairs of sides from the smaller and larger triangles. 4. Let's check the ratio for each pair: - Compare side 8 (smaller) to 15 (larger): $$\frac{8}{15}$$ - Compare side 5 (smaller) to 9 (larger): $$\frac{5}{9}$$ - Compare side 9 (smaller) to 24 (larger): $$\frac{9}{24} = \frac{3}{8}$$ 5. None of these ratios are equal, so let's check if the sides correspond differently. Since the triangles are similar, the ratios of corresponding sides must be equal. 6. Let's try matching 8 with 9, 5 with 15, and 9 with 24: - $$\frac{8}{9}$$ - $$\frac{5}{15} = \frac{1}{3}$$ - $$\frac{9}{24} = \frac{3}{8}$$ Still not equal. 7. Try matching 8 with 24, 5 with 15, and 9 with 27: - $$\frac{8}{24} = \frac{1}{3}$$ - $$\frac{5}{15} = \frac{1}{3}$$ - $$\frac{9}{27} = \frac{1}{3}$$ 8. All ratios equal $$\frac{1}{3}$$, so the ratio of the lengths of any two corresponding sides is $$\frac{1}{3}$$. Final answer: A) $$\frac{1}{3}$$