\( \triangle \mathrm{ABCN} \triangle \mathrm{PQR} \). If \( \operatorname{ar}(\mathrm{ABC}): \operatorname{ar}(\mathrm{PQR})=25: 9 \) then \( \mathrm{AB}: \mathrm{PQ}=-\quad \bullet 1: 225 \quad \bullet 3: 5 \) \( -225: 81 \quad \bullet 81: 225 \)

To determine the ratio of the corresponding sides \( \mathrm{AB} : \mathrm{PQ} \) given the ratio of the areas \( \operatorname{ar}(\mathrm{ABC}) : \operatorname{ar}(\mathrm{PQR}) = 25 : 9 \), we can use the property of similar triangles. ### Step-by-Step Solution: 1. **Understanding Similarity and Area Ratios:** For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Mathematically, if two triangles are similar, then: \[ \frac{\text{Area}_1}{\text{Area}_2} = \left( \frac{\text{Side}_1}{\text{Side}_2} \right)^2 \] 2. **Applying the Given Ratios:** Given: \[ \frac{\operatorname{ar}(\mathrm{ABC})}{\operatorname{ar}(\mathrm{PQR})} = \frac{25}{9} \] Let the ratio of the corresponding sides \( \mathrm{AB} : \mathrm{PQ} = x : y \). Then: \[ \left( \frac{x}{y} \right)^2 = \frac{25}{9} \] 3. **Solving for the Side Ratio:** Taking the square root of both sides: \[ \frac{x}{y} = \frac{5}{3} \] Therefore: \[ \mathrm{AB} : \mathrm{PQ} = 5 : 3 \] 4. **Interpreting the Options:** It seems there might have been a typographical error in the options presented. Based on the calculations, the correct ratio should be \( 5 : 3 \). Among the provided options: - \( 3 : 5 \) is the closest but is the inverse of the correct ratio. The intended correct answer is likely \( 5 : 3 \). ### **Final Answer:** \[ \boxed{5:\,3} \]