UPSC CDS I 2025 Mathematics Paper

Concept:In a right triangle, the altitude to the hypotenuse creates two smaller triangles that are similar to each other and to the original triangle. The ratio of their areas equals the square of the ratio of corresponding sides.Explanation:Triangle ABC is right-angled at A, with AB = 8 cm and BC = 10 cm. Using Pythagoras theorem, AC = √(BC² − AB²) = √(100 − 64) = 6 cm.AD is perpendicular to BC, so D is the foot of the altitude. Triangles ADC and ADB are similar (both have a right angle at D, and their other angles match the original triangle’s acute angles).The hypotenuses of the small triangles are AC and AB respectively. So the ratio of corresponding sides is AC : AB = 6 : 8 = 3 : 4.Therefore, the ratio of areas (ADC : ADB) = (3/4)² = 9/16.Answer:The required ratio is 9 : 16, which corresponds to option B.