UPSC CDS II 2024 Math Paper

Concept:The length of the perpendicular from the right angle vertex to the hypotenuse in a right triangle is found by equating two expressions for the area of the triangle.Explanation:The given triangle $ABC$ is right-angled at $B$.Side $AB = 63$ cm, side $BC = 16$ cm, and hypotenuse $AC = 65$ cm.$BD$ is drawn perpendicular from $B$ to $AC$.The area of triangle $ABC$ can be computed in two ways:(i) Using the two legs as base and height: $\text{Area} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 63 \times 16 = 504$ cm$^2$.(ii) Using $AC$ as base and $BD$ as height: $\text{Area} = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 65 \times BD$.Since both expressions represent the same area, we set them equal:$\frac{1}{2} \times 65 \times BD = 504$.Multiply both sides by 2: $65 \times BD = 1008$.Divide by 65: $BD = \frac{1008}{65}$ cm.Answer:$\frac{1008}{65}$ cm