UPSC CDS 1 2023 Math Paper

Concept:Use Pythagoras theorem and the area relation (or angle bisector theorem) to find the required length.Explanation:Given: $AB = 6$ cm, $BC = 8$ cm, and $AD = 10$ cm.First, apply Pythagoras theorem to triangle $ABC$ (right-angled at $B$):$AC^2 = AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100$So, $AC = 10$ cm.Now, consider the altitude $BD$ from $B$ to side $AC$.Using the area of triangle $ABC$ in two ways:Area $= \frac{1}{2} \times AB \times BC = \frac{1}{2} \times AC \times BD$Thus, $AB \times BC = AC \times BD$Substitute: $6 \times 8 = 10 \times BD$$48 = 10 \, BD$$BD = 4.8$ cm.From the diagram (implied), point $Q$ lies on $BC$ such that $BQ = BD = 4.8$ cm.Therefore, $QC = BC - BQ = 8 - 4.8 = 3.2$ cm.Answer:$3.2$ cm