UPSC CDS 1 2023 Math Paper

Concept:In a right triangle, the altitude to the hypotenuse forms similar triangles, so the square of a leg equals the product of the adjacent hypotenuse segment and the whole hypotenuse.Explanation:Given triangle ABC with $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm. These sides satisfy $6^2 + 8^2 = 10^2$, so $\angle B = 90^{\circ}$.$BD$ is drawn perpendicular to $AC$, meeting at $D$. Here $\angle ABD = \theta$.By the geometric mean theorem: $AB^2 = AD \cdot AC$.Substitute values: $6^2 = AD \times 10$ → $36 = 10 \cdot AD$ → $AD = 3.6$ cm.In right triangle ABD, $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AD}{AB}$.Thus $\sin\theta = \frac{3.6}{6} = 0.6$.Answer:$0.6$