UPSC CDS 1 2022 Math Paper

Concept:The rhombus has equal sides. Use Heron's formula to find the area of the triangle formed by two sides and one diagonal, then double to get the rhombus area, and finally divide by 4.Explanation:Perimeter = 100 m, so each side = $\frac{100}{4} = 25$ m.One diagonal = 40 m. This diagonal and two sides form a triangle with sides $25, 25, 40$ m.Semi-perimeter of triangle: $s = \frac{(25+25+40)}{2} = 45$ m.Area of triangle = $\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{45(45-25)(45-25)(45-40)} = \sqrt{45 \times 20 \times 20 \times 5}$.Simplify: $\sqrt{45 \times 20 \times 20 \times 5} = \sqrt{9 \times 5 \times 400 \times 5} = \sqrt{9 \times 25 \times 400} = 3 \times 5 \times 20 = 300$ m².The rhombus consists of two such triangles, so its area = $300 \times 2 = 600$ m².Dividing into four equal parts gives each part area = $\frac{600}{4} = 150$ m².Answer:150 square metre (Option A).