UPSC CDS 2 2019 Math Paper

Concept:In a parallelogram, if one diagonal forms an equilateral triangle with two adjacent sides, then the parallelogram is a rhombus, and its diagonals are perpendicular bisectors.Explanation:Given: $\mathrm{PQRS}$ is a parallelogram with diagonals $\mathrm{PR}$ and $\mathrm{QS}$ intersecting at $\mathrm{O}$.$\text{triangle} \mathrm{QRS}$ is equilateral with side $10\,\mathrm{cm}$. Therefore, $\mathrm{QR} = \mathrm{RS} = \mathrm{QS} = 10\,\mathrm{cm}$.In a parallelogram, opposite sides are equal: $\mathrm{PQ} = \mathrm{RS} = 10\,\mathrm{cm}$ and $\mathrm{PS} = \mathrm{QR} = 10\,\mathrm{cm}$.All four sides are $10\,\mathrm{cm}$, so $\mathrm{PQRS}$ is a rhombus.In a rhombus, the diagonals are perpendicular bisectors of each other. Hence, $\mathrm{PR} \perp \mathrm{QS}$ and $\mathrm{O}$ is the midpoint of both diagonals.Since $\text{triangle} \mathrm{QRS}$ is equilateral, the median from $\mathrm{R}$ to side $\mathrm{QS}$ is also the altitude. $\mathrm{O}$ is the midpoint of $\mathrm{QS}$, so $\mathrm{OR}$ is this median.Length of median in an equilateral triangle of side $a$ is $\frac{\sqrt{3}}{2}a$. Here $a = 10\,\mathrm{cm}$, so:$\mathrm{OR} = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3}\,\mathrm{cm}$.Diagonal $\mathrm{PR} = 2 \times \mathrm{OR} = 2 \times 5\sqrt{3} = 10\sqrt{3}\,\mathrm{cm}$.Answer:$10\sqrt{3}\,\mathrm{cm}$ (Option B).