CAT 2021 Slot 3 QA Q12 Geometry

Question 12 of the Quantitative Ability (QA) section, CAT 2021 Slot 3, with CAT's official answer and a worked solution.

A park is shaped like a rhombus and has area 96 sq m. If 40 m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹125 per m, is

TITA: type in the answer. There are no options and no negative marking.

Answer: 3500 Geometry TITA

Solution

Perimeter 40 gives side 10. Diagonals \(d_1, d_2\): \(\frac{d_1 d_2}{2} = 96 \Rightarrow d_1 d_2 = 192\), and \(\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 100 \Rightarrow d_1^2 + d_2^2 = 400\).
\((d_1 + d_2)^2 = 400 + 384 = 784 \Rightarrow d_1 + d_2 = 28\).
Cost \(= 28 \times 125 = 3500\).

Shortcut. Half-diagonals form a 6-8-10 triangle, so diagonals are 12 and 16.