CAT 2022 Slot 1 QA Q19 Geometry

Question 19 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

All the vertices of a rectangle lie on a circle of radius \(R\). If the perimeter of the rectangle is \(P\), then the area of the rectangle is

A
\(\frac{P^2}{2}-2 P R\)
B
\(\frac{P^2}{8}-2 R^2\)
C
\(\frac{P^2}{16}-R^2\)
D
\(\frac{P^2}{8}-\frac{R^2}{2}\)

Answer: (B) (P^2)/8-2 R^2 Geometry MCQ

Solution

Let sides be \(l, w\). The diagonal is a diameter: \(l^2 + w^2 = 4R^2\); perimeter gives \(l + w = \frac{P}{2}\).
\((l + w)^2 = l^2 + w^2 + 2lw \Rightarrow \frac{P^2}{4} = 4R^2 + 2lw\).
Area \(lw = \frac{P^2}{8} - 2R^2\).

Shortcut. Check with a square of side 1: \(R^2 = \frac12\), \(P = 4\); only option B gives \(2 - 1 = 1\).