CAT 2023 Slot 3 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

A rectangle with the largest possible area is drawn inside a semicircle of radius \(2 \mathrm{~cm}\). Then, the ratio of the lengths of the largest to the smallest side of this rectangle is

A
\(2 : 1\)
B
\(\sqrt{5} : 1\)
C
\(1 : 1\)
D
\(\sqrt{2} : 1\)

Answer: (A) 2 : 1 Geometry MCQ

Solution

Rectangle with base \(2x\) on the diameter and height \(y\): corner on the arc gives \(x^2+y^2=r^2\).
Area \(2xy\) is maximised when \(x=y=\frac{r}{\sqrt2}\).
Sides: \(2x = \sqrt2 r\) and \(y = \frac{r}{\sqrt2}\), ratio \(2:1\).

Shortcut. It is half of the square inscribed in the full circle, so the sides are 2:1.