CAT 2024 Slot 3 QA Q4 Geometry

Question 4 of the Quantitative Ability (QA) section, CAT 2024 Slot 3, with CAT's official answer and a worked solution.

A circular plot of land is divided into two regions by a chord of length \(10 \sqrt{3}\) meters such that the chord subtends an angle of \(120^{\circ}\) at the center. Then, the area, in square meters, of the smaller region is

A
\(25\left(\frac{4 \pi}{3}+\sqrt{3}\right)\)
B
\(20\left(\frac{4 \pi}{3}-\sqrt{3}\right)\)
C
\(25\left(\frac{4 \pi}{3}-\sqrt{3}\right)\)
D
\(20\left(\frac{4 \pi}{3}+\sqrt{3}\right)\)

Answer: (C) 25((4 π)/3-√(3)) Geometry MCQ

Solution

Chord \(= 2r\sin 60^\circ = r\sqrt3 = 10\sqrt3\), so \(r = 10\).
Smaller segment \(= \frac{120}{360}\pi r^2 - \frac12 r^2 \sin120^\circ = \frac{100\pi}{3} - 25\sqrt3 = 25\left(\frac{4\pi}{3} - \sqrt3\right)\).