CAT 2022 Slot 2 QA Q21 Geometry

Question 21 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

The length of each side of an equilateral triangle \(\mathrm{ABC}\) is \(3 \mathrm{~cm}\). Let \(\mathrm{D}\) be a point on \(\mathrm{BC}\) such that the area of triangle \(\mathrm{ADC}\) is half the area of triangle \(\mathrm{ABD}\). Then the length of \(\mathrm{AD}\), in \(\mathrm{cm}\), is

A
\(\sqrt{6}\)
B
\(\sqrt{5}\)
C
\(\sqrt{8}\)
D
\(\sqrt{7}\)

Answer: (D) √(7) Geometry MCQ

Solution

ADC and ABD share the height from A, so \(DC:BD=1:2\), giving \(BD=2\), \(DC=1\).
Foot of the altitude from A is the midpoint M of BC; \(DM=0.5\), \(AM=\frac{3\sqrt3}{2}\).
\(AD^2=\frac{27}{4}+\frac14=7\), so \(AD=\sqrt7\).