CAT 2022 Slot 2 QA Q1 Geometry

Question 1 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

In triangle \(A B C\), altitudes \(A D\) and \(B E\) are drawn to the corresponding bases. If \(\angle B A C=45^{\circ}\) and \(\angle A B C=\theta\), then \(\frac{A D}{B E}\) equals

A
\(\sqrt{2} \sin \theta\)
B
\(\sqrt{2} \cos \theta\)
C
\(\frac{(\sin \theta+\cos \theta)}{\sqrt{2}}\)
D
1

Answer: (A) √(2) θ Geometry MCQ

Solution

Altitude from A to BC: \(AD = AB\sin B = AB\sin\theta\).
Altitude from B to AC: \(BE = AB\sin A = AB\sin45^\circ = \frac{AB}{\sqrt2}\).
\(\frac{AD}{BE} = \sqrt2\sin\theta\).

Shortcut. Plug in \(\theta=90^\circ\): then AD = AB and \(BE=AB/\sqrt2\), so the ratio is \(\sqrt2\). Only option A gives \(\sqrt2\).