NinetynineCAT 2023 Slot 3 QA Q17 Geometry
Question 17 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.
Let \(\triangle A B C\) be an isosceles triangle such that \(A B\) and \(A C\) are of equal length. \(A D\) is the altitude from \(A\) on \(B C\) and \(B E\) is the altitude from \(\mathrm{B}\) on \(\mathrm{AC}\). If \(\mathrm{AD}\) and \(\mathrm{BE}\) intersect at \(\mathrm{O}\) such that \(\angle \mathrm{AOB}=105^{\circ}\), then \(\frac{A D}{B E}\) equals
- A\(2 \sin 15^{\circ}\)
- B\(\cos 15^{\circ}\)
- C\(2 \cos 15^{\circ}\)
- D\(\sin 15^{\circ}\)
Answer: (C) 2 15^ Geometry MCQ
Solution
- B, O, E are collinear, so in right triangle AOE (right angle at E), \(\angle AOE = 180^\circ-105^\circ=75^\circ\) and \(\angle OAE = 15^\circ\).
- AD bisects angle A, so \(\angle A = 30^\circ\) and \(\angle B=\angle C=75^\circ\).
- Area: \(BC\cdot AD = AC\cdot BE \Rightarrow \frac{AD}{BE} = \frac{AC}{BC} = \frac{\sin B}{\sin A} = \frac{\sin75^\circ}{\sin30^\circ} = 2\cos15^\circ\).