CAT 2025 Slot 3 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

In \( \triangle A B C , A B = A C = 12 \mathrm {~cm} \) and \( D \) is a point on side \( B C \) such that \( A D = 8 \mathrm {~cm} \). If \( A D \) is extended to a point \( E \) such that \( \angle A C B = \angle A E B \), then the length, in cm , of \( A E \) is

A
\( 14 \)
B
\( 20 \)
C
\( 18 \)
D
\( 16 \)

Answer: (C) 18 Geometry MCQ

Solution

Since \(AB = AC\), \(\angle ABC = \angle ACB = \angle AEB\).
Triangles \(ABD\) and \(AEB\) share \(\angle A\) and have \(\angle ABD = \angle AEB\), so they are similar.
\(\frac{AB}{AE} = \frac{AD}{AB} \Rightarrow AE = \frac{AB^2}{AD} = \frac{144}{8} = 18\).