CAT 2025 Slot 3 · Quant · Question 18

Since AB=AC, the base angles are equal, so \angle ABC=\angle ACB.

We are told \angle AEB=\angle ACB, so \angle AEB=\angle ABC as well.

Now compare triangles ABD and AEB. They share the angle at A, because D and E lie on the same ray from A.

And \angle ABD, which is \angle ABC, equals \angle AEB by the step above.

Two equal angles make the triangles similar: \triangle ABD\sim \triangle AEB.

Matching corresponding sides, \frac{AB}{AE}=\frac{AD}{AB}.

Cross-multiplying, A{B}^{2}=AD\times AE.

Substituting, 144=8\times AE, so AE=18 cm.

Shortcut. The configuration is the standard one in which A{B}^{2}=AD\times AE — AB is tangent to the circle through B, D and E. Recognising it gives AE=\frac{{12}^{2}}{8}=18 in a single step, and note that BC is never needed.