CAT 2025 Slot3 Quant: Geometry Triangle

Step 1: Identify given properties
In $\triangle ABC$, $AB = AC = 12$ cm, which means $\triangle ABC$ is an isosceles triangle.
Therefore, the base angles are equal: $\angle ABC = \angle ACB$.

Step 2: Establish Angle Equality
Given: $\angle ACB = \angle AEB$.
From Step 1, we know $\angle ABC = \angle ACB$.
By transitive property: $\angle ABD = \angle AEB$ (since $B, D, C$ are on the same line).

Step 3: Similarity of Triangles
Compare $\triangle ABD$ and $\triangle AEB$:
1. $\angle BAE = \angle BAE$ (Common angle)
2. $\angle ABD = \angle AEB$ (Proved above)
By AA Similarity criterion, $\triangle ABD \sim \triangle AEB$.

Step 4: Solve for AE
Using the property of similar triangles (ratio of corresponding sides):
$$\frac{AB}{AE} = \frac{AD}{AB}$$ $$AB^2 = AE \times AD$$ $$12^2 = AE \times 8$$ $$144 = 8 \times AE$$ $$AE = \frac{144}{8} = \mathbf{18} \text{ cm}.$$