CAT 2025 Question Paper | Quant Slot 3

CAT Questions

CAT 2025 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 18 : 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

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Video Explanation

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Explanatory Answer

In \(\triangle ABC\), \(AB=AC=12\) cm, so: \[ \angle ABC=\angle ACB \]
We are given: \[ \angle ACB=\angle AEB \]
Therefore: \[ \angle ABC=\angle AEB \]

Since side \(AB\) subtends equal angles at \(C\) and \(E\), the points \(A,C,E,B\) lie on a common circle, forming cyclic quadrilateral \(ACEB\).

In cyclic quadrilateral \(ACEB\):
\[ \angle AEC=\angle ABC \]
Since \(\triangle ABC\) is isosceles:
\[ \angle ABC=\angle ACB=\angle ACD \]
Therefore:
\[ \angle AEC=\angle ACD \]

Also, \(\angle DAC=\angle CAE\) because \(AD\) and \(AE\) lie on the same line.

Hence, by AA similarity:
\[ \triangle ADC\sim\triangle ACE \]

Corresponding sides give:
\[ \frac{AD}{AC}=\frac{AC}{AE} \]
Substituting \(AD=8\) and \(AC=12\):
\[ \frac{8}{12}=\frac{12}{AE} \]
\[ 2AE=36\Rightarrow AE=18\text{ cm} \]

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The answer is '\( 18 \)'

Choice 3 is the correct answer.

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