CAT 2024 Slot 2 QA Q5 Geometry

Question 5 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

\(A B C D\) is a trapezium in which \(A B\) is parallel to \(C D\). The sides \(A D\) and \(B C\) when extended, intersect at point \(E\). If \(A B=2 \mathrm{~cm}, C D=1 \mathrm{~cm}\), and perimeter of \(A B C D\) is 6 cm , then the perimeter, in cm , of \(\triangle A E B\) is
  1. A
    10
  2. B
    9
  3. C
    8
  4. D
    7

Answer: (C) 8 Geometry MCQ

Solution

  1. Since \(CD \parallel AB\), \(\triangle EDC \sim \triangle EAB\) with ratio \(CD:AB = 1:2\). So \(D\) and \(C\) are midpoints of \(EA\) and \(EB\).
  2. Hence \(AD = \frac{EA}{2}\) and \(BC = \frac{EB}{2}\).
  3. Perimeter of \(ABCD\): \(2 + 1 + \frac{EA+EB}{2} = 6\), so \(EA+EB = 6\).
  4. Perimeter of \(\triangle AEB = EA + EB + AB = 6 + 2 = 8\) cm.

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