CAT 2023 Slot 1 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals
  1. A
    1 : 2
  2. B
    5 : 8
  3. C
    8 : 5
  4. D
    2 : 1

Answer: (C) 8 : 5 Geometry MCQ

Solution

  1. In a cyclic quadrilateral, \(\triangle AEB \sim \triangle DEC\): \(\frac{AE}{DE} = \frac{AB}{DC} = 2\).
  2. Also \(\triangle AED \sim \triangle BEC\): \(\frac{DE}{CE} = \frac{AD}{BC} = \frac45\).
  3. \(\frac{AE}{CE} = \frac{AE}{DE}\cdot\frac{DE}{CE} = 2\times\frac45 = \frac85\).

Shortcut. For a cyclic quadrilateral, \(\frac{AE}{CE} = \frac{AB\cdot AD}{CB\cdot CD} = \frac{2\cdot4}{5\cdot1}\).

Same concept in other years

See every question on Chords in a circle: similar triangles and power of a point.

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