CAT 2021 Slot 2 · Quant · Question 17

CAT 2021 Slot 2 previous year question on Quadrilaterals, with the answer and a step-by-step solution.

QuadrilateralsGeometry and MensurationMultiple choiceMedium
The sides AB and CD of a trapezium ABCD are parallel, with AB being the smaller side. P is the midpoint of CD and ABPD is a parallelogram. If the difference between the areas of the parallelogram ABPD and the triangle BPC is 10 sq cm, then the area, in sq cm, of the trapezium ABCD is
  1. A
    20
  2. B
    25
  3. C
    40
  4. D
    30
Pick an option, then check.
Answer & solution

Answer: Option D

Since ABPD is a parallelogram, AB=DP. And P is the midpoint of CD, so DP=PC. Writing AB=a, we get DP=PC=a and hence CD=2a.

Let h be the distance between the two parallel sides. Then \left[ ABPD \right]=ah, and triangle BPC has base PC=a with the same height, so \left[ BPC \right]=\frac{1}{2}ah.

The difference is ah-\frac{1}{2}ah=\frac{1}{2}ah=10, so ah=20.

The trapezium has parallel sides a and 2a, so \left[ ABCD \right]=\frac{1}{2}\left( a+2a \right)h=\frac{3}{2}ah=\frac{3}{2}\times 20=30 sq cm.

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