UPSC Prelims 1995 · General Studies

Question 143: Basic Numeracy

In the Cartesian plane four points P, Q, R, S have coordinates (1, 1), (4, 2), (4, 4) and (1, 4). The area of the quadrilateral PQRS is

b7.5

c4.5

dImpossible to find unless the lengths of the diagonals are known

AnswerThe answer is (b).

Take the square with corners (1,1), (4,1), (4,4) and (1,4), whose area is 3 x 3 = 9. The quadrilateral PQRS cuts off the triangle with corners (1,1), (4,1) and Q (4,2), whose area is half of 3 x 1 = 1.5. So the area is 9 - 1.5 = 7.5. The shoelace formula gives the same result, 15/2 = 7.5.

Why each option is right or wrong

(a) 9 is the area of the full 3 x 3 square and ignores the cut-off triangle.
(b) Right: 7.5.
(c) 4.5 is the area of triangle PRS only.
(d) Coordinates fix the shape, so the area can be found without the diagonals.

Explanation: written by UPSC Grade, not by UPSC.

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Source: UPSC, Civil Services (Preliminary) Examination 1995, General Studies. UPSC's own paper isn't online; the words are from a published compilation. The paper on upsc.gov.in.