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

The four given points lie in the Cartesian coordinate system, so their polygonal area can be computed directly from coordinates [1]. Use the shoelace (Gauss) formula for a quadrilateral with vertices listed in order: area = 1/2 |(x1y2 + x2y3 + x3y4 + x4y1) - (y1x2 + y2x3 + y3x4 + y4x1)|. For P(1,1), Q(4,2), R(4,4), S(1,4): sum1 = 1*2 + 4*4 + 4*4 + 1*1 = 35; sum2 = 1*4 + 2*4 + 4*1 + 4*1 = 20. Thus area = 1/2 |35 - 20| = 7.5 square units. Hence option 2 is correct; diagonal lengths are not required when coordinates are known.

Sources

1. [1] Exploring Society:India and Beyond. Social Science-Class VI . NCERT(Revised ed 2025) > Chapter 1: Locating Places on the Earth > LET'S EXPLORE > p. 14