In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square. What is the area (in cm^2) of the rectangle PLMN? Note: The figure shown is representative.
A. sqrt(2) B. 2 C. 2*sqrt(2) D. 4
GATE 2025 · General Aptitude · Geometry · medium
Answer: Area of rectangle PLMN = 4 cm^2
Set up coordinates: P = (0,0), Q = (2,0), R = (2,2), S = (0,2). Let L = (2, t) on QR where 0 <= t <= 2.
Find the width of the rectangle using S on MN: Line PL has direction (2,t). Its equation through origin: t*x - 2*y = 0. Distance from S=(0,2) to this line = |t*0 - 2*2| / sqrt(t^2+4) = 4/sqrt(t^2+4). This distance = width of rectangle LM.
Compute PL and the area: PL = sqrt(2^2 + t^2) = sqrt(4 + t^2). Width = 4/sqrt(t^2+4). Area = PL x width = sqrt(4+t^2) x 4/sqrt(4+t^2) = 4 cm^2.