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A rhombus is formed by joining the midpoints of the sides of a unit square. What is the diameter of the largest circle that can be inscribed within the rhombus? A. 1/sqrt(2) B. 1/sqrt(2) C. sqrt(2) D. 2*sqrt(2)

GATE 2022 · General Aptitude · Geometry · medium

Answer: The diameter of the largest inscribed circle = 1/sqrt(2) cm.

  1. Identify the rhombus vertices: The midpoints of the four sides of the unit square are M1=(0.5,0), M2=(1,0.5), M3=(0.5,1), M4=(0,0.5). Joining these in order gives a rhombus.
  2. Compute the side length and diagonals: Side: distance from M1 to M2 = sqrt((1-0.5)^2+(0.5-0)^2) = sqrt(0.5) = 1/sqrt(2). Diagonals: M1 to M3 = 1, M2 to M4 = 1.
  3. Compute the area of the rhombus: Using the diagonal formula for the area of a rhombus: Area = (d1*d2)/2 = (1*1)/2 = 0.5 sq cm.
  4. Apply the inradius formula and compute diameter: Semi-perimeter = (4 * (1/sqrt(2)))/2 = 2/sqrt(2) = sqrt(2). Inradius r = 0.5/sqrt(2) = 1/(2*sqrt(2)). Diameter = 2r = 2/(2*sqrt(2)) = 1/sqrt(2).