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GATE 2022 ME (ME1) – Question 8

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

An equilateral triangle, a square and a circle have equal areas. What is the ratio of the perimeters of the equilateral triangle to square to circle?

  1. $3\sqrt{3} : 2 : \sqrt{\pi}$
  2. $\sqrt{3\sqrt{3}} : 2 : \sqrt{\pi}$
  3. $\sqrt{3\sqrt{3}} : 4 : 2\sqrt{\pi}$
  4. $\sqrt{3\sqrt{3}} : 2 : 2\sqrt{\pi}$

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Correct answer: (B) $\sqrt{3\sqrt{3}} : 2 : \sqrt{\pi}$

Explanation

Let the common area be $A$. Triangle: $\frac{\sqrt{3}}{4}a^2 = A$, so the perimeter is $3a = 2\sqrt{3\sqrt{3}}\sqrt{A}$. Square: perimeter $4\sqrt{A}$. Circle: $2\pi r = 2\sqrt{\pi}\sqrt{A}$. Dividing by $2\sqrt{A}$, the ratio is $\sqrt{3\sqrt{3}} : 2 : \sqrt{\pi}$.