CAT 2020 Quant Question [Slot 2] with Solution 01

Question

The sum of the perimeters of an equilateral triangle and a rectangle is 90 cm. The area, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy the relationship \(R = {T^2}\). If the sides of the rectangle are in the ratio 1: 3, then the length, in cm, of the longer side of the rectangle, is

  1. 24
  2. 27
  3. 21
  4. 18
Option: 2
Solution:

Let the breadth of the rectangle be b.

Length of the rectangle \( = 3b\)

Let a be the side of the equilateral triangle.

Given,

\(R = {T^2} \Rightarrow 3{b^2} = {\left( {\frac{{\sqrt 3 }}{4} \times {a^2}} \right)^2} \Rightarrow b = {a^2}/4\)

Given,

\(2(4b) + 3a = 90\)

\( \Rightarrow 8\left( {{a^2}/4} \right) + 3a - 90 = 0\)

\( \Rightarrow 2{a^2} + 3a - 90 = 0\)

\( \Rightarrow (a - 6)(2a + 15) = 0\)

\( \Rightarrow a = 6\)

\(\therefore b = 9\)

\( \Rightarrow 3b = 27\)

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