Question

The perimeters of an equilateral triangle and a square are same, the area of triangle: area of square is
$
  {\text{A}}{\text{. 1:1}} \\
  {\text{B}}{\text{. 4:3}}\sqrt 3 \\
  {\text{C}}{\text{. 4:3}} \\
  {\text{D}}{\text{. 3:2}} \\
$

Answer 1

Hint: To determine the ratio of area of triangle to the area of square, we consider two variables ‘a’ and ‘x’ for the sides of triangle and square respectively. And then we convert one variable in terms of the other to find the answer.

Complete step-by-step answer:
Let side of equilateral triangle = a
then Perimeter of an equilateral triangle = 3a.
Let side of square = x
then Perimeter of square = 4x.
Perimeter of equilateral triangle = Perimeter of square
3a = 4x
⟹a = $\dfrac{{{\text{4x}}}}{3}$
Area of Equilateral Triangle = $\dfrac{{\sqrt 3 }}{4}{{\text{a}}^2} = \dfrac{{\sqrt 3 }}{4}{\left( {\dfrac{4}{3}{\text{x}}} \right)^2} = \dfrac{4}{{3\sqrt 3 }}{{\text{x}}^2}$
Area of Square = ${{\text{x}}^2}$
Now, $\dfrac{{{\text{Area of Triangle}}}}{{{\text{Area of Square}}}} = \dfrac{{\dfrac{4}{{3\sqrt 3 }}{{\text{x}}^2}}}{{{{\text{x}}^2}}} = \dfrac{4}{{3\sqrt 3 }} = 4:3\sqrt 3 $

Note: In order to solve this type of questions the key is to have adequate knowledge in properties and formulae of squares and triangles. We calculate their perimeters and express them in the form of a single variable. Then we calculate their areas in terms of a single variable to find out the ratio.