An equilateral triangle is described on the diagonal of a square. What is the ratio of the area of the triangle to that of the square ?
An equilateral triangle is described on the diagonal of a square. What is the ratio of the area of the triangle to that of the square ? Correct Answer $$\sqrt 3 $$ : 2
Let the side of the square be a cm
Then, the length of its diagonal = $$\sqrt 2 $$ a cm
Area of equilateral triangle with side :
$$\eqalign{ & = \sqrt 2 a \cr & = \frac{{\sqrt 3 }}{4} \times {\left( {\sqrt 2 a} \right)^2} \cr & = \frac{{\sqrt 3 {a^2}}}{2} \cr} $$
∴ Required ratio :
$$\eqalign{ & = \frac{{\sqrt 3 {a^2}}}{2}:{a^2} \cr & = \sqrt 3 :2 \cr} $$
