Question

An equilateral triangle is drawn on the diagonal of a square. The ratio of the area of the triangle to that of the square is:

• A. $\sqrt{3}:2$
• B. $\sqrt{2}:\sqrt{3}$
• C. $2:\sqrt{3}$
• D. $1:\sqrt{2}$

Answer 1

The correct option is A $\sqrt{3}:2$

Area of square $=a^2$

Length of the diagonal $=\sqrt{2a}$

Area of equilateral triangle with side $\sqrt{2a}=\frac{\sqrt{3}}{4}\times (\sqrt{2a}{)}^2=\frac{\sqrt{3}{a}^2}{2}$

then,

Required ratio $=\frac{\sqrt{3}{a}^2/2}{a^2}$

$=\frac{\sqrt{3}}{2}$

Hence, ratio is $\sqrt{3}:2$