Question

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?

• A. $2:\sqrt{3}$
• B. $4:\sqrt{3}$
• C. $\sqrt{3}:2$
• D. $\sqrt{3}:4$

Answer 1

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

Let each side of the square $ABCD$ be $x$ units.

Then, diagonal of square $=x\sqrt{2}$ units
$\Rightarrow$ Each side of the equilateral $\Delta EDB=x\sqrt{2}$ units
Area of equilateral $\Delta =\frac{\sqrt{3}}{4}{\left(x\sqrt{2}\right)}^2$ $=\frac{\sqrt{3}}{2}{x}^2$
Area of square $=x^2$
$\therefore$ Required ratio $=\frac{\sqrt{3}}{2}{x}^2:x^2=\sqrt{3}:2$