Question

Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

Answer 1

Given: A square $ABCD$,
An equilateral $△BCE$ described on side $BC$ of the square.
An equilateral $△BDF$ described on the diagonal $BD$ of the square.
$△BCE\sim △BDF$ ($\because$ both are equiangular, each angle$={60}^{\circ }$)
$\therefore$ $\frac{ar\left(△BCE\right)}{ar\left(△BDF\right)}=\frac{B{C}^2}{B{D}^2}=\frac{B{C}^2}{{\left(\sqrt{2}BC\right)}^2}$ ($\because$ diagonal of a square$=\sqrt{2}$ side)
$=\frac{B{C}^2}{2B{C}^2}=\frac{1}{2}.$