Question

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Answer 1

R.E.F.Image

Let ABCD be a square of side a.

Therefore diagonal = $\sqrt{2}a$

Two equilateral triangles are

ABE , DBF

Length of one side of

$\Delta ABE=a$

$\Delta DBF=\sqrt{2}a$

We know equilateral triangle have all angles as ${60}^{\circ }$ ; so all equilateral triangles are similar to other.

$\therefore \frac{areaof\Delta ABE}{areaof\Delta DEF}={\left(\frac{a}{\sqrt{2a}}\right)}^2=\frac{1}{2}$

$\therefore$ proved.