Let PQRS is square whose side is ‘a’ units then PQ = QR = RS = SP = ‘a’ units.
Then the diagonal
\(\overline{PR}\) = \(\sqrt{a^2+a^2}\) = a√2 units.
Let △PRT is an equilateral triangle, then PR = RT = PT = a√2
∴ Area of equilateral triangle constructed on diagonal
Let △QRZ is another equilateral triangle whose sides are
\(\overline{QR}=\overline{RZ}=\overline{QZ}\)= ‘a’ units
Then the area of equilateral triangle constructed on one side of square = \(\frac{\sqrt{3}}{4}\)a2 ……. (2)
∴ 1/2 of area of equilateral triangle on diagonal \(\frac{1}{2}(\frac{\sqrt{3}}{2}a^2)=\frac{\sqrt{3}}{4}\) = a2 = area of equilateral triangle on the side of square.
Hence Proved.