Question

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

Answer 1

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.