(Q23) 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.

Solution :

PQRSZT

Let PQRS is square whose side is 'a' units then PQ = QR = RS = SP = 'a' units.

Then the diagonal PR = √(a2 + 2) = a√2

Let ∆PRT is an equilateral triangle, then PR = RT = PT = a√2

∵ Area of equilateral triangle constructed on diagonal =
√
 (a√2)2 =
√3
4
 a√2. √2 =
√3
 a2.....(1)

Let ∆QRZ is another equilateral triangle whose sides are

QR = RZ = QZ = 'a' units.

Then the area of equilateral triangle constructed on one side of square =
√3
4
 a2.....(2)
∴
1
2
 of area of equilateral triangle on diagonal =
1
 (
√3
2
 a2 ) =
√
 a2 = area of equilateral triangle on the side of square.

Hence Proved.