Prove that the area of an equilateral triangle described on one side of the square

• Geometry (C10)

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

Answer

Here, ABCD is a square, AEB is an equilateral triangle described on the side of the square and DBF is an equilateral triangle described on diagonal BD of the square.

To Prove: Area of ΔAEB = ½ Area of ΔDBF

Proof:

Let the side of the square be "a" units.

DB² = a² + a² = 2a²

DB = √2a units

Area of equilateral ΔAEB = √3/4 a²

Area of equilateral ΔDBF = √3/4 (√2a)² = √3/2 a²

Thus, Area of ΔAEB = ½ Area of ΔDBF