Question

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

Answer 1

Let ABCD be a square of side x units so that the diagonal

BD²=AB²+AD²

BD²=x²+x²=2x²

BD=x√2

Let APB be an equilateral triangle of side x, described on side AB of square ABCD, and BQD be an equilateral triangle of side x√2 , described on diagonal BD of square ABCD.

Then clearly

∆APB~∆BQD [By AAA similarity , each angle of both the triangle being 60°]

∴ ar(∆APB)/ar(∆BQD)=AB²/BD²

[Ratio of areas of two similar triangles is same as the ratio of the squares of their corresponding sides]

ar(∆APB)/ar(∆BQD)=x²/(x√2)²=1/2

or ar(∆APB)=1/2 ar(∆BQD)

Or 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