Question

Prove that 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.

Answer 1

Let `a` is the side of given triangle, then its diagonal will be,
`sqrt(a^2+a^2) = sqrt2a`
As, all equilateral triangles are similar, ratio of their ares will be squares of ratio of their sides.
So, ratio of area of given triangles,
`=a^2/(sqrt2a)^2 = a^2/(2a^2) = 1/2`
Thus, 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.