Question

AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed.
Prove that Area (∆ADE) : Area (∆ABC) = 3 : 4.

Answer 1

We have an equilateral triangle in which AD is altitude. An equilateral triangle is drawn using AD as base. We have to prove that,

Since the two triangles are equilateral, the two triangles will be similar also.

We know that according to the theorem, the ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

…… (1)

Nowis an equilateral triangle. So,

.

Therefore,

So,

We will now use this in equation (1). So,

Hence, proved.