Question

Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

Answer 1

Given: A square ABCD, 

An equilateral ΔBCE described on side BC of the square. 

An equilateral ΔBDF described on the diagonal BD of the square. 

ΔBCE ~ ΔBDF     (∵ both are equiangular, each angle = 60°)

∴ \(\frac{ar(ΔBCE)}{ar(ΔBDF)}=\frac{BC^2}{BD^2}=\frac{BC^2}{(\sqrt2BC)^2}\)        

(∵ diagonal of a square = √2 side)

= \(\frac{BC^2}{2\,BC^2}=\frac12.\)