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

GIVEN A squre ABCD and equilteral `Delta BCE and ACF` have been descibed on side BC and diagonal AC respectively.
TO PROOVE `ar (Delta BCE)=(1)/(2)ar (Delta ACF)`.
PROOF Since each of the `Delta BCE and Delta ACF` is an equilateral triangle, so each angle of each one of them is `60^(@)`. So, the triangles are equiangular, and hence similar.
`:. Delta BCE~ Delta ACE`.
`:.(ar (Delta BCE))/(ar (Delta ACF))= (BC^(2))/(AC^(2))=(BC^(2))/(2(BC)^(2))[ :. AC= sqrt(2)BC]`
`=(1)/(2)`
Hence, `ar (Delta BCE)=(1)/(2)xx ar (Delta ACF)`