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

Question

Philoid · Accepted Answer

Let ABCD be a square of side a

Hence, it’s diagonal =

Two desired equilateral triangles are formed as ΔABE and ΔDBF

Side of an equilateral triangle, ΔABE, described on one of its sides = a

Side of an equilateral triangle, ΔDBF, described on one of its diagonals =

We know that equilateral triangles have all its angles as 60° and all its sides of the same length Therefore, allequilateral triangles are similar to each other. Hence, the ratio between the areas of these triangles will be equal tothe square of the ratio between the sides of these triangles

= ²