Question

In an equilateral triangle,prove that three times the square of one side is equal to four times the square of one of its altitudes.with diagram

Answer 1

Let a is the length of the side of equilateral triangle and AE is the altitude.

So BE = EC = BC/2 = a/2

Now in triangle ABC,

From Pythagoras Theorem

AB2 = AE2 + BE2

=> a2 = AE2 + (a/2)2

=> AE2 = a2 - a2 /4

=> AE2 = 3a2 /4

=> 4AE2 = 3a2

=> 4*(square of altitude) = 3*(square of one side)

So three times the square of one side is equal to four times the square of one of its altitudes.