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.

Answer 1

 

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.
∴ BE = EC = BC/2 = a/2
Applying Pythagoras theorem in ΔABE, we get
AB2 = AE2 + BE2 

 

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

Answer 2

Solution: Let us assume that each side of the triangle is ‘a’, then its altitude AM is as follows:

Four times of square of altitude can be calculates as follows:

Hence; three times of square of a side = four times of square of altitude proved