Q27 of 32 Page 6

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. (CBSE 2002, 2007)

Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC



To Prove : 4 × (Square of altitude) = 3 × (Square of one side)

Proof:

Altitude of equilateral triangle divides the side in two equal parts. Therefore,



Pythagoras Theorem :  It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Applying Pythagoras theorem in ΔABE, we obtain

AB2 = AE2 + BE2

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

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