In an equilateral triangle with side a, prove that area = .

Let ABC be the equilateral triangle whose side is a cm.

We know that altitude bisects the opposite side.

So, BD = DC = a cm.

We know that the Pythagoras Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, by applying Pythagoras Theorem,

(a cm)2 = AD2 + (a/2 cm)2

a2 cm2 = AD2 + a2/4 cm2

AD = √3 a/2 cm = h

We know that area of a triangle = 1/2 × base × height

Ar(ΔABC) = 1/2 × a cm × √3 a/2 cm

ar(ΔABC) = √3 a2/4 cm2

Hence proved.

ar(ΔABC) = √3 a2/4 cm2

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