Q. 203.7( 3 Votes )

In an equilateral

Answer :


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


Let us draw altitude AD(h) such that AD BC.


We know that altitude bisects the opposite side.


So, BD = DC = a cm.


In ADC, ADC = 90°.


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,


AC2 = AD2 + DC2


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


a2 cm2 = AD2 + a2/4 cm2


AD2 = 3a2/4 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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