Question

Prove that area of an equilateral triangle is root 3/4 a 2 , where a is thee side of the triangle

Answer 1

if you know the formula use it or do by pythagoras theorem 
ex :
by Pythagoras theorem 
if sides of eq. tri. is a
hyp= a
base= a/2
and the area= √3a square/4

Answer 2

Solution :

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Derivation of Area of an equilateral triangle ;

Let ABC be an equilateral triangle with sides 'a'. Now, draw AD perpendicular to BC.

Here, we have ΔABD = ΔADC.

We will find area of ΔABD using pythagorean theorem, according to which, the square of hypotenuse is equal to the sum of the squares of the other two sides.

Here, we have ;

$\sf a {}^{2} = h {}^{2} + (\frac{a}{2} ) {}^{2} \\ \\ \sf h {}^{2} = a {}^{2} - \frac{a {}^{2} }{4} \\ \\ \sf h {}^{2} = \frac{3a {}^{2} }{4} \\ \\ \sf h = \frac{ \sqrt{3} }{2} a$
Now, we get the height ;

$\sf area \: of \: \Delta = \frac{1}{2} \times base \times height \\ \\ \sf area \: of \: \Delta = \frac{1}{2} \times a \times \frac{ \sqrt{3} }{2} a \\ \\ \sf area \: of \: \Delta = \frac{ \sqrt{3} }{4} a {}^{2}$

Hence, area of equilateral triangle is

$\sf area \: of \: \Delta = \frac{ \sqrt{3} }{4} a {}^{2}$

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Thanks for the question !