Question

show that altitude of a equilateral triangle is √3/2

Answer 1

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 ;

$\sf a {}^{2} = h {}^{2} + (\frac{a}{2} ) {}^{2} \\ \\ \sf h {}^{2} = a {}^{2} - \frac{a {}^{2} }{4} \\ \\ \sf h {}^{2} = \frac{3a {}^{2} }{4} \\ \\ \sf h = \frac{ \sqrt{3} }{2} a$

Hence, the altitudes of the triangle is proved.