Question

in an equilateral triangle with side 'a'. prove that the area of the triangle is √3 by 4 a square

Answer 1

To prove the area of a triangle with side a = √3a^2/4

Let a be the three sides of the triangle (Since its equilateral)

ie;

a = a

b = a

c = a

s (or perimeter) = a + b +c/2   (According to Heron's formula)

= a + a + a/2

= 3a/2

s - a = 3a/2 - a

= 3a/2 - a/1

= 3a/2 - a*2/1*2

= 3a/2 - 2a/a

= a/2

s - b = a/2

s - c = a/2  (Since its equilateral, all sides are equal)

A = √s(s - a)*(s - b)*(s - c)

= √3a/2 * a/2 * a/2 * a/2

= a/2 * a/2 * √3

= a^2/4√3

= √3a^2/4

Therefore, area of an equilateral triangle = √3a^2/4

Hence proved!

Hope it helps :)