Question

Prove that the area of a equilateral triangle is equal to √3/4 a^2 where a is the side of triangle

Answer 1

a is the side of traingle .
s=(a+a+a)/2=3a/2
now use hero 's formula ,
area of traingle =root {3a/2 {3a/2-a)(3a/2-a)(3a/2-a)}=root {3a^4/16}
=root3/4 .a^2 (hence proved)

Answer 2

Since all the 3 sides of the triangle are same,

AB = BC = CA = a

 Find the altitude of the △△ABC. 

Draw a perpendicular from point A to base BC, AD ⊥⊥ BC

By using Pythagoras theorem

In △ ADC

h² = AC² - DC²

= a² - (a/2)² [Because, DC = a²]

= a²- a²/4

h = √3a/2

We know that, Area of a triangle = 1/2 * Base * Height

= 1/2 * a * 3√a/2

= √3/4*a²

The area of a equilateral triangle = √3/4*a²

hope it helped you !!^_^

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