Prove that the area of an equilateral triangle is equal to `(sqrt(3))/4a^2,` where `a` is the side of the triangle. GIVEN : An equilateral triangle `A B C` such that `A B=B C=C A=adot` TO PROVE : `a r( A B C)=(sqrt(3))/4a^2` CONSTRUCTION : Draw `A D_|_B Cdot`

Explanation

In `/_\ABD and /_\ACD`

`AB=AC`....( sides of equilateral triangle)

`/_ADB=/_ADC`....(right angles)

`AD=AD...`(common side)

`therefore /_\ABD~=/_\ACD`...(by RHS congruency)

since `BC=BD+DC=a`

`therefore BD=DC=a/2`

`therefore 2BD=a`

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