Question

prove that area of equilateral triangle is root 3 by 4 into side square

Answer 1

Let the each side be ' a ' units.

Semiperimeter = a+a+a/2

=3a/2

Area of triangle, using heron's formula

$= \sqrt{s(s - a)(s - b)(s - c)}$

$= \sqrt{ \frac{3a}{2} ( \frac{3a}{2} - a)( \frac{3a}{2} - a)( \frac{3a}{2} - a) }$

$= \sqrt{ \frac{3a}{2}( \frac{3a - 2a}{2}) ( \frac{3a - 2a}{2}) ( \frac{3a - 2a}{2} )}$

$= \sqrt{ \frac{3a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2} }$

$= \frac{a}{2} \times \frac{a}{2} \sqrt{3}$

$= \frac{ {a}^{2} }{4} \sqrt{3}$

=√3/4 a^2

Hence proved.

Answer 2

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Here is your answer in the attachment

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