Question

Prove that the area of equilateral triangle is root 3 side^2 / 4.

Answer 1

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Here is your answer in the attachment.

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Answer 2

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Here is your answer
As we know that in equilateral triangle the height (altitude) will cut the middle of the triangle forming two right triangle and dividing the Base in two equal halves

Proof
1. Let the side of the triangle is a and height is h

2. Finding altitude, we draw the Perpendicular

3. By using Pythagoras Theorem
$h {}^{2} = {a}^{2} - ( \frac{a }{2} ) {}^{2} \\ {h}^{2} = {a}^{2} - \frac{ {a}^{2} }{4} \\ {h}^{2} = \frac{4 {a}^{2} - {a}^{2} }{4} \\ h = \sqrt{ \frac{3 {a}^{2} }{4} } \\ h = \frac{ \sqrt{3}a }{2}$
4. Now as we know that
The area of the triangle = 1/2(base × altitude)
$= \frac{1}{2} \times a \times \frac{ \sqrt{3}a }{4} \\ = \frac{ \sqrt{3} {a}^{2} }{4}$
Therefore the area of the equilateral triangle is
$= \frac{ \sqrt{3} (side) {}^{2} }{4}$
Hence Proved

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