Question

Show that area of equilateral triangle is√3\4a2 where a is the side of triangle​

Answer 1

Given :

• Length of each side of an equilateral triangle = a units

To prove :

• Area of equilateral triangle = (√3)/4 × a²

Proof :

To prove that area of equilateral triangle = (√3)/4 × a² , we will use heron's formula.

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So first we will find out semi-perimeter of triangle whose length of all side is a units.

$\sf s = \dfrac{a + a + a}{2}$

where s is semi-perimeter.

$\sf \implies s = \dfrac{3 a}{2}$

Now we will find area of triangle :-

$\sf \implies A = \sqrt{ \dfrac{3a}{2} \bigg( \dfrac{3a}{2} - a \bigg)\bigg( \dfrac{3a}{2} - a \bigg) \bigg( \dfrac{3a}{2} - a \bigg) }$

where A = area of triangle.

$\sf \implies A = \sqrt{ \dfrac{3a}{2} \bigg( \dfrac{3a - 2a}{2} \bigg)\bigg( \dfrac{3a - 2a}{2} \bigg) \bigg( \dfrac{3a - 2a}{2} \bigg) }$

$\sf \implies A = \sqrt{ \dfrac{3a}{2} \bigg( \dfrac{a}{2} \bigg)\bigg( \dfrac{a}{2} \bigg) \bigg( \dfrac{a}{2} \bigg) }$

$\sf \implies A = \sqrt{ \dfrac{3a \times a \times a \times a }{2 \times 2 \times 2 \times 2} }$

$\sf \implies A = \sqrt{ \dfrac{3 {a}^{4} }{ {2}^{4} } }$

$\sf \implies A = \dfrac{ {a}^{2} }{ {2}^{2} } \sqrt{ 3}$

$\sf \implies A = \dfrac{ {a}^{2} }{ 4 } \sqrt{ 3}$

$\sf \implies A = \dfrac{\sqrt{ 3} \: {a}^{2} }{ 4 }$

Hence proved

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$\bf \underline{ \red {\bf \: Additional \: Information}}$

Heron's formula :-

Heron's formula is used to find out area of triangle if all the sides of triangle are given.

Let's take a triangle whose sides are a, b and c.

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Now first we find semi-perimeter.

Let s be semi-perimeter.

$\sf s = \dfrac{a + b + c}{2}$

And now area (A) =

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