Question

prove that area of an equilateral triangle is equal to root 3/4 a square, where a is the side of the triangle

Answer 1

Solution :

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Derivation of Area of an equilateral triangle ;

Let ABC be an equilateral triangle with sides 'a'. Now, draw AD perpendicular to BC.

Here, we have ΔABD = ΔADC.

We will find area of ΔABD using pythagorean theorem, according to which, the square of hypotenuse is equal to the sum of the squares of the other two sides.

Here, we have ;

$\sf a {}^{2} = h {}^{2} + (\frac{a}{2} ) {}^{2} \\ \\ \sf h {}^{2} = a {}^{2} - \frac{a {}^{2} }{4} \\ \\ \sf h {}^{2} = \frac{3a {}^{2} }{4} \\ \\ \sf h = \frac{ \sqrt{3} }{2} a$
Now, we get the height ;

$\sf area \: of \: \Delta = \frac{1}{2} \times base \times height \\ \\ \sf area \: of \: \Delta = \frac{1}{2} \times a \times \frac{ \sqrt{3} }{2} a \\ \\ \sf area \: of \: \Delta = \frac{ \sqrt{3} }{4} a {}^{2}$

Hence, area of equilateral triangle is

$\sf area \: of \: \Delta = \frac{ \sqrt{3} }{4} a {}^{2}$

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Thanks for the question !

Answer 2

$\underline{\textbf{Hey mate Here is your answer }}$

$\underline{\textbf{To Prove }}$

$\boxed{\mathbf{Area\: Of\: an\: Equilateral \:Triangle \:= \:\frac{ \sqrt{3} }{4} {side}^{2} \:}}$

$\underline{\textbf{Proof ---}}$

Now let

$\textbf{∆ABC Be An Equilateral Triangle}$

$\textbf{And Measure Of Side = a }$

Now By $\textbf{Heron's Formula}$ That is

$\boxed{\mathbf{Area\:Of\; Triangle\:= \: \sqrt{s(s\: -\: a)(s\: - \:b)(s \:- \:c)} }}$

Therefore By This We Have

$\mathbf{ar(\triangle\:ABC\:)\:=\: \sqrt{s(s \:- \:a)(s\: -\: b)(s \:- \:c)} \:}$ ------ EQ ( 1 )

Where

S = Semi Perimeter And Sides = a , b , c

Therefore

$\mathbf{S \:= \:\frac{a+b+c}{2}}$

Since ∆ABC Is an Equilateral Triangle with a As side , So

$\mathbf{S \:= \:\frac{3a}{2}}$

Now Putting this value In EQ 1 We Have ➡️

$\mathbf{ar(\triangle\:ABC\:)\:=\: \sqrt{\frac{3a}{2}( \frac{3a}{2}\:- \:a)(\frac{3a}{2}\: -\: a)(\frac{3a}{2}\:- \:a)}}$

Now By Taking LCM And solving the Bracket We Have ➡️

$\mathbf{ar(\triangle\:ABC\:)\:=\: \sqrt{\frac{3a}{2}( \frac{a}{2}\:)(\frac{a}{2}\:)(\frac{a}{2}\:)}}$

That Is ➡️

$\mathbf{ar(\triangle\:ABC\:)\:=\: \sqrt{3 \times {( \frac{a}{2}) }^{2} {( \frac{a}{2}) }^{2} }}$

That Is Now Taking Squares Out We Have ➡️

$\mathbf{ar(\triangle\:ABC\:)\:=\: \frac{a}{2} \times \frac{a}{2} \: \sqrt{3 }}$

That Is

$\mathbf{ar(\triangle\:ABC\:)\:=\: \frac{ {a}^{2} }{4} \: \sqrt{3 }}$

Therefore We Have

$\boxed{\mathbf{ar(\triangle\: ABC\:)\: = \:\frac{ \sqrt{3} }{4} {a}^{2} }}$

$\underline{\mathbf{Hence\: Proved \: That }}$

$\boxed{\mathbf{ Area\: Of \: An\: Equilateral\:Triangle \:Is \:=\:\frac{ \sqrt{3} }{4} {a}^{2} }}$

$\large{\blue{Thanks...}}$

$\underline{\bold{\star\:\: BrainlyKing5\:\:\star}}$