Question

the area of an equilateral triangle is 16root3 m square .Find the height of the triangle
​

Answer 1

GIVEN :-

• The area of Equilateral ∆ = 16√3 m².

TO FIND :-

• The height of the equilateral ∆.

SOLUTION :-

As we know that the area of equilateral triangle is given by,

$\\ : \implies \displaystyle \sf \: Area_{(Equilateral \: \triangle)} = \frac{ \sqrt{3} }{4} \times (side) ^{2}$

$: \implies \displaystyle \sf \:16 \sqrt{3} = \frac{ \sqrt{3} }{4} \times (side) ^{2}$

$: \implies \displaystyle \sf \:4 \times 16 \sqrt{3} = \sqrt{3} \times (side) ^{2}$

$: \implies \displaystyle \sf \:64 \sqrt{3} = \sqrt{3} \times (side) ^{2}$

$: \implies \displaystyle \sf \:(side) ^{2} = \frac{64 \sqrt{3} }{ \sqrt{3} }$

$: \implies \displaystyle \sf \:(side) ^{2} = 64$

$: \implies \displaystyle \sf \:side = \sqrt{64}$

$: \implies \underline{ \boxed{ \displaystyle \sf \:side = 8 \: cm}}$

Now,

★ Base of equilateral ∆ = Side of equilateral ∆.

Now as we know that,

$\\ : \implies \displaystyle \sf \: Area_{(Equilateral \: \triangle)} = \frac{1}{2} \times base \times height$

$: \implies \displaystyle \sf \:16 \sqrt{3} = \frac{1}{2} \times 8 \times height$

$: \implies \displaystyle \sf \:16 \sqrt{3} = 4 \times height$

$: \implies \underline{ \boxed{ \displaystyle \sf \:height = 4\sqrt{3} \: m. }}$

Answer 2

$\underline{\underline{\sf{\clubsuit \:\:Question}}}$

• The area of equilateral Triangle is 16√3 m² . Find the height of the triangle

$\underline{\underline{\sf{\clubsuit \:\:Given}}}$

• The area of equilateral Triangle = 16√3 m²

$\underline{\underline{\sf{\clubsuit \:\:To\:Find}}}$

• Height of the triangle

$\underline{\underline{\sf{\clubsuit \:\:Answer}}}$

$\boxed{\bigstar\:\:\:\sf{Height\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:4\sqrt{3}\:m}}$

$\underline{\underline{\sf{\clubsuit \:\:Calculations}}}$

We know :

$\boxed{\boxed{\bf{Area\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:\dfrac{\sqrt{3}}{4}\times a^2\:\:sq.units}}}$

Firstly we need to find value of "a" for finding the height of the triangle .

$\bf{Area\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:\dfrac{\sqrt{3}}{4}\times a^2\:\:sq.units}$

$\implies\sf{16\sqrt{3} \:\:m^2\:\:=\:\:\dfrac{\sqrt{3}}{4}\times a^2}$

Divide both sides by √3

$\implies\sf{\dfrac{16\sqrt{3}}{\sqrt{3}} \:\:m^2\:\:=\:\:\dfrac{\sqrt{3}}{4}\times a^2\:\div \:\sqrt{3}}$

$\implies\sf{16\:\:m^2\:\:=\:\:\dfrac{\sqrt{3}}{4}\times a^2\:\times \:\dfrac{1}{\sqrt{3}}}$

$\implies\sf{16\:\:m^2\:\:=\:\:\dfrac{1}{4}\times a^2}$

Multiplying both sides by 4

$\implies\sf{16\:\:m^2\times\:4\:\:=\:\:\dfrac{1}{4}\times a^2\:\times\:4}$

$\implies\sf{64\:\:m^2\:\:=\:\:1\times a^2}$

$\implies\sf{64\:\:m^2\:\:=\:\:a^2}$

$\implies\sf{\sqrt{64\:\:m^2}\:\:=\:\:\sqrt{a^2}}$

$\implies\sf{8\:\:m\:\:=\:\:a\:\:}$

a = 8m

Also :

$\boxed{\boxed{\bf{Height\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:\dfrac{\sqrt{3}}{2}\times a\:\:sq.units}}}$

$\implies\sf{Height\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:\dfrac{\sqrt{3}}{2}\times 8m}$

$\implies\sf{Height\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:\sqrt{3}\times 4m}$

$\boxed{\implies\sf{Height\:\:of\:\:Equilateral\:\:Triangle\:\:=\:\:4\sqrt{3}\:m}}$