Question

Find the side , height and perimeter of an equilateral triangle whose area is 9√12

Answer 1

please find the attachment

Answer 2

$\large{\underline{\underline{\mathrm{\blue{APPROPRIATE \: QUESTION :-}}}}}$

Find the side , height and perimeter of an equilateral triangle whose area is 9√12 cm²

$\huge{\underline{\underline{\mathrm{\blue{GIVEN:-}}}}}$

• Area of an equilateral triangle is 9√12 cm²

$\huge{\underline{\underline{\mathrm{\blue{NEED\:TO\:FIND:-}}}}}$

Find the side, height and perimeter of Triangle = ?

$\huge{\underline{\underline{\mathrm{\blue{FORMULA \:USED:-}}}}}$

$\large{\boxed{\bf{\red{Area \:of\:Equilateral\:Triangle = \frac{\sqrt{3}}{4}\times{(Side)}^{2}}}}}$

$\large{\boxed{\bf{\red{Height\:of\:Equilateral\:Triangle = \frac{\sqrt{3}}{2}\times Side}}}}$

$\large{\boxed{\bf{\red{Perimeter \:of\:Equilateral\:Triangle = 3\times Side}}}}$

$\huge{\underline{\underline{\mathrm{\blue{SOLUTION:-}}}}}$

We have given that, the area of an equilateral triangle is 9√12 cm²

We need to find the side of an equilateral triangle

Putting the values in the formula

$\large\bf{Area\: of \: Equilateral \:Triangle = \frac{\sqrt{3}}{4}\times{(Side)}^{2}}$

$\implies$$\rm{9\sqrt{12} =\frac{\sqrt{3}}{4}\times{(Side)}^{2}}$

$\implies$$\rm{9\sqrt{2\times2\times3} =\frac{\sqrt{3}}{4}\times{(Side)}^{2}}$

$\implies$$\rm{9\times2\sqrt{3} =\frac{\sqrt{3}}{4}\times{(Side)}^{2}}$

$\implies$$\rm{18\sqrt{3} =\frac{\sqrt{3}}{4}\times{(Side)}^{2}}$

$\implies$$\rm{{(Side)}^{2} =\large\frac{18\sqrt{3}\times 4}{\sqrt{3}}}$

$\implies$$\rm{{(Side)}^{2} = 72\times\cancel\dfrac{\sqrt{3}}{\sqrt {3}}}$

$\implies$$\rm{{(Side)}^{2} = 72}$

$\implies$$\rm{Side = \sqrt{72}}$

$\implies$$\large\rm\green{Side =6 \sqrt{2}\: cm}$

Thus, the side of an equilateral triangle is 6√2 cm

∴ Now, we have to find the height of an equilateral triangle

Putting the values in the formula

$\large\bf{Height\: of \: Equilateral \:Triangle = \frac{\sqrt{3}}{2}\times Side}$

$\implies$$\rm{ Height = \frac{\sqrt{3}}{2}\times 6\sqrt{2}}$

$\implies$$\rm{ Height =\cancel\dfrac{6\sqrt{6}}{2}}$

$\implies$$\large\rm\green{ Height = 3\sqrt{6} \: cm}$

Thus, the height of an equilateral triangle is 3√6 cm

∴ Now, we have to find the Perimeter of an equilateral triangle

Putting the values in the formula

• Since, all sides of an equilateral triangle are equal in length, So

$\large\bf{Perimeter \: of \: Equilateral \:Triangle = 3\times Side}$

$\implies$$\rm{ Perimeter = 3\times 6\sqrt {2}}$

$\implies$$\large\rm\green{ Perimeter = 18\sqrt {2}\: cm}$

Thus, the perimeter of an equilateral triangle is 18√2 cm

$\large{\underline{\underline{\mathrm{\blue{FINAL \: ANSWER :-}}}}}$

$\large{\boxed{\boxed{\bf{\red{Perimeter = 18\sqrt{2}\: cm}}}}}$

$\large{\boxed{\boxed{\bf{\red{Side = 6\sqrt{2}\: cm}}}}}$

$\large{\boxed{\boxed{\bf{\red{Height = 3\sqrt6\: cm}}}}}$