Respuesta :
Correct answer is [tex]36\sqrt{3}[/tex]
The radius of an equilateral triangle is equal to 2/3 of the height.
[tex]R= \frac{2}{3} h= \frac{2}{3} \frac{a \sqrt{3} }{2} = \frac{a \sqrt{3} }{3} \\4 \sqrt{3}= \frac{a \sqrt{3} }{3} \\3\times 4\sqrt{3}=a \sqrt{3} \\a=12 \\ \\A= \frac{a^2 \sqrt{3} }{4} =\frac{12^2 \sqrt{3} }{4} =\frac{144 \sqrt{3} }{4} =36 \sqrt{3} [/tex]
The radius of an equilateral triangle is equal to 2/3 of the height.
[tex]R= \frac{2}{3} h= \frac{2}{3} \frac{a \sqrt{3} }{2} = \frac{a \sqrt{3} }{3} \\4 \sqrt{3}= \frac{a \sqrt{3} }{3} \\3\times 4\sqrt{3}=a \sqrt{3} \\a=12 \\ \\A= \frac{a^2 \sqrt{3} }{4} =\frac{12^2 \sqrt{3} }{4} =\frac{144 \sqrt{3} }{4} =36 \sqrt{3} [/tex]
Answer: The area of the equilateral triangle is 36√3 square meters.
Step-by-step explanation: We are to find the area of an equilateral triangle having the length of the radius equal to [tex]4\sqrt 3~\textup{meter}.[/tex]
We know that the area of an equilateral triangle having length of each side equal to 'a' units is given by
[tex]A=\dfrac{\sqrt3}{4}a^2~\textup{sq. units.}[/tex]
Now, the radius of an equilateral triangle is equal to two-third of the height of the trinagle.
So, if 'r' is the radius and 'h' is the height of the triangle, then we have
[tex]r=\dfrac{2}{3}h~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
And, the height of the equilateral triangle with side length 'a' units is given by
[tex]h=\dfrac{\sqrt3}{2}a.[/tex]
So, from equation (i), we have
[tex]r=\dfrac{2}{3}\times \dfrac{\sqrt3}{2}a\\\\\Rightarrow 4\sqrt3=\dfrac{a}{\sqrt3}\\\\\Rightarrow a=12.[/tex]
That is, side length, a = 12 meter.
Therefore, the area of the equilateral triangle is given by
[tex]A\\\\\\=\dfrac{\sqrt3}{4}a^2\\\\\\=\dfrac{\sqrt 3}{4}(12)^2\\\\\\=36\sqrt3~\textup{sq. meters.}[/tex]
Thus, the area of the equilateral triangle is 36√3 square meters.
