The perimeter of the equatorial triangle is 24 units
Given that,
An equilateral triangle has an height equal to [tex]4 \sqrt{3}[/tex]
The triangle is shown below
From Triangle ABC in the shown figure AD [tex]=4 \sqrt{3}[/tex]
Let the sides of the equilateral triangle be ‘a’
AB = BC = a
Since, it is an equilateral triangle we get,
BD = DC = a ÷ 2
Now, using Pythagoras Theorem in Triangle ABD,
The Pythagorean theorem is this: In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
[tex]\mathrm{AB}^{2}=\mathrm{BD}^{2}+\mathrm{AD}^{2}[/tex]
[tex]\begin{array}{l}{a^{2}=\left(\frac{a}{2}\right)^{2}+(4 \sqrt{3})^{2}} \\\\ {a^{2}-\left(\frac{a}{2}\right)^{2}=(4 \sqrt{3})^{2}}\end{array}[/tex]
[tex]\frac{4 a^{2}-a^{2}}{4}=16 \times 3[/tex]
[tex]\begin{array}{l}{\frac{3 a^{2}}{4}=16 \times 3} \\\\ {3 a^{2}=192} \\\\ {a^{2}=192 \div 3=64}\end{array}[/tex]
a = 8
Hence, the three sides of the triangle are 8 units each
In equilateral traingle, length of all three sides of triangle are equal
So, Perimeter = 3 [tex]\times[/tex] (Length of each side of triangle)
Perimeter = 3 [tex]\times[/tex] 8 = 24
Thus the perimeter of the equatorial triangle is 24 units
