To determine the perimeter of an equilateral triangle given its height, we can follow these steps:
1. Identify the properties of an equilateral triangle:
- All sides are equal.
- The height can be calculated using the formula for the height [tex]\( h \)[/tex] of an equilateral triangle with side length [tex]\( s \)[/tex]:
[tex]\[
h = \frac{\sqrt{3}}{2} s
\][/tex]
2. Given height:
- The height of the equilateral triangle is given as [tex]\( 4 \sqrt{3} \)[/tex].
3. Find the side length:
- Use the height formula to solve for the side length [tex]\( s \)[/tex]:
[tex]\[
4 \sqrt{3} = \frac{\sqrt{3}}{2} s
\][/tex]
- Isolate [tex]\( s \)[/tex] by multiplying both sides by 2:
[tex]\[
8 \sqrt{3} = \sqrt{3} s
\][/tex]
- Divide both sides by [tex]\( \sqrt{3} \)[/tex]:
[tex]\[
s = 8
\][/tex]
4. Calculate the perimeter:
- The perimeter [tex]\( P \)[/tex] of an equilateral triangle is three times the side length:
[tex]\[
P = 3s
\][/tex]
- Substitute the calculated side length:
[tex]\[
P = 3 \times 8 = 24
\][/tex]
So, the perimeter of the equilateral triangle is [tex]\(\boxed{24 \text{ units}}\)[/tex].
