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The height of an equilateral triangle is [tex]$4 \sqrt{3}$[/tex]. What is the perimeter of the equilateral triangle?

A. 12 units
B. [tex]$12 \sqrt{3}$[/tex] units
C. 24 units
D. [tex]$24 \sqrt{3}$[/tex] units


Sagot :

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].
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