Question

the height of an equilateral triangle is (4sqrt{3}). what is the perimeter of the equilateral triangle?
12 units
(12sqrt{3}) units
24 units
(24sqrt{3}) units

Explanation:

Step1: Recall the height formula for an equilateral triangle

For an equilateral triangle with side length \( s \), the height \( h \) is given by \( h=\frac{\sqrt{3}}{2}s \). We know \( h = 4\sqrt{3} \), so we can solve for \( s \).

$$ 4\sqrt{3}=\frac{\sqrt{3}}{2}s $$

Step2: Solve for the side length \( s \)

Multiply both sides of the equation by \( \frac{2}{\sqrt{3}} \) to isolate \( s \):

$$ s = 4\sqrt{3}\times\frac{2}{\sqrt{3}} $$

The \( \sqrt{3} \) terms cancel out, and \( 4\times2 = 8 \)? Wait, no, wait. Wait, \( 4\sqrt{3}\times\frac{2}{\sqrt{3}}=\frac{8\sqrt{3}}{\sqrt{3}} = 8 \)? Wait, that can't be right. Wait, no, wait, the formula for height of equilateral triangle is \( h=\frac{\sqrt{3}}{2}s \), so solving for \( s \): \( s=\frac{2h}{\sqrt{3}} \). Let's substitute \( h = 4\sqrt{3} \):

$$ s=\frac{2\times4\sqrt{3}}{\sqrt{3}} $$

The \( \sqrt{3} \) in the numerator and denominator cancels, and \( 2\times4 = 8 \)? Wait, but then perimeter would be \( 3\times8 = 24 \). Wait, let's check again. Wait, maybe I made a mistake in the formula. Wait, the height of an equilateral triangle: if we split the equilateral triangle into two 30 - 60 - 90 triangles, the height is opposite the 60 - degree angle, the side of the equilateral triangle is the hypotenuse, and half of the side is adjacent to the 60 - degree angle. So in a 30 - 60 - 90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \), where the side opposite 30 is \( \frac{s}{2} \), opposite 60 is \( h \), and hypotenuse is \( s \). So \( h=\frac{\sqrt{3}}{2}s \), so \( s=\frac{2h}{\sqrt{3}} \). Substituting \( h = 4\sqrt{3} \):

$$ s=\frac{2\times4\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{\sqrt{3}} = 8 $$

Wait, but then perimeter is \( 3s=3\times8 = 24 \). So that's correct.

Wait, let's do the algebra again. Starting with \( h=\frac{\sqrt{3}}{2}s \), solve for \( s \):

Multiply both sides by \( \frac{2}{\sqrt{3}} \):

\( s=\frac{2h}{\sqrt{3}} \)

Substitute \( h = 4\sqrt{3} \):

\( s=\frac{2\times4\sqrt{3}}{\sqrt{3}} \)

The \( \sqrt{3} \) cancels, so \( s = 8 \). Then perimeter \( P=3s = 3\times8=24 \).

Answer:

24 units (corresponding to the option "24 units")