An equilateral triangle has a height measuring $12\sqrt{3}$ cm. Determine the area and perimeter of the triangle.

The correct option is Area = $144\sqrt{3}$ cm², Perimeter = 72 cm.

To find the area and perimeter of an equilateral triangle, we first need to determine its side length, denoted as $a$.

In an equilateral triangle, all three sides are equal, and the height ($h$) divides the triangle into two congruent right-angled triangles.

The relationship between the height $h$ and side length $a$ of an equilateral triangle is given by the formula:

$h=\frac{\sqrt{3}}{2}a$

We are given that the height is $12\sqrt{3}$ cm. Substituting this value into the height formula:

$12\sqrt{3}=\frac{\sqrt{3}}{2}a$

To solve for $a$, divide both sides of the equation by $\sqrt{3}$:

$12=\frac{1}{2}a$

Multiply both sides by 2:

$a=24$ cm

Now that we have found the side length $a=24$ cm, we can calculate both the perimeter and the area of the triangle.

Step 1: Calculate the Perimeter
The perimeter of an equilateral triangle is the sum of its three equal sides:

$\text{Perimeter}=3\times a$

$\text{Perimeter}=3\times 24=72$ cm

Step 2: Calculate the Area
The formula for the area ($A$) of an equilateral triangle is:

$A=\frac{\sqrt{3}}{4}{a}^2$

Substitute $a=24$ cm into the area formula:

$A=\frac{\sqrt{3}}{4}\times {24}^2$

$A=\frac{\sqrt{3}}{4}\times 576$

$A=144\sqrt{3}$ cm²

Therefore, the area of the triangle is $144\sqrt{3}$ cm² and the perimeter is 72 cm.