If the altitude of an equilateral triangle is `12sqrt(3) cm`, then its area would be :

Options

A. `12 cm^(2)`
B. `144 sqrt(3) cm^(2)`
C. `72 cm^(2)`
D. `36 sqrt(3) cm^(2)`

Answer

To find the area of an equilateral triangle given its altitude, we can follow these steps:

### Step 1: Understand the relationship between the altitude and the side length of the triangle.
The altitude (h) of an equilateral triangle can be expressed in terms of its side length (s) using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]

### Step 2: Substitute the given altitude into the formula.
We know that the altitude \( h = 12\sqrt{3} \) cm. So, we can set up the equation:
\[ 12\sqrt{3} = \frac{\sqrt{3}}{2} s \]

### Step 3: Solve for the side length (s).
To isolate \( s \), we can multiply both sides of the equation by 2:
\[ 2 \times 12\sqrt{3} = \sqrt{3} s \]
\[ 24\sqrt{3} = \sqrt{3} s \]

Now, divide both sides by \( \sqrt{3} \):
\[ s = 24 \text{ cm} \]

### Step 4: Calculate the area of the equilateral triangle.
The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]

Substituting \( s = 24 \) cm into the area formula:
\[ A = \frac{\sqrt{3}}{4} (24)^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 576 \]
\[ A = 144\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the equilateral triangle is \( 144\sqrt{3} \text{ cm}^2 \).

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