If the altitude of an equilateral triangle is ` 2 sqrt(3)` , then its area is :

Options

A. `4 sqrt(3) cm^(2) `
B. `12 sqrt(3) cm^(2) `
C. `(8)/(sqrt(3)) cm^(2) `
D. none of these

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 of the equilateral triangle.
The altitude (h) of an equilateral triangle can be expressed in terms of its side length (s) using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]
Given that the altitude \( h = 2\sqrt{3} \), we can set up the equation:
\[ 2\sqrt{3} = \frac{\sqrt{3}}{2} s \]

### Step 2: Solve for the side length (s).
To isolate \( s \), we can multiply both sides of the equation by 2:
\[ 4\sqrt{3} = \sqrt{3} s \]
Now, divide both sides by \( \sqrt{3} \):
\[ s = 4 \]

### Step 3: Use the side length to find 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 \( s = 4 \) into the formula:
\[ A = \frac{\sqrt{3}}{4} \times 4^2 \]
Calculating \( 4^2 \):
\[ A = \frac{\sqrt{3}}{4} \times 16 \]
Now, simplify:
\[ A = 4\sqrt{3} \]

### Final Answer:
The area of the equilateral triangle is \( 4\sqrt{3} \, \text{cm}^2 \).

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