The length of a median of an equilateral triangle is `12sqrt(3)` cms. Then the area of the triangle is :

Options

A. 144 sq. cm.
B. `288 sqrt(3)` sq. cm.
C. `144 sqrt(3)` sq. cm
D. 288 sq. cm.

Answer

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

### Step-by-Step Solution:

1. **Understanding the Median of an Equilateral Triangle**:
The length of the median (m) of an equilateral triangle can be expressed in terms of the side length (a) as:
\[ m = \frac{\sqrt{3}}{2} a \]

2. **Setting Up the Equation**:
We are given that the length of the median is \( 12\sqrt{3} \) cm. Therefore, we can set up the equation:
\[ 12\sqrt{3} = \frac{\sqrt{3}}{2} a \]

3. **Solving for Side Length (a)**:
To isolate \( a \), we can first eliminate \( \sqrt{3} \) from both sides:
\[ 12 = \frac{1}{2} a \]
Now, multiply both sides by 2:
\[ a = 24 \text{ cm} \]

4. **Calculating the Area of the Triangle**:
The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 24 \) cm into the formula:
\[ A = \frac{\sqrt{3}}{4} \times (24)^2 \]

5. **Calculating \( (24)^2 \)**:
\[ (24)^2 = 576 \]
So, substituting this back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times 576 \]

6. **Dividing by 4**:
\[ A = 144\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the triangle is \( 144\sqrt{3} \) cm².

---