In an equilateral triangle, there is a point inside the triangle whose perpendicular distance from each side is `3sqrt3 , 4 sqrt3 & 5sqrt 3` respectively . Find the area of triangle .

Answer

To find the area of the equilateral triangle given the perpendicular distances from a point inside the triangle to each of its sides, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have an equilateral triangle \( ABC \).
- The perpendicular distances from a point inside the triangle to the sides \( BC, AC, \) and \( AB \) are given as \( h_1 = 3\sqrt{3}, h_2 = 4\sqrt{3}, h_3 = 5\sqrt{3} \).

2. **Use the Area Formula for a Triangle**:
- The area \( A \) of triangle \( ABC \) can be expressed in terms of the side length \( a \) and the heights from the point to the sides:
\[ A = \frac{1}{2} \times a \times h_1 + \frac{1}{2} \times a \times h_2 + \frac{1}{2} \times a \times h_3 \]
- This simplifies to:
\[ A = \frac{1}{2} a (h_1 + h_2 + h_3) \]

3. **Calculate the Sum of Heights**:
- Substitute the values of \( h_1, h_2, \) and \( h_3 \):
\[ h_1 + h_2 + h_3 = 3\sqrt{3} + 4\sqrt{3} + 5\sqrt{3} = (3 + 4 + 5)\sqrt{3} = 12\sqrt{3} \]

4. **Relate the Area to the Side Length**:
- The area of an equilateral triangle can also be expressed as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- Setting the two area expressions equal gives:
\[ \frac{1}{2} a (12\sqrt{3}) = \frac{\sqrt{3}}{4} a^2 \]

5. **Simplify the Equation**:
- Multiply both sides by 4 to eliminate the fraction:
\[ 2a(12\sqrt{3}) = \sqrt{3} a^2 \]
- This simplifies to:
\[ 24a\sqrt{3} = \sqrt{3} a^2 \]

6. **Divide by \( \sqrt{3} \)** (assuming \( a \neq 0 \)):
\[ 24a = a^2 \]

7. **Rearrange the Equation**:
- Rearranging gives:
\[ a^2 - 24a = 0 \]
- Factor out \( a \):
\[ a(a - 24) = 0 \]
- Thus, \( a = 0 \) or \( a = 24 \). Since \( a \) represents the side length of the triangle, we take \( a = 24 \).

8. **Calculate the Area**:
- Now substitute \( a = 24 \) back into the area formula:
\[ A = \frac{\sqrt{3}}{4} (24^2) = \frac{\sqrt{3}}{4} \times 576 = 144\sqrt{3} \]

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