From a point in the interior of an equilateral triangle, the length of the perpendiculars to the three sides are 6 cm, 8 cm and 10 cm respectively. The area of the triangle is

Options

A. `48 cm^(2)`
B. `16sqrt(3) cm^(2)`
C. `192sqrt(3) cm^(2)`
D. `192 cm^(2)`

Answer

To find the area of the equilateral triangle given the lengths of the perpendiculars from a point inside the triangle to its sides, we can use the following steps:

### Step-by-Step Solution:

1. **Identify the lengths of the perpendiculars**:
Let the lengths of the perpendiculars from the point to the three sides of the equilateral triangle be \( h_1 = 6 \, \text{cm} \), \( h_2 = 8 \, \text{cm} \), and \( h_3 = 10 \, \text{cm} \).

2. **Use the formula for the side of the triangle**:
The formula for the side \( a \) of an equilateral triangle in terms of the perpendiculars is given by:
\[ a = \frac{2}{\sqrt{3}} (h_1 + h_2 + h_3) \]

3. **Calculate the sum of the perpendiculars**:
\[ h_1 + h_2 + h_3 = 6 + 8 + 10 = 24 \, \text{cm} \]

4. **Substitute the sum into the formula**:
\[ a = \frac{2}{\sqrt{3}} \times 24 = \frac{48}{\sqrt{3}} \, \text{cm} \]

5. **Rationalize the denominator**:
\[ a = \frac{48 \sqrt{3}}{3} = 16 \sqrt{3} \, \text{cm} \]

6. **Calculate the area of the equilateral triangle**:
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substitute \( a = 16 \sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (16 \sqrt{3})^2 \]

7. **Calculate \( (16 \sqrt{3})^2 \)**:
\[ (16 \sqrt{3})^2 = 256 \times 3 = 768 \]

8. **Substitute back to find the area**:
\[ A = \frac{\sqrt{3}}{4} \times 768 = \frac{768 \sqrt{3}}{4} = 192 \sqrt{3} \, \text{cm}^2 \]

### Final Answer:
The area of the equilateral triangle is \( 192 \sqrt{3} \, \text{cm}^2 \).

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