From a point in the interior of an equilateral triangle, the perpendicular distance of the sides are `sqrt(3) cm, 2sqrt(3) cm` and `5sqrt(3) cm`. The perimeter (in cm) of the triangle is

Options

A. 64
B. 32
C. 48
D. 24

Answer

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

### Step-by-Step Solution:

1. **Identify the Perpendicular Distances**:
Let the perpendicular distances from the point to the sides of the triangle be:
- \( p_1 = \sqrt{3} \, \text{cm} \)
- \( p_2 = 2\sqrt{3} \, \text{cm} \)
- \( p_3 = 5\sqrt{3} \, \text{cm} \)

2. **Sum the Perpendicular Distances**:
We need to find the sum of these perpendicular distances:
\[ p_1 + p_2 + p_3 = \sqrt{3} + 2\sqrt{3} + 5\sqrt{3} \]
Combine the terms:
\[ p_1 + p_2 + p_3 = (1 + 2 + 5)\sqrt{3} = 8\sqrt{3} \, \text{cm} \]

3. **Calculate the Side Length of the Triangle**:
The formula for the side length \( s \) of an equilateral triangle in terms of the sum of the perpendicular distances is:
\[ s = \frac{p_1 + p_2 + p_3}{\frac{\sqrt{3}}{2}} = \frac{8\sqrt{3}}{\frac{\sqrt{3}}{2}} = 8 \cdot 2 = 16 \, \text{cm} \]

4. **Calculate the Perimeter of the Triangle**:
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3 \times s = 3 \times 16 = 48 \, \text{cm} \]

### Final Answer:
The perimeter of the triangle is \( 48 \, \text{cm} \).

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