The lengths of the perpendiculars drawn from any point in the interior of an equilateral triangle to the respective sides are `P_1`. `P_2` and `P_3` . The length of each side of the triangle is

Options

A. `(2)/(sqrt3) (p_1 + p_2 + p_3)`
B. `1/3 (p_1 + p_2 + p_3)`
C. `(1)/(sqrt3) (p_1 + p_2 + p_3)`
D. `(4)/(sqrt3) (p_1 + p_2 + p_3)`

Answer

To find the length of each side of an equilateral triangle given the lengths of the perpendiculars drawn from any point in the interior of the triangle to the respective sides, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have an equilateral triangle ABC with side length 'a'. From a point O inside the triangle, perpendiculars are drawn to the sides BC, CA, and AB, which have lengths \( P_1, P_2, \) and \( P_3 \) respectively.

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

3. **Area using Perpendiculars**: The area can also be expressed as the sum of the areas of the three smaller triangles formed by the point O and the sides of the triangle:
- Area of triangle OBC = \( \frac{1}{2} \times BC \times P_1 = \frac{1}{2} \times a \times P_1 \)
- Area of triangle OCA = \( \frac{1}{2} \times CA \times P_2 = \frac{1}{2} \times a \times P_2 \)
- Area of triangle OAB = \( \frac{1}{2} \times AB \times P_3 = \frac{1}{2} \times a \times P_3 \)

Therefore, the total area using the perpendiculars is:
\[ A = \frac{1}{2} a P_1 + \frac{1}{2} a P_2 + \frac{1}{2} a P_3 \]
Simplifying this gives:
\[ A = \frac{1}{2} a (P_1 + P_2 + P_3) \]

4. **Equating the Two Area Expressions**: Since both expressions represent the area of triangle ABC, we can set them equal to each other:
\[ \frac{\sqrt{3}}{4} a^2 = \frac{1}{2} a (P_1 + P_2 + P_3) \]

5. **Solving for 'a'**: To isolate 'a', we can multiply both sides by 4 to eliminate the fraction:
\[ \sqrt{3} a^2 = 2 a (P_1 + P_2 + P_3) \]
Next, divide both sides by 'a' (assuming \( a \neq 0 \)):
\[ \sqrt{3} a = 2 (P_1 + P_2 + P_3) \]
Finally, solve for 'a':
\[ a = \frac{2 (P_1 + P_2 + P_3)}{\sqrt{3}} \]

6. **Final Result**: Thus, the length of each side of the equilateral triangle is:
\[ a = \frac{2}{\sqrt{3}} (P_1 + P_2 + P_3) \]