The length of the perpendicular 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. (a) `(2)/(sqrt(3))(p_(1) + p_(2) + p_(3))`
B. (b) `(1)/(3)(p_(1) + p_(2) + p_(3))`
C. (c) `(1)/(sqrt(3))(p_(1) + p_(2) + p_(3))`
D. (d) `(4)/(sqrt(3))(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 an interior point to the respective sides, we can follow these steps:

### Step 1: Understand the relationship between the area and the perpendiculars
The area of an equilateral triangle can be expressed in two ways:
1. Using the side length \( a \):
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
2. Using the perpendicular distances \( p_1, p_2, p_3 \) from an interior point to the sides:
\[ \text{Area} = \frac{1}{2} \times a \times (p_1 + p_2 + p_3) \]

### Step 2: Set the two area expressions equal
Since both expressions represent the area of the same triangle, we can set them equal to each other:
\[ \frac{\sqrt{3}}{4} a^2 = \frac{1}{2} \times a \times (p_1 + p_2 + p_3) \]

### Step 3: Simplify the equation
To eliminate \( a \) from the right side, we can multiply both sides by 4:
\[ \sqrt{3} a^2 = 2a (p_1 + p_2 + p_3) \]

### Step 4: Divide both sides by \( a \) (assuming \( a \neq 0 \))
\[ \sqrt{3} a = 2 (p_1 + p_2 + p_3) \]

### Step 5: Solve for \( a \)
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{2 (p_1 + p_2 + p_3)}{\sqrt{3}} \]

### Final Answer
Thus, the length of each side of the equilateral triangle is:
\[ a = \frac{2 (p_1 + p_2 + p_3)}{\sqrt{3}} \]