Let A be a point inside a regular polygon of 10 sides. Let `p_(1), p_(2)...., p_(10)` be the distances of A from the sides of the polygon. If each side is of length 2 units, then find the value of `p_(1) + p_(2) + ...+ p_(10)`

Explanation

To find the value of \( p_1 + p_2 + \ldots + p_{10} \) where \( p_i \) represents the perpendicular distances from a point \( A \) inside a regular polygon with 10 sides (decagon), we can follow these steps:

### Step 1: Understanding the Geometry
We have a regular decagon (10-sided polygon) with each side of length 2 units. The distances \( p_1, p_2, \ldots, p_{10} \) are the perpendicular distances from point \( A \) to each of the sides of the decagon.

### Step 2: Area Calculation Using Triangles
We can divide the decagon into 10 triangles by drawing lines from point \( A \) to each vertex of the decagon. Each triangle will have a base equal to the length of the side of the decagon (which is 2 units) and a height equal to the perpendicular distance from point \( A \) to that side.

### Step 3: Area of Each Triangle
The area of each triangle formed by point \( A \) and two consecutive vertices \( A_i \) and \( A_{i+1} \) of the decagon can be expressed as:
\[ \text{Area of triangle } A A_i A_{i+1} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times p_i = p_i \]

### Step 4: Total Area of the Decagon
The total area of the decagon can be calculated by summing the areas of all 10 triangles:
\[ \text{Total Area} = \sum_{i=1}^{10} \text{Area of triangle } A A_i A_{i+1} = p_1 + p_2 + \ldots + p_{10} \]

### Step 5: Area of the Regular Decagon
The area \( A \) of a regular polygon can be calculated using the formula:
\[ \text{Area} = \frac{n \cdot a^2}{4 \cdot \tan\left(\frac{\pi}{n}\right)} \]
where \( n \) is the number of sides and \( a \) is the length of each side. For our decagon:
- \( n = 10 \)
- \( a = 2 \)

Substituting these values into the formula:
\[ \text{Area} = \frac{10 \cdot 2^2}{4 \cdot \tan\left(\frac{\pi}{10}\right)} = \frac{10 \cdot 4}{4 \cdot \tan\left(\frac{\pi}{10}\right)} = \frac{10}{\tan\left(\frac{\pi}{10}\right)} \]

### Step 6: Relating Area to Distances
From the earlier steps, we established that:
\[ p_1 + p_2 + \ldots + p_{10} = \text{Area of the decagon} \]
Thus, we have:
\[ p_1 + p_2 + \ldots + p_{10} = \frac{10}{\tan\left(\frac{\pi}{10}\right)} \]

### Conclusion
The value of \( p_1 + p_2 + \ldots + p_{10} \) is equal to the area of the decagon, which can be calculated using the area formula for a regular polygon.