The number of diagonals of a polygon of n sides is equal to:

Question

Accepted Answer

To determine the number of diagonals in a polygon with $ n $ sides, we use the formula $\frac{n(n-3)}{2}$. Let's break down why this formula is correct and why the other options are incorrect.

### Explanation for Option 1: $\frac{n(n-3)}{2}$

1. **Understanding Diagonals**: A diagonal is a line segment connecting two non-adjacent vertices in a polygon. In a polygon with $ n $ sides, each vertex can connect to $ n-3 $ other vertices to form a diagonal (since it cannot connect to itself or its two adjacent vertices).

2. **Total Connections**: Initially, if we consider all possible connections (including sides), each vertex connects to $ n-1 $ other vertices. This gives us a total of $\frac{n(n-1)}{2}$ connections (using the combination formula for choosing 2 vertices out of $ n $).

3. **Subtracting Sides**: Since the polygon has $ n $ sides, we subtract these $ n $ sides from the total connections to get the number of diagonals. Thus, the formula becomes:
   $$
   \frac{n(n-1)}{2} - n = \frac{n(n-1) - 2n}{2} = \frac{n(n-3)}{2}
   $$
   Therefore, Option 1 is the correct formula for calculating the number of diagonals in a polygon.

### Explanation for Incorrect Options

- **Option 2: $\frac{n(n-1)}{2}$**

This formula calculates the total number of line segments that can be formed by connecting any two vertices in the polygon, including the sides. It does not account for the fact that sides are not diagonals. Therefore, it overcounts the number of diagonals by including the sides.

- **Option 3: $\frac{n(n-2)}{2}$**

This formula does not correctly account for the exclusion of both the vertex itself and its adjacent vertices when calculating diagonals. It incorrectly assumes each vertex connects to $ n-2 $ other vertices, which would include one adjacent vertex, thus overcounting the diagonals.

- **Option 4: $\frac{n}{2}$**

This formula is not related to the calculation of diagonals in a polygon. It does not consider the combinatorial nature of connecting vertices and is not derived from any logical reasoning related to polygon diagonals.

In conclusion, the correct formula for the number of diagonals in a polygon with $ n $ sides is $\frac{n(n-3)}{2}$, as given in Option 1. This formula accurately accounts for the exclusion of sides and adjacent vertices, providing the correct count of diagonals.