If a polygon has 20 diagonals, then what is the number of sides ?

Options

A. 6
B. 10
C. 12
D. 8

Explanation

To find the number of sides of a polygon given that it has 20 diagonals, we can use the formula for the number of diagonals in a polygon:

\[ D = \frac{n(n-3)}{2} \]

where \(D\) is the number of diagonals and \(n\) is the number of sides of the polygon.

### Step 1: Set up the equation
Given that the number of diagonals \(D\) is 20, we can substitute this into the formula:

\[ 20 = \frac{n(n-3)}{2} \]

### Step 2: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 2:

\[ 40 = n(n-3) \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ n(n-3) - 40 = 0 \]

This simplifies to:

\[ n^2 - 3n - 40 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to -40 and add to -3. The numbers -8 and 5 work:

\[ (n - 8)(n + 5) = 0 \]

### Step 5: Solve for \(n\)
Setting each factor equal to zero gives us:

1. \(n - 8 = 0 \Rightarrow n = 8\)
2. \(n + 5 = 0 \Rightarrow n = -5\)

Since the number of sides cannot be negative, we discard \(n = -5\).

### Conclusion
Thus, the number of sides of the polygon is:

\[ \boxed{8} \]