A polygon has 54 diagonals. The number of sides in the polygon is:

Options

A. 7
B. 9
C. 12
D. none of these

Answer

To find the number of sides in a polygon given that it has 54 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 in the polygon.

### Step 1: Set up the equation
Given that the polygon has 54 diagonals, we can set up the equation:

\[ \frac{n(n-3)}{2} = 54 \]

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

\[ n(n-3) = 108 \]

### Step 3: Expand the equation
Now, expand the left side of the equation:

\[ n^2 - 3n = 108 \]

### Step 4: Rearrange the equation
Rearrange the equation to set it to zero:

\[ n^2 - 3n - 108 = 0 \]

### Step 5: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to -108 and add to -3. The numbers -12 and 9 work:

\[ (n - 12)(n + 9) = 0 \]

### Step 6: Solve for \( n \)
Now, set each factor equal to zero:

1. \( n - 12 = 0 \) → \( n = 12 \)
2. \( n + 9 = 0 \) → \( n = -9 \) (not a valid solution since the number of sides cannot be negative)

Thus, the only valid solution is:

\[ n = 12 \]

### Conclusion
The polygon has **12 sides**.

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