A polygon has `27` diagonals . How many sides it have ?

Options

A. `7`
B. `8`
C. `9`
D. `12`

Answer

To find the number of sides of a polygon that has 27 diagonals, we can use the formula for the number of diagonals in a polygon, which is given by:

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

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

### Step-by-Step Solution:

1. **Set up the equation**: Since we know that the polygon has 27 diagonals, we can set up the equation:
\[ 27 = \frac{n(n - 3)}{2} \]

2. **Multiply both sides by 2**: To eliminate the fraction, multiply both sides of the equation by 2:
\[ 54 = n(n - 3) \]

3. **Rearrange the equation**: Rearranging gives us:
\[ n(n - 3) - 54 = 0 \]
or
\[ n^2 - 3n - 54 = 0 \]

4. **Factor the quadratic equation**: We need to find two numbers that multiply to -54 and add up to -3. The numbers -9 and 6 work because:
\[ -9 \times 6 = -54 \quad \text{and} \quad -9 + 6 = -3 \]
Thus, we can factor the equation as:
\[ (n - 9)(n + 6) = 0 \]

5. **Solve for \( n \)**: Setting each factor to zero gives us:
\[ n - 9 = 0 \quad \Rightarrow \quad n = 9 \]
\[ n + 6 = 0 \quad \Rightarrow \quad n = -6 \]

6. **Select the valid solution**: Since the number of sides cannot be negative, we discard \( n = -6 \). Therefore, the number of sides \( n \) is:
\[ n = 9 \]

### Final Answer:
The polygon has **9 sides**.

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