A polygon has 44 diagonals , then the number of its sides is

Options

A. 11
B. 7
C. 8
D. None of these

Answer

To find the number of sides of a polygon given that it has 44 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.

### Step-by-Step Solution:

1. **Set up the equation using the formula for diagonals**:
Given that the number of diagonals \(D = 44\), we can set up the equation:
\[ \frac{n(n-3)}{2} = 44 \]

2. **Multiply both sides by 2 to eliminate the fraction**:
\[ n(n-3) = 88 \]

3. **Rearrange the equation**:
\[ n^2 - 3n - 88 = 0 \]

4. **Use the quadratic formula to solve for \(n\)**:
The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -3\), and \(c = -88\).

5. **Calculate the discriminant**:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-88) = 9 + 352 = 361 \]

6. **Substitute back into the quadratic formula**:
\[ n = \frac{3 \pm \sqrt{361}}{2} \]

7. **Calculate the square root of 361**:
\[ \sqrt{361} = 19 \]

8. **Substitute the value back**:
\[ n = \frac{3 \pm 19}{2} \]

9. **Calculate the two possible values for \(n\)**:
- \(n = \frac{3 + 19}{2} = \frac{22}{2} = 11\)
- \(n = \frac{3 - 19}{2} = \frac{-16}{2} = -8\) (not valid as \(n\) must be positive)

10. **Conclusion**:
The number of sides of the polygon is \(n = 11\).

### Final Answer:
The number of sides of the polygon is **11**.