Question

A polygon has 44 diagonals. Then the number of sides of the polygon are 
 

• A. 11
• B. 12
• C. 10
• D. 13

Hint:

Instead of setting up and factoring the full quadratic expression, plug the given multiple-choice numbers directly into the factored template $n(n-3) = 88$. Testing option (A) gives $11 \times (11-3) = 11 \times 8 = 88$, validating the solution instantly!
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Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given that a polygon contains exactly 44 diagonals and need to figure out its total number of sides ($n$).

Step 2: Key Formula or Approach:
Connecting any two vertices out of $n$ available points yields a total of ${}^n\mathrm{C}_2$ possible straight lines. Of these lines, exactly $n$ form the boundary edges of the polygon. The remaining interior paths are the diagonals. The formula for the total number of diagonals is: $$\text{Number of diagonals} = {}^n\mathrm{C}_2 - n = \frac{n(n-1)}{2} - n = \frac{n(n-3)}{2}$$

Step 3: Detailed Explanation:
Equate our diagonal formula to the given problem value of 44: $$\frac{n(n-3)}{2} = 44$$ Multiply by 2 to remove the denominator fraction: $$n(n-3) = 88$$ $$n^2 - 3n - 88 = 0$$ Factor the quadratic equation by looking for two integers that multiply to $-88$ and sum to $-3$. Those numbers are $-11$ and $+8$: $$(n - 11)(n + 8) = 0$$ This yields two possible mathematical solutions: $$n = 11 \quad \text{or} \quad n = -8$$ Since the number of physical sides of a geometric polygon must be a positive integer value, we discard the negative root. Thus, $n = 11$.

Step 4: Final Answer:
The total number of sides of the polygon is 11, which matches option (A).