A polygon has 44 diagonals. Find the number of its sides.

Introduction / Context:
The number of diagonals in a convex n-gon is given by the standard combinatorial formula n(n − 3)/2. This counts all ways to join non-adjacent vertices by a segment, avoiding sides. Given the number of diagonals, invert the formula to find n.

Given Data / Assumptions:

• Diagonals D = 44.
• Formula: D = n(n − 3)/2 for integer n ≥ 3.

Concept / Approach:
Set n(n − 3)/2 = 44 ⇒ n(n − 3) = 88. Solve the quadratic equation n^2 − 3n − 88 = 0 and take the positive root.

Step-by-Step Solution:
n^2 − 3n − 88 = 0Factor: (n − 11)(n + 8) = 0n = 11 (discard n = −8)

Verification / Alternative check:
Plug back: D = 11(11 − 3)/2 = 11*8/2 = 44, consistent.

Why Other Options Are Wrong:
n = 9, 10, 12 yield diagonals 27, 35, and 54 respectively, not 44.

Common Pitfalls:
Using n(n − 1)/2 (which counts all vertex pairs) without subtracting edges; the correct formula removes the n polygon sides, giving n(n − 3)/2.

Final Answer:
11