A polygon has exactly 44 diagonals. Using the relationship between the number of sides of a polygon and its diagonals, how many sides does this polygon have?

Introduction / Context:
This problem tests the formula connecting the number of sides of a polygon and the number of diagonals it has. It is a very common question in geometry based aptitude tests and helps reinforce the idea of counting segments between vertices and excluding edges of the polygon.

Given Data / Assumptions:

• The polygon is simple and has n sides.
• The total number of diagonals in this polygon is 44.
• A diagonal is a line segment connecting two non adjacent vertices of the polygon.
• We assume n is at least 3 because a polygon must have at least 3 sides.

Concept / Approach:
The formula for the number of diagonals of an n sided polygon is n(n minus 3) / 2. This comes from counting all possible line segments between pairs of vertices and then subtracting the sides of the polygon. We set this formula equal to 44 and solve the resulting quadratic equation to find n. Only the positive integer solution that is meaningful for polygons is accepted.

Step-by-Step Solution:
Step 1: Use the diagonal formula: number of diagonals = n(n - 3) / 2.Step 2: Set n(n - 3) / 2 equal to 44, so n(n - 3) / 2 = 44.Step 3: Multiply both sides by 2 to remove the denominator: n(n - 3) = 88.Step 4: Expand: n^2 - 3n - 88 = 0.Step 5: Solve the quadratic equation n^2 - 3n - 88 = 0 using factorisation or the quadratic formula. The discriminant is 9 + 352 = 361, whose square root is 19.Step 6: Compute n = (3 ± 19) / 2, so n = (3 + 19) / 2 = 22 / 2 = 11 or n = (3 - 19) / 2 = -16 / 2 = -8. Discard the negative value as invalid for a number of sides.

Verification / Alternative check:
You can directly verify the result by substituting n = 11 into the diagonals formula. For an 11 sided polygon, diagonals = 11(11 - 3) / 2 = 11 * 8 / 2 = 88 / 2 = 44. This matches the given condition exactly, so n = 11 is correct. Trying any other option, such as 9, would give 9(9 - 3) / 2 = 9 * 6 / 2 = 54 / 2 = 27 diagonals, which does not match the required value.

Why Other Options Are Wrong:

• 13: Gives diagonals = 13(10) / 2 = 130 / 2 = 65, which is more than 44.
• 9: Gives diagonals = 9(6) / 2 = 54 / 2 = 27, less than 44.
• 7: Gives diagonals = 7(4) / 2 = 28 / 2 = 14, much less than 44.

Common Pitfalls:
Students sometimes confuse the diagonal formula with nC2 or make sign errors when forming the quadratic equation. Another frequent error is to forget to discard the negative root, even though a polygon cannot have a negative number of sides. Some may also misinterpret the problem and try to count edges instead of diagonals. Carefully applying the correct formula and checking results against the given diagonal count avoids these mistakes.

Final Answer:
The polygon has 11 sides, because an 11 sided polygon has exactly 44 diagonals.