A polygon has 77 diagonals.Determine the number of sides?

Formula: The number of diagonals of an n-sided polygon is n(n - 3)/2.

• Set up the equation: n(n - 3)/2 = 77.

• Multiply both sides by 2: n(n - 3) = 154.

• Expand and rearrange into a quadratic: n^2 - 3n - 154 = 0.

• Compute the discriminant and solve: discriminant = (-3)^2 - 4·1·(-154) = 9 + 616 = 625, sqrt = 25, so n = (3 ± 25)/2.

• Evaluate the roots: (3 + 25)/2 = 14 and (3 - 25)/2 = -11. Discard -11 because the number of sides must be positive.

• Therefore n = 14.

Check: 14 × 11 / 2 = 77, so the solution is correct.

Answer: The polygon has 14 sides.