Question

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

Answer 1

Let n be the number of sides in a polygon.

Since, Polygon of n sides has (ⁿC₂ – n) number of diagonals

∴ ⁿC₂ – n = 44

= `(n!)/(2!(n - 2)!)` – n = 44

= `(n(n - 1)(n - 2)!)/(2*(n - 2)!)` – n = 44

⇒ `(n(n - 1))/2` – n = 44

= `(n^2 - n - 2n)/2` = 44

⇒ n² – 3n = 44

⇒ n² – 3n – 88 = 0

= n² – 11n + 8n – 88 = 0

⇒ n(n – 11) + 8(n – 11) = 0

= (n – 11(n + 8) = 0

∴  n = 11 and n = – 8  ....[∵ n ≠ – 8]

So n = 11

Hence, the required number of sides = 11.