A polygon of n sides has n vertices. By joining any two vertices of a polygon, we obtain either a side or a diagonal of the polygon.
A number of line segments obtained by joining the vertices of a n sided polygon taken two at a time = Number of ways of selecting 2 out of n.
= nC2
= \(\frac{n(n-1)}{2}\)
Out of these lines, n lines are the sides of the polygon, sides can’t be diagonals.
∴ Number of diagonals of the polygon = \(\frac{n(n-1)}{2}\) - n = \(\frac{n(n-3)}{2}\)
Given that a polygon has 44 diagonals.
Let n be the number of sides of the polygon.
\(\frac{n(n-3)}{2}\) = 44
⇒ n(n – 3) = 88
⇒ n2 – 3n – 88 = 0
⇒ (n + 8) (n – 11)
⇒ n = -8 (or) n = 11
n cannot be negative.
∴ n = 11 is number of sides of polygon is 11.
