Question

If a polygon has 44 diagonals, find the number of its sides.

Answer 1

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 

= `("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 

= `("n"("n" - 1))/2 - "n" = ("n"("n" - 3))/2`

Given that a polygon has 44 diagonals.

Let n be the number of sides of the polygon.

`("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.