If a polygon of ‘n’ sides has 1/2 n(n – 3) diagonals. How many s

1.	If a polygon of ‘n’ sides has 1/2 n(n – 3) diagonals. How many sides will a polygon having 65 diagonals? Is there a polygon with 50 diagonals?
Answer» Given: Number of diagonals of a polygon with n-sides = \(\frac{n(n-3)}{2}\) No. of diagonals of a given polygon = 65 i.e., \(\frac{n(n-3)}{2}\) = 65 where n is number of sides of the polygon ⇒ n² – 3n = 2 × 65 ⇒ n² – 3n – 130 = 0 ⇒ n² – 13n + 10n – 130 = 0 ⇒ n(n – 13) + 10(n – 13) = 0 ⇒ (n – 13) (n + 10) = 0 ⇒ n – 13 = 0 (or) n + 10 = 0 ⇒ n = 13 (or) n = -10 But n can’t be negative. ∴ n = 13 (i.e.) number of sides = 13. Also to check 50 as the number of diagonals of a polygon ∴ \(\frac{n(n-3)}{2}\) = 50 ⇒ n² – 3n = 100 ⇒ n² – 3n – 100 = 0 There is no real value of n for which the above equation is satisfied. ∴ There can’t be a polygon with 50 diagonals.

If a polygon of ‘n’ sides has 1/2 n(n – 3) diagonals. How many sides will a polygon having 65 diagonals? Is there a polygon with 50 diagonals?

Answer»

Given: Number of diagonals of a polygon with n-sides = \(\frac{n(n-3)}{2}\)

No. of diagonals of a given polygon = 65 i.e., \(\frac{n(n-3)}{2}\) = 65

where n is number of sides of the polygon

⇒ n² – 3n = 2 × 65

⇒ n² – 3n – 130 = 0

⇒ n² – 13n + 10n – 130 = 0

⇒ n(n – 13) + 10(n – 13) = 0

⇒ (n – 13) (n + 10) = 0

⇒ n – 13 = 0 (or) n + 10 = 0

⇒ n = 13 (or) n = -10

But n can’t be negative.

∴ n = 13 (i.e.) number of sides = 13.

Also to check 50 as the number of diagonals of a polygon

∴ \(\frac{n(n-3)}{2}\) = 50

⇒ n² – 3n = 100

⇒ n² – 3n – 100 = 0

There is no real value of n for which the above equation is satisfied.

∴ There can’t be a polygon with 50 diagonals.