Answer :

No. of diagonals =

n(n - 3) = 2 × 65

⇒ n^{2} - 3n = 130

⇒ n^{2} - 3n - 130 = 0

Performing factorization we get:

⇒ n^{2} - 13n + 10n - 130 = 0

⇒ n(n - 13) + 10(n - 13) = 0

⇒ (n + 10)(n - 13) = 0

n = 13, - 10

Since no. of sides cannot be negative so

No. of Sides = 13

When No. of Diagonals is 50

n(n - 3) = 50 × 2

⇒ n^{2} - 3n - 150 = 0

Discriminant = (9 - 4 × 1 × ( - 150)) = 609

Since 609 is not a perfect square so n can never be a whole number.

Hence 50 diagonals are not possible

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