Q13 of 58 Page 118

If a polygon of ‘n’ sides has diagonals. How many sides will a polygon having 65 diagonals? Is there a polygon with 50 diagonals?

No. of diagonals =

n(n - 3) = 2 × 65


⇒ n2 - 3n = 130


⇒ n2 - 3n - 130 = 0


Performing factorization we get:


⇒ n2 - 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


⇒ n2 - 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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