Question

if a polygon of 'n' sides has 1/2n(n-3) diagnols. how many sides will a polygon having 65 diagnols? is there a polygon with 50 diagnols?

Answer 1

no there is no polygon with 50 digonals

Answer 2

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Given :

• Number of diagonals with n-sides = n(n-3)/2
• No. of diagonals of a given polygon = 65
• I.e., 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

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

⛬ 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.

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