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प्रश्न
A polygon has 90 diagonals. Find the number of its sides?
उत्तर
Let the number sides of the polygon be n.
The number of diagonals of a polygon with n sides = `("n"("n" - 3))/2`
Given `("n"("n" - 3))/2` = 90
n2 – 3n = 180
n2 – 3n = 180
n2 – 3n – 180 = 0
n2 – 15n + 12n – 180 = 0
n(n – 15) + 12(n – 15) = 0
(n + 12)(n – 15) = 0
n + 12 = 0 or n – 15 = 0
n = – 12 or n = 15
But n = – 12 is not possible
∴ n = 15
∴ The number of sides of the polygon having 90 diagonals is 15.
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