Question

A polygon has 90 diagonals. Find the number of its sides?

Answer 1

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.