# All the diagonals of the polygon were drawn and there were 15 of them. How many vertices does this polygon have?

August 22, 2021 | education

| Let us prove that a convex n-gon has n * (n – 3) / 2 diagonals.

Let’s choose any of n vertices. Obviously, it can be connected by diagonals with n – 3 vertices, since there are no diagonals for the vertex itself and two neighboring vertices. Then there will be n * (n – 3) diagonals in total, but each will be counted 2 times, since counting was carried out for each diagonal from 2 vertices. Therefore, there are only n * (n – 3) / 2 diagonals.

By the statement of the problem, the polygon has 15 diagonals. Then:

n * (n – 3) / 2 = 15,

n ^ 2 – 3 * n – 30 = 0,

D = 9 + 4 * 30 = 129.

√D = √129 is not an integer, so there is no integer n solution to the equation.

Hence, there is no polygon with 15 diagonals.

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