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

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. One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.