All diagonals are drawn in the polygon. there were 27 of them. What is this polygon?

1. Each vertex of an n-gon is connected by a diagonal with n – 3 vertices (not counting the neighboring vertices with which it forms two adjacent sides, and itself).
2. Since each diagonal involves two vertices, then for the number of diagonals of an n-gon we obtain:
N (n) = n (n – 3) / 2.
3. By the condition of the problem, we have:
N (n) = 27;
n (n – 3) / 2 = 27;
n (n – 3) = 54;
n ^ 2 – 3n – 54 = 0;
D = 3 ^ 2 + 4 * 54 = 9 + 216 = 225;
n = (3 ± √225) / 2 = (3 ± 15) / 2;
1) n = (3 – 15) / 2 = -12/2 = -6 – does not fit;
2) n = (3 + 15) / 2 = 18/2 = 9.
Answer: 9-gon.