Answer 1

A polygon of n sides has n vertices. By joining any two vertices we obtain either a side or a diagonal.
∴ Number of ways of selecting 2 out of 9 \[=^n C_2 = \frac{n\left( n - 1 \right)}{2}\]

Out of these lines, n lines are the sides of the polygon.

∴ Number of diagonals =\[\frac{n\left( n - 1 \right)}{2} - n = \frac{n\left( n - 3 \right)}{2}\]

1. In a hexagon, there are 6 sides.
∴ Number of diagonals =\[\frac{6\left( 6 - 3 \right)}{2} = 9\]