QUESTION

Describe the relationship between the number of sides and the number of diagonals of a polygon.

Hint: - A polygon is simply a plain figure enclosed by straight lines.
Diagonal: A line segment in a polygon that joins two non-consecutive vertices.
With the help of above 2 statements one can deduce the relationship between the number of sides and no of diagonals.

Let consider a regular polygon of $n$ vertices.
Since, to make diagonal we need to choose number of pairs of two vertices that can be formed from $n$vertices i.e.
$\Rightarrow _{}^n{C_2}$ eq.1
But one thing to be noted here we need to subtract $n$ from eq.1 since adjacent vertices cannot make a diagonal
$\Rightarrow {\text{ }}_{}^n{C_2} - n \\ \Rightarrow {\text{ }}\dfrac{{n!}}{{2!(n - 2)!}}{\text{ }} - {\text{ }}n \\ \Rightarrow {\text{ }}\dfrac{{n(n - 1)}}{2}{\text{ }} - n \\ \Rightarrow {\text{ }}\dfrac{{n(n - 1) - 2n}}{2} \\ {\text{On taking }}n{\text{ common from expression}} \\ \Rightarrow {\text{ }}\dfrac{{n(n - 3)}}{2}{\text{ }} \\$
Is $\dfrac{{n(n - 3)}}{2}$.