Question

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

Answer 1

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.

Complete step by step answer:
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
Then, eq.1 becomes
   $
   \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{ }} \\
$
Hence, the relation between the number of sides and the number of diagonals of a polygon
Is $\dfrac{{n(n - 3)}}{2}$.

Note: -Whenever you get this kind of question the key concept is to find a result that you have knowledge about polygons, diagonals. You need to know about basic properties of polygon, diagonals like to form a diagonal; two non-consecutive vertices are required.For eg:A triangle has 3 vertices therefore n=3, no of diagonals will be 3(3-3)/2=0. hence this can be implied to polygon of any no of sides.