Question

A 14 sided regular polygon has how many diagonals:
A) 14
B) 28
C) 77
D) 154
E) 182

Answer 1

Hint:First find the possible number of lines that can be made by 14 vertices which also contains the sides of the polygon. We know that there are 14 sides in the polygon. Then we obtain the number of diagonals by subtracting the number of sides from the obtained value to get the desired result.

Complete step-by-step answer:
It is given to us that there is a polygon with 14 sides and we have to find the number of possible diagonals that can be drawn in it.
We know that there are 14 sides in a polygon. It means that there are 14 vertices in the polygon and we also know that if we join any two vertices of the polygon, either we obtain a side or a diagonal of the polygon. Any line can take two points so the number of ways to select two points out of 14 points is given as $C\left( {14,2} \right)$.
Now, expand the combination using the formula:
$C\left( {n,r} \right) = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Then the obtained combination can be expanded as:
$C(14,2) = \dfrac{{14!}}{{(14 - 2)!2!}}$
$C(14,2) = \dfrac{{14!}}{{12!2!}}$
$C(14,2) = \dfrac{{14 \times 13 \times 12!}}{{12!2!}}$
$C(14,2) = 13 \times 7$
$C(14,2) = 91$
We know that these obtained values also contain the diagonals of the polynomial so we subtract the number of sides of the polygon to get the total number of possible diagonals.
Number of diagonals$ = 91 - \left( {{\text{Number of sides}}} \right)$
Number of diagonals $ = 91 - 14$
Number of diagonals $ = 77$
So, there are 77 possible numbers of diagonals in the polygon with 14 sides.

So, the correct answer is “Option C”.

Note:There are n sides so there will be n vertices. We know that any line has two vertices so one line can be obtained by the joining of two lines. When we calculated the all possible combination of 2 points out of 14 points, this value also involves the sides of the polygon which is not a diagonal so do not forget to subtract the number of sides out of the obtained value.