Question

How many diagonals does a regular pentagon have?
(a) 3
(b) 4
(c) 5
(d) 6

Answer 1

Hint: For solving this question first we will see the that diagonal is the line segment that joins two no-adjacent vertices so, we will derive the formula ${}^{n}{{C}_{2}}-n$ for the number of diagonals of a $n$ sided polygon by the simple concept of combinations. After that, we will put $n=5$ in ${}^{n}{{C}_{2}}-n$ and calculate the correct answer.

Complete step by step answer:
Given:
We have to find the number of diagonals for a regular pentagon.
Now, before we proceed we should know how to select $r$ objects from the $n$ distinct objects. The formula for the number of different possible ways is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ .
Now, we should know that in a polygon, the diagonal is the line segment that joins two non-adjacent vertices. Moreover, as we know that in case of any $n$ sided polygon we have $n$ vertices and we can select any two vertices by total ${}^{n}{{C}_{2}}$ different ways. Which means we can make total ${}^{n}{{C}_{2}}$ line segments from the $n$ vertices. And as we know that there are $n$ sides between adjacent points so, ${}^{n}{{C}_{2}}-n$ will be the line segments between non-adjacent points.
Now, from the above discussion, we conclude that for any $n$ sided polygon there will be ${}^{n}{{C}_{2}}-n$ diagonals.
Now, we come back to our question in which we have to find the number of diagonals for a regular pentagon. As we know that in a regular pentagon there are a total of 5 sides which means the value of $n=5$ . Then,
Number of diagonals for the regular pentagon $={}^{n}{{C}_{2}}-n={}^{5}{{C}_{2}}-5=\dfrac{5\times 4}{2}-5=10-5=5$ .
For more clarity look at the figure given below:

In the above figure, we have a regular pentagon ABCDE with its five diagonals AD, AC, BD, BE and CE.
Thus, a regular pentagon has 5 diagonals.
Hence, (c) is the correct option.

Note: Here, the student before solving first try to understand the problem and we should apply the formula of the selection of $r$ objects from the $n$ distinct objects and try to derive the result ${}^{n}{{C}_{2}}-n$ for the number of diagonals of a $n$ sided polygon it will strengthen our concepts of combinations. Moreover, though the problem is very easy, and we could have simply counted the number of diagonals but we should remember the formula ${}^{n}{{C}_{2}}-n$ and it might be useful for some problems.