Chapter 8.5, Problem 76E

To determine

To find: the number of diagonals of the polygon.

Answer to Problem 76E

The number of diagonals of the decagon is 5.

Explanation of Solution

Given Information:

Decagon

Concept used:

A line segment connection any two nonadjacent vertices is called a diagonal of a polygon.

Calculation:

The number of combinations of $n$ elements taken $r$ at a time is

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Here,

  $\begin{array}{l}n=10\\ n=2\end{array}$

Put value both sides in equation ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$ ,

  $\begin{array}{c}{}_{10}{C}_2=\frac{10!}{2!\left(10-2\right)!}\\ =\frac{10\times 9\times 8!}{2!8!}\\ =45\end{array}$

Remember 10 of the above 15 are sides.

The number of diagonals of the Hexagon

  $\begin{array}{l}15-10\\ =5\end{array}$

Hence, the number of diagonals of the decagon is 5.