Chapter 6.1, Problem 53PS

Solution

a.

To find: The expression for the number of line segments that join pairs of vertices on an $n$ -sided polygon.

  $\frac{n\left(n-1\right)}{2}$ .

Given:

Convex polygon with $n$ sides.

Calculation:

The number of line segments is the number of combinations of $n$ sides taken 2 at a time. That is,

  $\begin{array}{c}C{}_2=\frac{n!}{2!\left(n-2\right)!}\\ =\frac{n\cdot \left(n-1\right)\cdot \cancel{\left(n-2\right)!}}{2\cdot \cancel{\left(n-2\right)!}}\\ =\frac{n\left(n-1\right)}{2}\end{array}$

Hence, the number of line segments is $\frac{n\left(n-1\right)}{2}$ .

b.

To find: The formula for the number of diagonals of an $n$ -sided convex polygon.

  $\frac{n\left(n-1\right)}{2}-n$ .

Given:

Convex polygon with $n$ sides.

Calculation:

From subpart (a), the number of line segments on an $n$ -sided convex polygon is $\frac{n\left(n-1\right)}{2}$ .

In these line segments, $n$ is the number of sides of the polygon.

So, the number of diagonals will be the difference between the total number of line segments and the number of sides of the polygon. That is,

  $\frac{n\left(n-1\right)}{2}-n$