Chapter 3.6, Problem 29WE

To determine

Tocalculate: number of diagonals corresponding to given number of sides

Answer to Problem 29WE

Number of sides	3	4	5	6	7	8	n
Number of diagonals	0	2	4	6	8	10	$2(n-3)$

Explanation of Solution

Given information: The following table is given

Number of sides	3	4	5	6	7	8	n
Number of diagonals	0	2	?	?	?	?	?

Calculation:

First find the relationship between the number of sides and number of diagonals.

Let x denotes number of sides and y denotes corresponding number of diagonals.

For $x=3,y=0$

For $x=4,y=2$

So, take the two points as $(3,0),(4,2)$

Let $\left(x_1,y_1\right)=(3,0),\left(x_2,y_2\right)=(4,2)$

Equation of a line passing through points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ is given by $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Put $\left(x_1,y_1\right)=(3,0),\left(x_2,y_2\right)=(4,2)$

  $\begin{array}{l}y-0=\frac{2-0}{4-3}\left(x-3\right)\\ \Rightarrow y=2(x-3)\end{array}$

Now find number of diagonals $(y)$ corresponding to given number of sides $(x)$

At $x=5,y=2(5-3)=2(2)=4$

At $x=6,y=2(6-3)=2(3)=6$

At $x=7,y=2(7-3)=2(4)=8$

At $x=8,y=2(8-3)=2(5)=10$

At $x=n,y=2(n-3)$

So,

Number of sides	3	4	5	6	7	8	n
Number of diagonals	0	2	4	6	8	10	$2(n-3)$