Chapter 1, Problem 51RE

To determine

To show:The given points lie on a line

Explanation of Solution

Given:

  $A=\left(2,5\right),B=\left(6,1\right)$ and $C=\left(8,-1\right)$

Proof:

In order to find if the points lie on the same line, find the slope considering two points at a time. Choose the points $A=\left(2,5\right)$ and $B=\left(6,1\right)$ first. The slope of line containing these two points can be computed using $m=\frac{y-y_1}{x-x_1}$ .

  $m=\frac{1-5}{6-2}$

  $=-1$

So, the slope of the line is $-1$ .

Consider the points $\left(2,5\right)$ and $\left(8,-1\right)$ next. Compute the slope using $m=\frac{y-y_1}{x-x_1}$ .

  $m=\frac{y-y_1}{x-x_1}$

  $=\frac{-1-5}{8-2}$

  $=-1$

So, the slope of the line is $-1$ .

The slope is $-1$ in both cases. Both lines contain the point $\left(2,5\right)$ . Since the lines have the same slope and a common point, they meet at the point $\left(2,5\right)$ . So, all three points lie on the same line.

Therefore, the points $\left(2,5\right)$ , $\left(6,1\right)$ , and $\left(8,-1\right)$ lie on a line.

Conclusion:

Therefore,theslope of the line is $-1$ and the points $\left(2,5\right)$ , $\left(6,1\right)$ , and $\left(8,-1\right)$ lie on a line.