Chapter 8.3, Problem 31WE

To determine

To Explain: whether the points $\left(1,4\right),\left(3,0\right),\left(5,-4\right)$ are collinear or not.

Answer to Problem 31WE

The points are collinear.

Explanation of Solution

Given information:

The points of a plane are given as:

  $\left(1,4\right),\left(3,0\right),\left(5,-4\right)$

Formula used:

To check the collinear points,

  $\frac{x_1-x_2}{x_1-x_3}=\frac{y_1-y_2}{y_1-y_3}=\frac{z_1-z_2}{z_1-z_3}$

The three or more points are collinear when they are in straight line.

Consider the points,

  $\left(1,4\right),\left(3,0\right),\left(5,-4\right)$

Recall that to check the collinear points,

  $\frac{x_1-x_2}{x_1-x_3}=\frac{y_1-y_2}{y_1-y_3}=\frac{z_1-z_2}{z_1-z_3}$

Therefore, the points are given by,

  $\left(1,4\right),\left(3,0\right),\left(5,-4\right)$

Make a table of the coordinates in increasing order $x$ -coordinates then change in the $x$ -coordinates and $y$ -coordinates as move one point to the next.

If the ratio of vertical change divide by horizontal change is constant, the points are collinear. Other wise they are not.

Check the points are collinear or not.

  $\begin{array}{cc}x& y\\ 1& 4\\ 3& 0\\ 5& -4\end{array}$

Here, first we check with the help of formula:

  $\begin{array}{c}\frac{x_1-x_2}{x_1-x_3}=\frac{y_1-y_2}{y_1-y_3}\\ \frac{1-3}{1-5}=\frac{4-0}{4+4}\\ \frac{-2}{-4}=\frac{4}{8}\\ \frac{1}{2}=\frac{1}{2}\end{array}$

The points are collinear because the lines are in straight line having same results.

Thus, the points are collinear.