Chapter 1.4, Problem 29E

To determine

To find: whether the given points are collinear and also justify the reason.

Answer to Problem 29E

The points $\left(5,9\right),\left(3,3\right)$ and $\left(1,6\right)$ are collinear

Explanation of Solution

Given information:

The given points are $\left(5,9\right),\left(3,3\right)$ and $\left(1,6\right)$ because they lie on a straight line.

Calculation:

Let the point-coordinates A, B and C be $\left(5,9\right),\left(3,3\right)$ and $\left(1,6\right)$ respectively.

If the line from A to B has the same slope as the line from A to C, then these points form a straight line and are collinear.

Put the point coordinates $\text{A}\left(5,9\right)\text{ and B}\left(3,3\right)$ to evaluate the slope of line AB as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{3-9}{-3-5}\\ =\frac{-6}{-8}\\ =\frac{3}{4}\end{array}$

Thus, the slope of AB is $\frac{3}{4}$ .

Put the point coordinates $\text{A}\left(5,9\right)\text{ and C}\left(3,3\right)$ to evaluate the slope of line AC as follows:

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{6-9}{-1-5}\\ =\frac{-3}{-4}\\ =\frac{3}{4}\end{array}$

Thus, the slope of AC is $\frac{3}{4}$ .

The line from A to B has the same slope as the line from A to C. This means that it is a straight line. So, the given points are collinear.