Chapter 9.1, Problem 31WE

To determine

To find: The given points are collinear or not.

Answer to Problem 31WE

The given points are not collinear.

Explanation of Solution

Given information: the given points are $A\left(-5,-2\right),B\left(-2,1\right)$ and $C\left(1,3\right)$ .

Formula used: Distance formula $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation: Take three pairs of points $AB,BC$ and $CA$ .

Now, apply the formula to calculate the distance between $A\left(-5,-2\right),B\left(-2,1\right)$ and $C\left(1,3\right)$ .

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-2-\left(-5\right)\right)}^2+{\left(1-\left(-2\right)\right)}^2}\\ =\sqrt{{\left(-2+5\right)}^2+{\left(1+2\right)}^2}\\ =\sqrt{3^2+3^2}\\ =\sqrt{18}\end{array}$

  $\begin{array}{l}BC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1-\left(-2\right)\right)}^2+{\left(3-1\right)}^2}\\ =\sqrt{9+4}\\ =\sqrt{13}\end{array}$

  $\begin{array}{l}CA=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1+5\right)}^2+{\left(3+2\right)}^2}\\ =\sqrt{36+36}\\ =\sqrt{72}\end{array}$

Since, if the sum of the above two distances is equal to the third distance, the given points are said to be collinear.

  $\begin{array}{l}\because AB+BC=CA\\ \therefore \sqrt{18}+\sqrt{13}\ne \sqrt{72}\end{array}$

Therefore, the given points are not collinear.