Chapter 9.1, Problem 29WE

To determine

To find: The given points are collinear or not.

Answer to Problem 29WE

The given points are collinear.

Explanation of Solution

Given information: Consider the given points as $A\left(1,2\right),B\left(7,4\right)$ and $C\left(-2,1\right)$ .

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 $A\left(1,2\right),B\left(7,4\right)$ and $C\left(-2,1\right)$ .

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(7-1\right)}^2+{\left(4-2\right)}^2}\\ =\sqrt{36+4}\\ =\sqrt{40}\end{array}$

  $\begin{array}{l}BC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-2-7\right)}^2+{\left(1-4\right)}^2}\\ =\sqrt{81+9}\\ =\sqrt{90}\end{array}$

  $\begin{array}{l}CA=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1-\left(-2\right)\right)}^2+{\left(1-2\right)}^2}\\ =\sqrt{9+1}\\ =\sqrt{10}\end{array}$

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

  $\begin{array}{l}\because AB+CA=BC\\ \therefore \sqrt{40}+\sqrt{10}=\sqrt{90}\end{array}$

Therefore, the given points are collinear.