Chapter 9.1, Problem 30WE

To determine

To find: The given points are collinear or not.

Answer to Problem 30WE

The given points are collinear.

Explanation of Solution

Given information: the given points are $A\left(5,0\right),B\left(-7,3\right)$ and $C\left(1,1\right)$ .

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 $A\left(5,0\right),B\left(-7,3\right)$ and $C\left(1,1\right)$ .

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-7-5\right)}^2+{\left(3-0\right)}^2}\\ =\sqrt{144+9}\\ =\sqrt{153}\end{array}$

  $\begin{array}{l}BC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1-\left(-7\right)\right)}^2+{\left(1-3\right)}^2}\\ =\sqrt{64+4}\\ =\sqrt{68}\end{array}$

  $\begin{array}{l}CA=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1-5\right)}^2+{\left(1-0\right)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\end{array}$

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

  $\begin{array}{l}\because BC+CA=AB\\ \therefore \sqrt{68}+\sqrt{17}=\sqrt{153}\end{array}$

Therefore, the given points are collinear.