Chapter 9.1, Problem 32WE

To determine

To find: The given points are collinear or not.

Answer to Problem 32WE

The given points are collinear.

Explanation of Solution

Given information: the given points are $A\left(1,5\right),B\left(2,0\right)$ and $C\left(4,-10\right)$ .

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 $A\left(1,5\right),B\left(2,0\right)$ and $C\left(4,-10\right)$ .

  $\begin{array}{l}AB=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(2-1\right)}^2+{\left(0-5\right)}^2}\\ =\sqrt{1+25}\\ =\sqrt{26}\end{array}$

  $\begin{array}{l}BC=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(4-2\right)}^2+{\left(-10-0\right)}^2}\\ =\sqrt{4+100}\\ =\sqrt{104}\end{array}$

  $\begin{array}{l}CA=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(1-4\right)}^2+{\left(5-\left(-10\right)\right)}^2}\\ =\sqrt{9+225}\\ =\sqrt{234}\end{array}$

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

  $\begin{array}{l}\because AB+BC=CA\\ \therefore \sqrt{26}+\sqrt{104}=\sqrt{234}\end{array}$

Therefore, the given points are collinear.