Lesson Objectives
• Demonstrate an understanding of the distance formula
• Learn how to determine if three points are collinear using the distance formula

## How to Determine if Three Points are Collinear Using the Distance Formula

In our last lesson, we used the distance formula to determine if three points were the vertices of a right triangle. Now, we will look at another application of our distance formula. We can also use the distance formula to determine if three points are collinear, which means they lie on the same line. Now, there are a few different methods we can use for this task and this is normally the slowest method. It’s usually faster to use a method that involves slopes for this process or to use a formula that involves determinants, but for the sake of completeness, we will cover this method. As we get into matrices later on in the course, we will discover a much faster way to perform this task.
Three points will be collinear if the sum of the distances between two pairs of points is equal to the distance between the remaining pair of points. The basic idea here is to find the distance between each pair of points. With three points, this will give us three distances. Then we want to sum the two smaller distances and see if this equals the larger.
Example #1: Determine if the three points are collinear. $$(2, -7), (6, -6), (10, -5)$$ We will use d1 for the distance between (2, -7) and (6, -6), d2 for the distance between (6, -6) and (10, -5), and d3 for the distance between (2, -7) and (10, -5). $$d_1=\sqrt{(6 - 2)^2 + (-6 + 7)^2}$$ $$d_1=\sqrt{16 + 1}=\sqrt{17}$$ $$d_2=\sqrt{(10 - 6)^2 + (-5 + 6)^2}$$ $$d_2=\sqrt{16 + 1}=\sqrt{17}$$ $$d_3=\sqrt{(10 - 2)^2 + (-5 + 7)^2}$$ $$d_3=\sqrt{64 + 4}=\sqrt{68}=2\sqrt{17}$$ Now we will sum the two smaller distances and set this equal to the largest distance. If these two sides are equal, we can say that our points are collinear. $$\sqrt{17}+ \sqrt{17}=2\sqrt{17}$$ $$2\sqrt{17}=2\sqrt{17}$$ Since this equation is true, we can say our points are collinear. Example #2: Determine if the three points are collinear. $$(2, 2), (-4, 8), (-2, 3)$$ We will use d1 for the distance between (2, 2) and (-4, 8), d2 for the distance between (-4, 8) and (-2, 3), and d3 for the distance between (2, 2) and (-2, 3). $$d_1=\sqrt{(-4 - 2)^2 + (8 - 2)^2}$$ $$d_1=\sqrt{36 + 36}=6\sqrt{2}$$ $$d_2=\sqrt{(-2 + 4)^2 + (3 - 8)^2}$$ $$d_2=\sqrt{4 + 25}=\sqrt{29}$$ $$d_3=\sqrt{(-2 - 2)^2 + (3 - 2)^2}$$ $$d_3=\sqrt{16 + 1}=\sqrt{17}$$ Now we will sum the two smaller distances and set this equal to the largest distance. If these two sides are equal, we can say that our points are collinear. $$\sqrt{29}+ \sqrt{17}\ne 6\sqrt{2}$$ Since the two sides are not equal, we can say our points are not collinear. Looking at the graph below, we can see these points form a triangle and do not lie on the same line.

#### Skills Check:

Example #1

Determine if the points are collinear.

Please choose the best answer. $$(5, 9), (3, 7), (4, 9)$$

A
Yes
B
No

Example #2

Determine if the points are collinear.

Please choose the best answer. $$(2, -9), (1, -6), (-3, 6)$$

A
Yes
B
No

Example #3

Determine if the points are collinear.

Please choose the best answer. $$(1,5), (2, 3), (-2, -11)$$

A
Yes
B
No

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