Lesson Objectives

- Learn how to determine if three points are collinear using determinants

## How to Determine if Three Points are Collinear Using Determinants

In the last lesson, we learned how to determine the area of a triangle using determinants. Now, we will see another application of this formula. If our formula yields a result of zero, the three points given are collinear. This means they lie on the same line.

Example #1: Determine if the points given are collinear. $$(4, 0)$$ $$(9, 3)$$ $$(-1, -3)$$ Let's label our points and then plug into the formula: $$\text{Point 1}: (4, 0)$$ $$\text{Point 2}: (9, 3)$$ $$\text{Point 3}: (-1, -3)$$ $$\left| \begin{array}{ccc}4&0&1\\9&3&1\\-1&-3& 1\end{array}\right|=0$$ Since our formula gives us a result of zero, we know these three points are collinear.

### Test for Collinear Points

$$\left| \begin{array}{ccc}x_{1}&y_{1}&1\\ x_{2}& y_{2}& 1\\ x_{3}& y_{3}& 1\end{array}\right|=0$$ Let's look at an example.Example #1: Determine if the points given are collinear. $$(4, 0)$$ $$(9, 3)$$ $$(-1, -3)$$ Let's label our points and then plug into the formula: $$\text{Point 1}: (4, 0)$$ $$\text{Point 2}: (9, 3)$$ $$\text{Point 3}: (-1, -3)$$ $$\left| \begin{array}{ccc}4&0&1\\9&3&1\\-1&-3& 1\end{array}\right|=0$$ Since our formula gives us a result of zero, we know these three points are collinear.

#### Skills Check:

Example #1

Determine if collinear. $$(-2, -2), (1, -1), (7, 5)$$

Please choose the best answer.

A

Yes

B

No

Example #2

Determine if collinear. $$(3, 0), (0, -12), (1, -8)$$

Please choose the best answer.

A

Yes

B

No

Example #3

Determine if collinear. $$(15, 3), (2, -1), (-3, 5)$$

Please choose the best answer.

A

Yes

B

No

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