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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