Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.
In this lesson, you will learn the definition of collinear points in Geometry. Also, explore how collinear points relate to the real-world by looking at some examples.
What Are Collinear Points?
Collinear points are points that lie on the same line. The word 'collinear' breaks down into the prefix 'co-' and the word 'linear.' 'Co-' indicates togetherness, as in coworker or cooperate. 'Linear' refers to a line. So, collinear basically means points that hang out on the same line together.
A good way to picture the concept of collinear points is to think about food on skewers, like in the following picture.
Each skewer represents part of a line, and the tomatoes are points. So, the tomatoes labeled A, B, and C are collinear because they have all been lined up on the same skewer. All of the tomatoes on the plate are not collinear because no single straight skewer can poke through all of them the way they are arranged.
If we take two of our tomato skewers and simplify them to a geometry diagram, it might look something like this:
We can see that one line contains points A, B, and C, so those three points are collinear. A different line contains points T, O, and M, so those three points are collinear, but they are not collinear to points A, B, and C. When points are not collinear, we call them noncollinear. So, for example, points A, T, and O are noncollinear because no line can pass through the three of them together.
We are going to look at two example questions that relate to the following diagram:
A) Name two points that are collinear to point O.
B) Name two points that are noncollinear to point U.
For the first question, we are going to locate point O and follow any lines that pass through it. The line along the top of the figure passes through O and H, as highlighted in the figure below.
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So, O and H are collinear. However, the question asked for two collinear points to O, so just H is not sufficient. There are no other labeled points on the top line, so we have to look somewhere else. Point O is also on the line that goes from the top right diagonally to the bottom left, as highlighted here:
This line passes through O, U, and G. So, U and G are the two points that are collinear to O.
The second question asked us to find two points that are NOT collinear to U. So, we are looking for any pair of points that are not on the same line that also passes through U. There are many correct answers to this question. For instance, points L and R are noncollinear to U because a single straight line cannot pass through U, L, and R. Some other correct answers to this question would be: G and A, S and O, and H and G.
Let's review. Collinear points are points that lie on the same line. The word 'collinear' breaks down into the prefix 'co-' and the word 'linear.' 'Co-' indicates togetherness, and 'linear' refers to a line. So, collinear basically means points that hang out on the same line together. When points are not collinear, we call them noncollinear because they cannot be connected with a single line.
Co- Togetherness or Cooperation
Collinear - points that lie on the same line
Noncollinear - points that don't all lie on the same line and can't be connected with a single line
Completing this lesson should help you do the following:
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