Learn the collinearity definition and understand what collinear points are. Also, see a comparison between collinear and non-collinear points. Also, learn about methods to determine collinearity.
Collinear Points in Geometry | Definition & Examples
Table of Contents
- Collinearity in Practical Life
- Collinearity vs. Non-Collinearity
- Methods to Determine if Points Are Collinear
- Lesson Summary
Frequently Asked Questions
How can one prove that points are collinear?
Points can be mathematically shown to be collinear by checking to see if the area of the triangle formed by the three points is equal to 0 or not. If a triangle has an area of 0, then that means all three points are on the same line; they do not form a triangle.
What are collinear and noncollinear points?
Collinear points are points that are all on the same straight line. Non-collinear points are points that are not all on the same straight line.
When are three points collinear?
Three points are collinear if they are all on the same straight line. If the points are not in a straight line, then they are non-collinear. One straight line will run through all three points if they are collinear.
What are examples of things that are collinear?
Food items on the same skewer are collinear. Points along any straight line such as the corner formed by two walls are collinear.
Table of Contents
- Collinearity in Practical Life
- Collinearity vs. Non-Collinearity
- Methods to Determine if Points Are Collinear
- Lesson Summary
The term collinear relates to points in geometry. The collinear points definition in geometry is when three or more points lie on the same line. When three or more points satisfy this definition, the points are said to be collinear. The points can be individual dots such as this line made with three dots in a row.
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The dots can also be three or more points in a shape that form a single line like the three dots in this star shape.
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The three orange dots are collinear since all three are on the same line.
Collinearity in Practical Life
Collinear points are also seen in the real world outside of geometry. Examples of collinear points in practical life include the following:
- Bricks that form a straight line at the end of a wall will create collinear points.
- A skewer of roasted bell peppers and roasted steak form collinear points since all the food is located on the same skewer line.
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- The measuring points on a ruler are all collinear points as well.
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When points do not exist on the same line, then they are referred to as non-collinear. When points are collinear, they have collinearity. When points are non-collinear, they are said to have non-collinearity.
Take this star, for example. Points that are not on the same line are non-collinear.
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The purple points of this star are non-collinear since they are not all on the same line. The orange points are collinear and have collinearity since they are all on the same straight line. One way to test for collinearity versus non-collinearity is to see if the points can be connected with a single straight line. If the line is curved or has to be turned in any way, then the points are non-collinear and have non-collinearity.
In math and geometry, there are a few formal ways to show whether a collection of three or more points are collinear. One uses the distance formula, another uses the slope, while another uses the area of a triangle.
Using Distance Formula
Using the distance formula to check whether three points are collinear requires calculating the distance between the first segment and the second segment and then seeing if the sum equals the distance of a line connecting the first and last point. If they are equal, then the three points are collinear. If the points are non-collinear, then this would not be equal. Compare this to a triangle. The sum of two sides of a triangle will always be larger than the third side.
Example
Let's look at an example.
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The three points in question are (1, 1), (2, 1), and (3, 1). Checking for collinearity with the distance formula requires finding the distance between point A and point B as well as between point B and point C. The sum needs to equal the distance between point A and point C.
Remember, the distance formula is this one:
{eq}d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} {/eq}
Calculating the distance between point A and point B, the result is
{eq}d = \sqrt{(2 - 1)^2 + (1 - 1)^2} \\ d = \sqrt{(1)^2 + (0)^2} \\ d = \sqrt{1} \\ d = 1 {/eq}
Calculating the distance between point B and point C, the result is
{eq}d = \sqrt{(3 - 2)^2 + (1 - 1)^2} \\ d = \sqrt{(1)^2 + (0)^2} \\ d = \sqrt{1} \\ d = 1 {/eq}
Adding the above two distances, the sum of the two distances is 1 + 1 =2.
This needs to equal the distance between point A and point C. This distance is
{eq}d = \sqrt{(3 - 1)^2 + (1 - 1)^2} \\ d = \sqrt{(2)^2 + (0)^2} \\ d = \sqrt{4} \\ d = 2 {/eq}
Yes, this equals the sum of the two earlier distances. Therefore, points A, B, and C are collinear and have collinearity.
Slope Method
The slope method checks for collinearity by checking to see if the slope of the first segment is equal to the slope of the second segment. If they are the same, then the points are collinear.
Example
Let's check the following three points for collinearity using the slope method. The points are (1, 1), (2, 2), and (3, 3).
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The formula for calculating slope with two points is:
{eq}m = \frac{(y_2 - y_1)}{(x_2 - x_1)} {/eq}
Using this formula, the slope between points A and B is the following.
{eq}m = \frac{(2 - 1)}{(2 - 1)} \\ m = \frac{1}{1} \\ m = 1 {/eq}
Does this equal the slope between the points B and C?
{eq}m = \frac{(3 - 2)}{(3 - 2)} \\ m = \frac{1}{1} \\ m = 1 {/eq}
Yes. Both slopes are 1. Therefore, the points A, B, and C are collinear and have collinearity. If the results are different, then the points are non-collinear.
Area of Triangle Method
The area of a triangle method checks for collinearity by seeing if the three points form a triangle or not. If they form a triangle, then the three points will have an area and are therefore non-collinear. If the area equals 0, then the three points are collinear and do not form a triangle.
The formula for the area of a triangle with three points is the following.
{eq}A = \frac{1}{2}|(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))| {/eq}
This needs to equal 0 for the points to be collinear.
Example
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Using the area of a triangle method, let's check these same three points for collinearity.
{eq}A = \frac{1}{2}|(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2))| \\ A = \frac{1}{2}|(1(2 - 3) + 2(3 - 1) + 3(1 - 2))| \\ A = \frac{1}{2}|(1(-1) + 2(2) + 3(-1))| \\ A = \frac{1}{2}|(-1 + 4 - 3)| \\ A = \frac{1}{2}|(0)| \\ A = 0 {/eq}
The area equals 0, so, therefore, the points A, B, and C are collinear and have collinearity.
In review, the collinear points definition in geometry is when three or more points lie on the same line. When three or more points are collinear, they are said to have collinearity. When the points are not on the same line, they are non-collinear and have non-collinearity.
Examples of collinearity in the real world include foods on the same skewer stick and numbers on a ruler.
Three formal methods to check for collinearity of three points are the following:
- Using the distance formula to check if the sum of the two segments equals the distance between the first and last point.
- Using the slope formula to check if the slope of the first segment is equal to the slope of the second segment.
- Using the area of a triangle method to see if the area formed by the three points is equal to 0.
Video Transcript
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.
Real-World Examples
A good way to picture the concept of collinear points is to think about food on skewers, like in the following picture.
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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:
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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.
Geometry Problems
We are going to look at two example questions that relate to the following diagram:
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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:
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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.
Lesson Summary
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.
Key Terms
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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
Learning Outcomes
Completing this lesson should help you do the following:
- Define collinear and noncollinear
- Identify sets of collinear points in a diagram
- Name noncollinear points in a diagram
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