In geometry, collinearity is an important concept that describes the relationship between three or more points on a straight line. Three points are said to be collinear if they lie on the same straight line.
But how can we prove that three points are collinear? Read on to find out what exactly collinearity is and some of the methods that you can use to prove collinearity of three points. You might also enjoy reading: How to Find Vertical Asymptotes of Rational Functions.
What Is Collinearity?
Collinearity can be defined as the condition of lying on the same straight line. Thus, three points are said to be collinear if they all lie on the same line. There are several methods to prove collinearity, the simplest of which involves using the slope of the line that the points lie on.
To prove that three points are collinear using this method, we need to calculate the slopes of the lines that are formed by connecting each pair of points. If the slopes of any two lines are the same, then the three points are collinear.
Keep reading to explore how to prove that three points are collinear:
Method 1: Distance Formula
One of the ways to prove that three points are collinear is by using the distance formula. The distance formula can calculate the distance between two points in a Cartesian plane. If the sum of the two shorter distances is equal to the longest distance, then the three points are collinear.
Therefore, if you can show that the distance between points A and B, points B and C, and points A and C are equal, then you can prove that the three points are collinear.
For example, to prove that A, B, and C are collinear, you will need to prove that the sum of the lengths of any two line segments among AB, CA, and BC is equal to the length of the other line segment, meaning either AB + BC = AC or AC +CB = AB or BA + AC = BC.
This method may take a little longer, but it is always useful when dealing with more complex geometric figures. If you want to learn more about how to determine if three points are collinear using the distance formula, I encourage you to check out this video.
Method 2: Slope Formula
Another method to prove collinearity of three points is by using the slope formula. The slope formula provides a way to calculate the slope of a line that connects two points in a Cartesian plane.
If you can show that the slope between points A and B is equal to the slope between points B and C, then you can conclude that the three points are collinear.
To find the slope (m) between points A and B (xA, yA) and (xB, yB) and points B and C (xC, yC), we use this formula mAB = yB − yA /xB − xA and mBC = yC − yB /xC − xB If the slope between points A and B (mAB) is equal to the slope between points B and C (mBC), then the three points must all be on the same line.
However, this method may not be applicable if the points lie on a vertical line, as the slope between two points on a vertical line is undefined. Check out this video to learn more about how to prove that points are collinear using the Slope Method.
Method 3: Vector Algebra
A third method to prove collinearity of three points is by using vector algebra. Vector algebra can be used to determine whether three points are collinear or not by finding the cross product of two vectors that connect the three points.
Three points with position vectors a, b, and c are collinear if the vectors (a−b) and (a−c) are parallel. We must show (a−b)=k(a−c) for some constant k to prove collinearity (Source: undergroundmathematics.org).
However, this method may require advanced knowledge of vector algebra, making it more suitable for advanced students. I encourage you to check out this video to explore how to prove three points are collinear using vectors.
Method 4: Using the Intersection of Lines
You can also use the intersection of lines to prove collinearity of three points. By drawing lines through pairs of points, you can find the intersection point of these lines. If the intersection point of two lines is the same as the third point, then the three points are collinear.
Using the intersection of lines method requires a good understanding of line geometry, but it can be an efficient and easy way to prove collinearity. Check out this video to learn more about how to find the intersection point of two linear equations.
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Wrapping Up
Collinearity is a fundamental concept in geometry that describes the linear relationship between three or more points.
While there are different methods to prove collinearity of three points, I encourage you to learn all of the methods and apply the one that suits you best.
By using the distance formula, slope formula, vector algebra, or the intersection of lines, you can easily prove that three points are collinear.