Collinear Points: Definition, How to Prove & Examples
Collinear Points: Definition, How to Prove & Examples
TL;DR
Collinear points are three or more points that all lie on the same straight line. This article covers the definition, collinear versus non-collinear points, the three methods to prove collinearity (slope equality, zero triangle area, and the distance test), how each one works, and six worked examples.
What Are Collinear Points?
Collinear points are a set of three or more points that all lie on the same straight line. The word splits into col- ("together") and -linear ("line"): points sitting together on one line. Any two points are trivially collinear, since a single straight line always passes through any pair, so the idea only becomes a real question once you have three or more.
The points do not need to be evenly spaced or close together. As long as one ruler laid on the page touches every point at once, they are collinear. Stretch them across a kilometre or bunch them in a centimetre; the only thing that matters is that no point strays off the line.
Collinear vs Non-Collinear Points
Points that do not all fall on one line are non-collinear points. Three non-collinear points are special: they are exactly what you need to define a triangle, and to fix a single flat plane. That connection runs deep, since the cleanest test for collinearity, coming up next, asks whether three points form a triangle at all.
- Collinear: (1,1), (2,2), (3,3) all lie on the line y=x.
- Non-collinear: (0,0), (4,0), (0,3) form a right triangle, so they bend away from any single line.
How Do You Prove That Points Are Collinear?
There are three standard methods, and each one is really the same idea checked a different way: if the points were collinear, a certain "bend" measurement would come out to zero. Pick whichever fits the information you have.
Method 1: The Slope Test
The slope of the segment between two points measures its steepness, the rise over the run: m=\frac{y_2 - y_1}{x_2 - x_1}.
If three points A, B, C lie on one straight line, then walking from A to B has the same steepness as walking from B to C, because a straight line has one fixed slope everywhere. So the test is: slope of AB = slope of BC.
Method 2: The Area-of-Triangle Test
Three non-collinear points form a triangle with some positive area. Three collinear points are squashed flat onto a line, so the "triangle" they would form has zero area. That gives a clean test using the coordinate area formula:
Area = \frac{1}{2},\lvert x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \rvert.
Method 3: The Distance Test
If three points lie on a line in the order A, B, C, then travelling A to B and then B to C covers exactly the same ground as going straight from A to C:
AB + BC = AC.
Examples of Collinear Points
Example 1 - Are P(1,2), Q(2,3), and R(3,4) collinear? Use the slope test
Slope of PQ = \dfrac{3 - 2}{2 - 1} = 1. Slope of QR = \dfrac{4 - 3}{3 - 2} = 1. The slopes match, so there is no bend at Q.
Final answer: collinear.
Example 2 - Check whether A(2,3), B(4,7), and C(6,11) are collinear
Done correctly: slope of BC = \dfrac{11 - 7}{6 - 4} = 2. Now both segments through B have slope 2, so the points are collinear.
Example 3 - Use the area test on A(-3,-1), B(-1,0), C(1,1)
Area = \tfrac{1}{2},\lvert (-3)(0 - 1) + (-1)(1 - (-1)) + (1)((-1) - 0) \rvert = 0. The area is 0, so no triangle forms.
Example 4 - Use the distance test on A(1,1), B(2,2), C(4,4)
Check: AB + BC = AC. The two shorter pieces add to the longest.
Final answer: collinear.
Example 5 - Are (0,0), (2,1), and (4,3) collinear? Use the slope test
Final answer: non-collinear.
Example 6 - Find the value of k so that (2,k), (4,8), and (6,12) are collinear
Final answer: k = 4.
Where Collinearity Matters
Collinearity is vital in various fields, such as Surveying and construction, Computer graphics and GPS, Astronomy, and Physics and engineering.
Where Students Lose the Mark on Collinear Points
Mistake 1: Checking only one slope
Don't do this: Treat a single segment's slope as proof for a third point that segment never touched.
Mistake 2: Forgetting the absolute value in the area formula
Don't do this: Skip the modulus bars and judge collinearity off a signed value.
Mistake 3: Misordering points in the distance test
Don't do this: Assume the order the points were listed in is the order they sit on the line.
The Short Version
- Collinear points are three or more points lying on a single straight line; any two points are always collinear.
- The slope test checks that slope AB = slope BC (no bend at the shared point).
- The area test checks that the triangle the points would form has zero area.
- The distance test checks that the two shorter segments add up to the longest, AB + BC = AC.
Work Through These Problems to Solidify Your Understanding
- Are (1,3), (2,5), and (3,7) collinear? Use the slope test.
- Use the area test on (0,0), (1,2), and (2,5).
- Find k so that (1,2), (3,k), and (5,10) are collinear.
Answer to Question 1: collinear; both slopes equal 2. Answer to Question 2: non-collinear; the bracketed expression is 1≠0, so the area is positive. Answer to Question 3: k = 6.