Collinear Points - Definition, Formula, Examples - Cuemath

Collinear Points

Collinear points are those points that lie on the same straight line. It is not necessary that they should be co-planar but they must lie on the same straight line. The word collinear is derived from the Latin words 'col' and 'linear' where 'col' stands for together and 'linear' means in the same line. The property of the points being collinear is known as collinearity. Let us learn more about collinear points in this article.

1.	What are Collinear Points?
2.	Non-Collinear Points
3.	Collinear Points Formula
4.	FAQs on Collinear Points

What are Collinear Points?

Collinear points are a set of three or more points that exist on the same straight line. Collinear points may exist on different planes but not on different lines. The property of points being collinear is known as collinearity. So any three points or more will only be collinear if they are in the same straight line. Only one line is possible that can go through three different points which are collinear. Observe the figure given below in which points P, Q, and R are the collinear points.

Non-Collinear Points

If three or more points do not lie on the same straight line, then they are said to be non-collinear points. If any point of all the points is not on the same line, then as a group they are non-collinear points. In the figure given below, points M, N, O, P, and Q are non-collinear points since they do not lie on the same straight line.

Collinear Points Formula

The collinear points formula is used to find out whether three points are collinear or not. There are various methods that can determine whether three points are collinear or not. The three most common formulas that are used to find if points are collinear or not are the Slope Formula, the Area of Triangle Formula, and the Distance Formula. Let us discuss all these formulas one by one.

Slope Formula

We apply the slope formula to find the slope of lines formed by the 3 points under consideration. If the 3 slopes are equal, then the three points are collinear.

For example, if we have three points X, Y, and Z, the points will be collinear only if the slope of line XY = slope of line YZ = slope of line XZ. To calculate the slope of the line joining two points, we use the slope formula.

The slope of the line joining points P(x1, y1) and Q(x2, y2) is:

\(m = \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

Area of Triangle Formula

In this method, we use the fact that a triangle cannot be formed by three collinear points. This means if any 3 points are collinear they cannot form a triangle. Therefore, we check the points of the triangle by using them in the formula for the area of a triangle. If the area is equal to 0, then those points will be considered to be collinear. In other words, the triangle formed by three collinear points will have no area since it will just be a line joining the three points. The formula for the area of a triangle that is used to check the collinearity of points is expressed as:

Area of the triangle with the given points (vertices) A(x1, y1), B(x2, y2), and C(x3, y3) is:

\( \text{A} =\frac{1}{2}\left|\left(x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right)\right|=0\)

Distance Formula

Using the distance formula, we find the distance between the first and the second point, and then the distance between the second and the third point. After this, we check if the sum of these two distances is equal to the distance between the first and the third point. This will only be possible if the three points are collinear points. To calculate the distance between two points whose coordinates are known to us, we use the distance formula.

The distance between two points A(x1, y1) and B(x2, y2) is:

\(d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

So, if we have three collinear points in the order A, B, and C, then these points will be collinear if AB + BC = CA.

All the three methods can be understood with the help of the solved examples given under the Practice Section.

Tips on Collinear Points:

• Three points will be collinear, only if they fall in the same straight line.
• This property of points being collinear is known as collinearity.
• Collinear points can exist on different planes.

☛ Related Topics

• X and Y Graph
• Coordinate Plane
• Equation of a Straight Line
• Area of triangle in coordinate geometry
• Distance between Two Points
• Equation of a Line