Collinear Points

In Euclidean geometry, if two or more than two points lie on a line close to or far from each other, then they are said to be collinear. Collinear points are points that lie on the straight line. The word 'collinear' is the combined word of two Latin names ‘col’ + ‘linear’. ‘prefix ' 'co' and the word 'linear.' 'Co' indicates togetherness, as in coworker or cooperate. 'Linear' refers to a line. You may see many real-life examples of collinearity such as a group of students standing in a straight line, eggs in a carton are kept in a row, next to each other, etc.

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In this article, we will learn the collinear points definition and how to find collinear points.

Collinear Points Definition

In a given plane, three or more points that lie on the same straight line are called collinear points. Two points are always in a straight line. In geometry, the collinearity of a set of points is the property of the points lying on a single line. A set of points with this property is said to be collinear. In general, we can say that points are aligned in a line or a row.

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• Consider a straight line in the above cartesian plane formed by x-axis and y-axis.

• The three points A (2, 4), B (4, 6), and C (6, 8) are lying on the same straight line L.

• These three points are said to be collinear points.

How to Prove if Points are Collinear?

There are two methods to find whether the three points are collinear or not, they are: 

• One is the slope formula method and 

• The other is the area of the triangle method.

Slope Formula Method

Three points are collinear, if the slope of any two pairs of points is the same.

With three points R, S, and T, three pairs of points can be formed, they are: RS, ST, and RT.

If the slope of RS = slope of ST = slope of RT, then R, S, and T are collinear points.

Example

Prove that the three points R(2, 4), S (4, 6), and T(6, 8) are collinear.

Solution:

If the three points R (2, 4), S (4, 6), and T (6, 8) are collinear, then the slopes of any two pairs of points will be equal.

Use slope formula to find the slopes of the respective pairs of points:

Slope of RS = (6 – 4) / (4 – 2) = 1

Slope of ST = (8 – 6) / (6 – 4) = 1

Slope of RT = (8 – 4) / (6 – 2) = 1

Since slopes of any two pairs out of three pairs of points are the same, this proves that R, S, and T are collinear points.

Area of Triangle Method

Three points are collinear if the value of the area of the triangle formed by the three points is zero.

Substitute the coordinates of the given three points in the area of the triangle formula. If the result for the area of the triangle is zero, then the given points are said to be collinear.

Formula for area of a triangle formed by three points is

\[\frac{1}{2}\begin{bmatrix}x_{1}-x_{2} & x_{2}-x_{3}\\ y_{1}-y_{2}& y_{2}-y_{3}\end{bmatrix}\]

Let us substitute the coordinates of the above three points R, S, and T in the determinant formula above for the area of a triangle to check if the answer is zero.

\[\frac{1}{2}\begin{bmatrix}2-4 & 4-6\\ 4-6& 6-8\end{bmatrix} = \frac{1}{2}\begin{bmatrix}-2 & -2\\ -2&-2 \end{bmatrix} = \frac{1}{2}(4-4)= 0\]

Since the result for the area of the triangle is zero, therefore R (2, 4), S (4, 6), and T (6, 8) are collinear points.

Non-Collinear Points Definition

The set of points which do not lie on the same straight line are said to be non-collinear points. In the below figure, points X, Y, and Z do not make a straight line, so they are called the non-collinear points in a plane.

Examples of Non-collinear Points

Let's consider P, Q, and R are non-collinear points, draw lines joining these points.

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Number of lines through these three non-collinear points is 3.

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If P, Q, R, & S are non-collinear points then

Number of lines through these four non-collinear points is 6.

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So in general, we can say number of lines through “n” non-collinear points = \[\frac{n(n-1)}{2}\]

Solved Examples

Example 1: Identify the collinear and non-collinear point from the below figure.

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Solution: Points A, B, and C are collinear. And points D, E, and F are non-collinear points in a plane.

Example 2: Check whether the points (2, 5) (24, 7), and (12, 4) are collinear or not?

Solution: Using slope formula to solve this problem.

Let the points be A (2, 5), B (24, 7), and C (12, 4).

Slope of AB = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{7-5}{10-2} = \frac{2}{8} = \frac{1}{4}\]

Slope of BC = \[\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{4-7}{12-24} = \frac{-3}{-12} = \frac{1}{4}\]

As slope of AB = slope of BC. We can say that the given points A (2, 5), B (24, 7), and C (12, 4) are collinear.

Example 3: Given that the points P (8, 1), Q (15, 7), and R (x, 3) are collinear. Find x.

Solution: As given the points P, Q, and R are collinear, we have:

Slope of PQ = Slope of QR

\[\frac{7-1}{15-8} = \frac{3-7}{x-15}\]

\[\frac{6}{7} = \frac{-4}{x-15}\]

6 (x - 15) = - 4 * 7

6x - 90 = - 28

6x = 62

=> x = 31/3