Collinear Points: Definition, Formula & Solved Examples

Collinear points are three or more points that lie on the same straight line. The word ‘Collinear’ is a compound word that is made of two words: ‘co’ meaning togetherness and ‘linear’ meaning a line. Collinear points might exist on different planes but not on different lines. Since the points are collinear, it is known as collinearity. Non-collinear points are points which do not lie on a straight line. 

Collinear points applications in real life include, students standing in a straight line in an assembly, cars parked on a straight line in the parking lot or a set of balls kept on a straight line. Collinear Points are important in solving real life problems from Euclidean Geometry. Only one straight line can be drawn from a pair of say 3 collinear points. The collinear point formula can be estimated using method of slope, area of triangle or the distance formula. 

Read More:  NCERT Solutions Class 9 Maths

Key Terms: Collinear, Collinearity, Straight Lines, Points, Geometry, Collinear Point Formula, Slope, Area of Triangle, Distance, Non-Collinear

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Collinear Points

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When three or more points lie on the same straight line on a given plane, they are known as Collinear points. These points are aligned in a row or a line. Real-life collinear points include people standing in queue, and bottles kept in a row in the fridge.

Collinear Points

The video below explains this:

Collinear Point Formulae Detailed Video Explanation:

Also Read: Intersecting & Non-Intersecting Lines

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Collinear Points in Maths

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In Mathematics, Collinear Points are points that are positioned on the same straight line. In the figure below, points (A, B, C,D, E) and points (I, H, C, F, G) are collinear.

Collinear Points in Maths

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Formulas of Collinear Points

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There are three basic ways of finding if three points are collinear or not.

Slope Formula Method

If the slope of any two pairs of points is the same, then the three points are definitely on the same line and they are collinear.

For example, Imagine there are three points- A, B, and C. So the pairs are AB, BC, and AC.

So, if Slope of AB= Slope of BC, then A, B, and C are collinear points.

Slope Formula Method

Also Read: Three Dimensional Geometry

Area of the Triangle Method

Another method to prove whether the points are collinear points or not is by finding the area of the triangle formed by the three points. If the area is 0, the points are collinear.

Area of the Triangle Method

For Example, The three points A(x1, y1), B(x2, y2), and C(x3, y3) are collinear, then by remembering the formula of area of the triangle formed by three points we get;

(1/2) | [x1(y2 – y3) + x2(y3 – y1) + x3[y1 – y2]| = 0

Also Read: Lines and Angles MCQs

Distance Method

If the distance between the 1st point and 2nd point added to the distance between 2nd point and 3rd point is equal to the distance between 1st and 3rd point, then all the three points are collinear.

For example, If we take A, B, and C as any three collinear points, then,

Distance from A to B + Distance from B to C = Distance from A to C

So, A, B and C are collinear points

Now, by the distance formula we know, the distance between two points (x1, y1) and (x2, y2) is given by;

D=√(x2−x1)2+(y2−y1)2

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Non-Collinear Points

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Points that are not on the same line and through which a straight line can never be formed, are known as non-collinear points.

For Example, Let consider the following figure, we cannot draw lines combining the points P, Q, R, S & T. Thus, they are non-collinear points.

Also Read: Corresponding Angles Axiom

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Things to Remember

• Three or more points lying on the same straight line are known as the collinear points.
• Points that are positioned at non-linear positions on which a straight line cannot be formed are known as non-linear points.
• There are three main ways of finding whether points are collinear or not.
• If three points are collinear, the slopes formed from any two points are the same as the slope formed by the other two.
• The area of the triangle formed by any three collinear points will always be zero.

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