Collinear Points: Formulas, Definition, Sample Questions

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

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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Important Concepts Related to Collinear Points
Distance between Two Points	Horizontal and Vertical Lines	Lines and Angles
Vertex	Transversal	Properties of Parallel Lines
Angle Formula	Obtuse Angle	Linear Pair of Angles

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

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.

Non-Linear Points

Also Read: Corresponding Angles Axiom

Solved Examples Ques: Prove that points A(5, -2), B(4, -1) and C(1, 2) are collinear points using the Distance Method. Solution : Distance between any two points (x1, y1) and (x2, y2) is  d = √[(x2 - x1)2 + (y2 - y1)2] To find the lengths AB, BC and AC using the formula, AB = √[(4 - 5)2 + (-1 + 2)2] AB = √[(-1)2 + (1)2] AB = √[1 + 1] AB = √2 BC = √[(1 - 4)2 + (2 + 1)2] BC = √[(-3)2 + (3)2] BC = √[9 + 9] BC = √18 BC = 3√2 AC = √[(1 - 5)2 + (2 + 2)2] AC = √[(-4)2 + (4)2] AC = √[16 + 16] AC = √32 AC = 4√2 Therefore, AB + BC = √2 + 3√2 = 4√2 = AC Thus, AB + BC = AC This proves that given three points A, B, and C are collinear. Ques: Prove that the given three points (4, 4), (-2, 6), and (1, 5) are collinear points using Slope Formula Method. Solution: Formula: m = (y2 - y1)/(x2-x1) Step 1 : AB’s Slope : (x1, y1) ==> (4 , 4) and (x2, y2) ==> (-2 , 6) m = (6 - 4) / (-2 - 4) = 2/(-6) = -1/3 Step 2 : BC’s Slope : (x1, y1) ==> (-2, 6) and (x2, y2) ==> (1, 5) m = (5 - 6) / (1 - (-2)) = (-1 )/(1 + 2) = -1/3 Step 3 : Slope of ‘AB’ = Slope of ‘BC’ Hence, the given points are collinear

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.

Also Read: 

Important Concepts Related to Lines & Angles
Types of Vectors	Pascal’s Triangle	Angle between Line and Plane
Vectors	Angles between Two Lines	Equation of Plane
Addition of Vectors	Angles between Two Planes	Section Formula in Coordinate Geometry
Hyperbola	Distance Formula and Derivation of Coordinate Geometry	Standard Equation of a Circle
Conic Sections Parabola	Parabola	Latus Rectum of Hyperbola