Collinear Points-Definition, Formula, And Methods To Find Collinear Points

Collinear points are three or more points that lie on the same straight line. The word collinear is a compound term derived from two words: “co,” meaning togetherness, and “linear,” meaning a line. Collinear points geometry refers to the arrangement of these points on a single line.

These points may exist on different planes, but they cannot lie on different lines. Therefore, when points are collinear, it is referred to as collinearity. On the other hand, non collinear points are those that do not lie on a straight line. Understanding the collinear points formula and collinear lines is essential for deeper exploration of this concept.

What is Collinear Points Definition?

The collinear definition geometry refers to points that lie on the same straight line. The term collinear comes from the Latin words “col” (together) and “linear” (line), which gives the collinear points meaning as points that align along a single line.

So, what is collinear points? In simple terms, they are points that lie on the same straight line. According to the collinear points definition, two or more points are considered collinear if they fall on the same straight line.

To define collinear points, you can say that they are a set of points that lie along a single straight line. The collinear math definition emphasizes this geometric relationship, where points are collinear if they all share the same line.

What is Collinear Points in Math

In Mathematics, collinear points are points that lie on the same straight line. As you can see in the figure below, points P, Q, and R are collinear — a classic collinear points example.

Non-Collinear Points

A non collinear point is a point that doesn’t lie in a straight line. The definition of non collinear points in geometry states that these are points that do not share the same straight line. The non collinear points meaning refers to points positioned such that no single straight line can pass through all of them.

The non collinear points definition geometry helps distinguish them from collinear points, which lie on the same line. Examples of non collinear points in real life can be seen in scenarios like the corners of a triangle or scattered stars in the sky.

The example of non-collinear points is given below:

Collinear Points Formula

The collinear point’s formula is a mathematical way of determining whether three points are collinear, or in a straight line. There are various methods of calculating this – some more complex than others – but the collinear point’s formula is a reliable way to check whether three points are indeed in a straight line. There are three methods to find the collinear points. They are:

• Area of triangle
• Distance Formula
• Slope Formula

Area of triangle Formula

This method relies on the fact that three collinear points cannot form a triangle. This means that if any three points are collinear, they cannot form a triangle. To check the points of the triangle for collinearity, we use 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, a triangle formed by three collinear points will have no area since it would 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 as follows:

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

A=1/2|(x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂))|=0

Distance Formula

The three points are collinear if the sum of the distance between the first and second points plus the distance between the second and third points is equal to the distance between the first and third points.

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

Slope Formula

We can use the slope formula to find out whether or not the three points are collinear. If the three points have equal slopes, then they are collinear.

For example, if we have three points X, Y, and Z, they will be collinear only if the slope of line XY is equal to the slope of line YZ is equal to the 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=y₂−y₁/x₂−x₁

Summary

Three points are said to be collinear if they lie on the same straight line. This property is known as collinearity. Collinear points can exist on different planes.

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