Collinear and Non-collinear Points

Three points A, B, and C are said to be collinear if they lie on the same straight line. If the points do not lie on a straight line then the points are said to be Non-collinear points. Three Non-collinear points can form a triangle. Here, in the figure 1, A, B, and C are collinear as they lie in a straight line,where as points A, B, and D are non collinear.

collinear

Important Points about Non-collinear points:

The points that do not lie on the same line are called non-collinear points.
Three non-collinear points form a triangle.

Test for Co linearity:

To show if the three points A, B, and C are collinear or Non-collinear we find the distance between the points Ab, BC, and CA. If  the sum of any two distance of these three points equals the third distance then the points A, B, and C are said to be collinear otherwise if there is no relation between the distance AB, BC, and CA, we say that the points A,B, and C are non-collinear.

To find the distance between the points A(a,b) and B(c,d) we use the formula.,

AB =sqrt((c-a)^2+(d-b)^2)

Solved Example on Collinear Points and Non-collinear points.

Problem1:Identify the three collinear points from the diagram.

three collinear points

options:
a. A, B, and C
b. B, D, and e
c. C, B, and D
d. none of these


Correct Answer: A
Solution:
Step 1: If the points lie on the same line, then the points are called collinear points.
Step 2: In the figure, A, B, and C lie on the same line.
Step 3: So, A, B, and C are the three collinear points.

Similarly, points BDE and CBD are Non-collinear points

Problem2: check if the given three points A(1,1), B(2,3),and C(3,1) are collinear or non collinear?

Solution:

To check if the given three points A(1,1), B(2,3), and C(3,1) are collinear or non-collinear, we need to find the distance AB,BC and CA.

Therefore,

AB=sqrt((2-1)^2+(3-1)^2)

AB=sqrt(1^2+2^2)

AB=sqrt(1+4)



ABsqrt(5)

BC=sqrt((3-2)^2+(1-3)^2)

BC=sqrt(1^2+(-2)^2)

BC=sqrt(1+4)

BC=sqrt(5)

CA=sqrt((1-3)^2+(1-1)^2)

CA=sqrt((-2)^2+0)

CA=sqrt(4)

Since, the sum of distance AB and BC which is 2sqrt(5), has no relation to the distance of CA which is sqrt(4)

Therefore, the points A(1,1),B(2,3), and C(3,1) are non-collinear.

My forthcoming post is on what is a system of linear equations and sample paper of class 9th cbse will give you more understanding about Algebra.

Practice problems on collinear and non collinear points:

Q1: Show if the points A(5,1), B(8,5) and C( 6,3) are collinear or non-collinear?

(Answer: Distance: AB=5, BC=sqrt(8),and CA=sqrt(5). Therefore, points A, B, and C are Non-collinear points ).

Q2: Show if the following points are Collinear or Non-collinear A(1,1), B(-2,7), and C(3, -3)

(Answer: Distance: AB= sqrt(5), BC=sqrt(5), and CA=sqrt(5). Therefore, points  A, B, and C are collinear points. Since, AB + CA = BC ).