How to prove if points are Collinear in coordinate geometry?

Collinear points definition:

- Three or more points that lie on a same straight line are called collinear points.
- Consider a straight line L in the above Cartesian coordinate plane formed by x axis and y axis.
- This straight line L is passing through three points A, B and C whose coordinates are (2, 4), (4, 6) and (6, 8) respectively.
- {We may also say, alternatively, that the three points A (2, 4), B (4, 6) and C (6, 8)are lying on a same straight line L}
- Three or more points which lie on a same straight line are called collinear points.

How to find if three points are collinear?:

- There are two methods to find if three points are collinear.
- One is slope formula method and the other is area of triangle method.
- Slope formula method to find that points are collinear.
- Three or more points are collinear, if slope of any two pairs of points is same.
- With three points A, B and C, three pairs of points can be formed, they are:
*AB, BC and AC.* - If Slope of AB = slope of BC = slope of AC, then A, B and C are
*collinear points.*

Example

Show that the three points A (2, 4), B (4, 6) and C (6, 8) are collinear.

Solution:

- If the three points A (2, 4), B (4, 6) and C (6, 8) are collinear, then
- slopes of any two pairs of points will be equal.
- Now, apply slope formula to find the slopes of the respective pairs of points:
- Slope of AB = (6 – 4)/ (4 – 2) = 1,
- Slope of BC = (8 – 6)/ (6 – 4) = 1, and
- Slope of AC = (8 – 4) /(6 – 2) = 1
- Since slopes of any two pairs out of three pairs of points are same, this proves that A, B and C are collinear points.
- Area of triangle to find if three points are collinear.
- Three points are collinear if the value of area of triangle formed by the three points is
*zero.* - Apply the coordinates of the given three points in the area of triangle formula. If the result for area is zero, then the given points are said to be collinear.
- First of all, recall the formula for area of a triangle formed by three points.

It is

In the formula above, the two vertical bars enclosing the variables represent a determinant.

Let us apply the coordinates of the above three points A, B and C in the determinant formula above for area of a triangle to check if the answer is zero.

Since the result for area of triangle is zero, therefore A (2, 4), B (4, 6) and C (6, 8) are collinear points.