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How To Determine If Points Are Collinear In Coordinate Geometry?

R
Ravi Kota
24/06/2017 0 1

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.

 

 

 

 

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Pragya Mishra | 08 Mar

nice

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