HOW TO PROVE IF THE GIVEN POINTS ARE COLLINEAR USING SLOPE

The points which lie on the same line are known as collinear points.

In case we have three point and we need to prove that the given points are collinear, we may follow the steps given below.

Working Rule

Step 1 :

Choose the points A and B and find the slope.

Step 2 :

Choose the points B and C and find the slope.

Step 3 :

If the slopes found in step 1 and step 2 are equal, then the points A, B and C are collinear.

 m = (y2 - y1)/(x2-x1)

Examples

Example 1 :

Using the concept of slope, show that each of the following set of points are collinear.

(2 , 3), (3 , -1) and (4 , -5)

Solution :

Formula to find the slope of the line passing through two points

 m = (y2 - y1)/(x2-x1)

Step 1 :

Slope of AB :

(x1, y1)  ==>  (2, 3) and (x2, y2)  ==> (3, -1)

m  =  (-1 - 3) / (3 - 2)

=  (-4 )/1

=  -4

Step 2 :

Slope of BC :

(x1, y1)  ==>  (3 , -1) and (x2, y2)  ==> (4, -5)

m  =  (-5 - (-1)) / (4 - 3)

=  (-5 + 1 )/1

=  -4

Step 3 :

Slope of AB  =  Slope of BC

So, the given points are collinear.

Example 2 :

Using the concept of slope, show that each of the following set of points are collinear.

(4 , 4), (-2 , 6) and (1 , 5)

Solution :

Formula to find the slope of the line passing through two points

 m = (y2 - y1)/(x2-x1)

Step 1 :

Slope of AB :

(x1, y1)  ==> (4 , 4) and (x2, y2)  ==> (-2 , 6)

m  =  (6 - 4) / (-2 - 4)

=  2/(-6)

=  -1/3

Step 2 :

Slope of BC :

(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

So, the given points are collinear.

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