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.
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)
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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