Plot the following points and check whether they are collinear or not:
(i) (1, 3), (-1, -1), (-2, -3)
(ii) (1, 1), (2, -3), (-1, -2)
(iii) (0, 0), (2, 2), (5, 5)
Answer (1)
To determine if the given points are collinear, we can check if the slopes between each pair of points are the same. If they are, the points are collinear.
Let's go through each set of points:
(i) Points: ( 1 , 3 ) , ( − 1 , − 1 ) , ( − 2 , − 3 )
First, we calculate the slope between the first two points ( 1 , 3 ) and ( − 1 , − 1 ) :
m 1 = − 1 − 1 − 1 − 3 = − 2 − 4 = 2
Now, let's calculate the slope between the second point ( − 1 , − 1 ) and the third point ( − 2 , − 3 ) :
m 2 = − 2 − ( − 1 ) − 3 − ( − 1 ) = − 2 + 1 − 3 + 1 = − 1 − 2 = 2
Since both slopes, m 1 and m 2 , are equal, these points are collinear.
(ii) Points: ( 1 , 1 ) , ( 2 , − 3 ) , ( − 1 , − 2 )
First, calculate the slope between ( 1 , 1 ) and ( 2 , − 3 ) :
m 1 = 2 − 1 − 3 − 1 = 1 − 4 = − 4
Next, calculate the slope between ( 2 , − 3 ) and ( − 1 , − 2 ) :
m 2 = − 1 − 2 − 2 − ( − 3 ) = − 1 − 2 − 2 + 3 = − 3 1 = − 3 1
Since m 1 = m 2 , these points are not collinear.
(iii) Points: ( 0 , 0 ) , ( 2 , 2 ) , ( 5 , 5 )
Calculate the slope between ( 0 , 0 ) and ( 2 , 2 ) :
m 1 = 2 − 0 2 − 0 = 2 2 = 1
Calculate the slope between ( 2 , 2 ) and ( 5 , 5 ) :
m 2 = 5 − 2 5 − 2 = 3 3 = 1
Since both slopes, m 1 and m 2 , are equal, these points are collinear.
In summary:
Points (1, 3), (-1, -1), (-2, -3) are collinear.
Points (1, 1), (2, -3), (-1, -2) are not collinear.
Points (0, 0), (2, 2), (5, 5) are collinear.
