Question

Determine whether the following points are collinear.

i. A (-1, -1), B (0, 1), C (1, 3) 

ii. D (- 2, -3), E (1, 0), F (2, 1)

Answer 1

i. Slope of line AB = (y₂ - y₁)/(x₂ - x₁)

= (1-(-1))/(0 - (-1)) = (1 + 1)/(0 + 1) = 2

Slope of line BC = (y₂ - y₁)/(x₂ - x₁)

= (3-1)/(1 - 0) = 2

∴ slope of line AB = slope of line BC 

∴ line AB || line BC 

Also, point B is common to both the lines. 

∴ Both lines are the same. 

∴ Points A, B and C are collinear.

ii. Slope of line DE = (y₂ - y₁)/(x₂ - x₁)

= (0 - (-3))/(1-(-2))

= (0 + 3)/(1 + 2) = 3/3 = 1

Slope of line EF = (y₂ - y₁)/(x₂ - x₁)

= (1 - 0)/(2 - 1) = 1

∴ slope of line DE = slope of line EF 

∴ line DE || line EF 

Also, point E is common to both the lines. 

∴ Both lines are the same. 

∴ Points D, E and F are collinear