Question

Show that the following sets of points are collinear. 

 (1, −1), (2, 1) and (4, 5)

Answer 1

The formula for the area ‘A’ encompassed by three points, (x₁,y₁) (x₂,y₂)and (x₃,y₃) is given by the formula,

We know area o triangle formed by three points (x₁,y₁),(x_2,y₂),(x₃,y₃) is given by  

`Δ=1/2[x_1(y_2-y_3)+x_2 (y_3-y_1)+x_3(y_1-y_2)] ` 

The three given points are A(1, −1), B(2, 1) and C(4, 5). Substituting these values in the earlier mentioned formula we have,  

A`=1/2[1(1-5)+2(5+1)+4(-1-1)]` 

`=1/2[1(-4)+2(6)+4(-2)]` 

`=1/2[-4+12+8]`

`=1/2[-12+12]` 

`=0`

Since the area enclosed by the three points is equal to 0, the three points need to be collinear.