Question

Show that the following sets of points are collinear. 

(a) (2, 5), (4, 6) and (8, 8) 

(b) (1, -1), (2, 1) and (4, 5).

Answer 1

(a) Let three given points be A(2, 5), B(4, 6) and C(8, 8). 

Area of the triangle having vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃) 

= \(\frac{1}2\) |x₁(y₂ - y₃)+x₂(y₃ - y₁)+x₃(y₁ - y₂)| 

Area of ∆ABC 

= \(\frac{1}2\) |2(6 – 8) + 4(8 -5) + 8(5 – 6)| 

= \(\frac{1}2\) |-4 + 12 - 8| 

= \(\frac{1}2\) 0 sq. units 

We know that if area enclosed by three points is zero, then points are collinear. 

Hence, 

given three points are collinear. 

(b) Let three given points be A(1, −1), B(2, 1) and C(4, 5) 

Area of the triangle having vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃) 

= \(\frac{1}2\) |x₁(y₂ - y₃)+x₂(y₃ - y₁)+x₃(y₁ - y₂)| 

Area of ∆ABC 

= \(\frac{1}2\) |1(1 – 5) + 2(5 + 1) + 4(-1 – 1)| 

= \(\frac{1}2\) |-4 + 12 - 8| 

= \(\frac{1}2\) 0 sq. units 

We know that if area enclosed by three points is zero, then points are collinear. 

Hence, 

given three points are collinear.