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).
(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)
= 
|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|
Area of ∆ABC
= 
|2(6 – 8) + 4(8 -5) + 8(5 – 6)|
= 
|-4 + 12 - 8|
= 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)
= 
|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|
Area of ∆ABC
= 
|1(1 – 5) + 2(5 + 1) + 4(-1 – 1)|
= 
|-4 + 12 - 8|
= 0 sq. units
We know that if area enclosed by three points is zero, then points are collinear.
Hence, given three points are collinear.
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