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Question
Show that the following sets of points are collinear.
(1, −1), (2, 1) and (4, 5)
Solution
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),(x2,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.
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