Let A, B and C be the three points.
We have to find the three lengths AB, BC and AC among the given three points A, B and C.
The three points A, B and C are collinear, if the sum of the lengths of any two line segments among AB, BC and AC is equal to the length of the remaining line segment.
That is,
AB + BC = AC
(or)
AB + AC = BC
(or)
AC + BC = AB
Example :
Using the concept of distance between two points, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.
Solution :
We know the distance between the two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is
d = √[(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]
Let us find the lengths AB, BC and AC using the above distance formula.
AB = √[(4 - 5)^{2} + (-1 + 2)^{2}]
AB = √[(-1)^{2} + (1)^{2}]
AB = √[1 + 1]
AB = √2
BC = √[(1 - 4)^{2} + (2 + 1)^{2}]
BC = √[(-3)^{2} + (3)^{2}]
BC = √[9 + 9]
BC = √18
BC = 3√2
AC = √ [(1 - 5)^{2} + (2 + 2)^{2}]
AC = √ [(-4)^{2} + (4)^{2}]
AC = √ [16 + 16]
AC = √32
AC = 4√2
Therefore, AB + BC = √2 + 3√2 = 4√2 = AC
Thus, AB + BC = AC
Hence, the given three points A, B and C are collinear.
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