How to determine if points are collinear using distance formula :
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.
AB + BC = AC
AB + AC = BC
AC + BC = AB
Example 1 :
Using the concept of distance between two points, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.
We know the distance between the two points (x₁, y₁) and (x₂, y₂) is
d = √ (x₂ - x₁)² + (y₂ - y₁)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 = √[2X9]
BC = 3√2
AC = √ [(1 - 5)² + (2 + 2)²]
AC = √ [(-4)² + (4)²]
AC = √ [16 + 16]
AC = √[2x16]
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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