Coordinate Geometry-17

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Examine the set of points (1/2, -2), (-2, 1/3), (5/2, 0) for collinearity.

Solution: If three points are collinear, then the area of the triangle formed by the points is equal to zero.

Let (x₁, y₁) = (1/2, -2); (x₂, y₂) = (-2, 1/3); (x₁, y₁) = (5/2, 0)

Area of Triangle = ± 1/2 [ x₁ (y₂ - y₃) + x₂ (y₃ - y₁) +

x₃ (y₁ - y₂) ]

= ± 1/2 [1/2 (1/3 - 0) - 2(0 + 2) + 5/2 (-2 - 1/3)] = + / - 1/2 [- 58 / 6] = 29 / 6

Therefore, the three points are not collinear.

Note If (1/2, -2), (-2, 1/3), (5/2, 0) be denoted by A, B. C.

We can also show that the sum of any line segments is not equal to the third segment.