Question

verify whether the points (-2,3),(4,2)and (10,1)are collinear or not​

Answer 1

Answer:

Yes, the points (-2, 3), (4, 2) and (10, 1) are collinear.

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Step-by-step explanation:

Collinear points are those points that lie on the same line.

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For any three points to be collinear, the area formed on joining the three points must be equal to 0 sq.units as collinear points lie on the same line, and joining them will not result in an area being formed.

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For any three points (x₁, y₁), (x₂, y₂) and (x₃, y₃), the area formed by them is given by;

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$\boxed{\sf Area \ of \ the \ \Delta = \dfrac{1}{2}\bigg\{\sf x_1\left(y_2 - y_3\right) + x_2\left(y_3 - y_1\right) + x_3\left(y_1 - y_2\right)\bigg\}}$

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In order for these three points to be collinear, the area formed by them must be equal to 0, so we'll assume the Area of the Δgle to be equal to 0.

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Here;

• x₁ = -2

• x₂ = 4

• x₃ = 10

• y₁ = 3

• y₂ = 2

• y₃ = 1

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$\sf \dashrightarrow \ Area \ of \ the \ \Delta = \dfrac{1}{2}\bigg\{\sf x_1\left(y_2 - y_3\right) + x_2\left(y_3 - y_1\right) + x_3\left(y_1 - y_2\right)\bigg\}$

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$\sf \dashrightarrow \ 0 = \dfrac{1}{2}\bigg\{\sf -2\left(2 - 1\right) + 4\left(1 - 3\right) + 10\left(3 - 2\right)\bigg\}$

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$\dashrightarrow \ \sf 0 = \dfrac{1}{2}\bigg\{\sf -2\left(1\right) + 4\left(-2\right) + 10\left(1\right)\bigg\}$

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$\dashrightarrow \ \sf 0 = \dfrac{1}{2}\bigg\{\sf -2 - 8 + 10\bigg\}$

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$\dashrightarrow \ \sf 0 = \dfrac{1}{2}\bigg\{\sf -10 + 10\bigg\}$

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$\dashrightarrow \ \sf 0 = \dfrac{1}{2}\bigg\{0\bigg\}$

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$\dashrightarrow \ \sf 0 = 0$

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$\dashrightarrow \ \sf LHS = RHS$

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Since the area formed by the three points is 0, the given set of points are collinear.

[Attached a screenshot of the three points plotted on a graph]