CONDITION FOR COLLINEARITY OF THREE POINTS

Example 1 :

Using the concept of area of triangle, show that the points (5, -2), (4, -1) and (1, 2) are collinear.

Solution :

Using the concept of area of triangle, if the three points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, then 

x₁y₂ + x₂y₃ + x₃y₁ = x₂y₁ + x₃y₂ + x₁y₃

Here,

(x₁, y₁)  =  (5, -2) 

(x₂, y₂)  =  (4, -1)

(x₃, y₃)  =  (1, 2)

x₁y₂ + x₂y₃ + x₃y₁  =  5(-1) + 4(2) + 1(-2)

=  -5 + 8 -2

=  1 ----(1)

x₂y₁ + x₃y₂ + x₁y₃  =  4(-2) + 1(-1) + 5(2)

=  -8 - 1 + 10

=  1 ----(2)

From (1) and (2), 

x₁y₂ + x₂y₃ + x₃y₁ = x₂y₁ + x₃y₂ + x₁y₃

Hence, the three points A, B and C are collinear.

Example 2 :

Using the concept of slope of a line, show that the following points are collinear.

(5, -2), (4, -1), (1, 2)

Solution :

Let A(5, -2), B(4, -1) and C(1, 2).

Slope of the line joining (x₁, y₁) and (x₂, y₂) is,

Slope of the line joining the points A(5, - 2) and B(4, -1) substitute (x₁, y₁) = (5, -2) and (x₂, y₂) = (4, -1) into the above formula.

= ⁽⁻¹ ⁺ ²⁾⁄₍₄ ₋ ₅₎

= -1

Slope of the line joining the points B(4, -1) and C(1, 2) substitute (x₁, y₁) = (4, -1) and (x₂, y₂) = (1, 2) into the above formula.

 = ⁽² ⁺ ¹⁾⁄₍₁ ₋ ₄₎

= -1

slope of AB = slope of BC

And also, B is the common point.

Hence, the points A, B and C are collinear.

Example 3 :

Using the concept of distance between two points, show that the points (5, -2), (4, -1) and (1, 2) are collinear.

Solution :

Let A(5, -2), B(4, -1) and C(1, 2).

Distance between two points (x₁, y₁) and (x₂, y₂) :

Distance between the two points A(5, - 2) and B(4, -1) : 

AB = √[(4 - 5)² + (-1 + 2)²]

= √[(-1)² + (1)²]

= √[1 + 1]

= √2

Distance between B and C :

BC = √[(1 - 4)² + (2 + 1)²]

= √[(-3)² + (3)²]

= √[9 + 9]

= √18

= 3√2

Distance between A and C :

AC = √[(1 - 5)² + (2 + 2)²]

= √[(-4)²  + (4)²]

= √[16 + 16]

= √[32]

= 4√2

AB + BC = √2 + 3√2

= 4√2

= AC

AB + BC = √2 + 3√2

AB + BC = 4√2

AB + BC = AC

Hence, the three points A, B and C are collinear.

Example 4 :

Using the concept of equation of line, show that the points (5, -2), (4, -1) and (1, 2) are collinear.

Solution :

Equation of the straight line in two-points form :

Find the equation of the line through the points

(5, -2) and (4, -1)

Substitute (x₁, y₁) = (5, -2) and (x₂, y₂) = (4, -1).

⁽ʸ ⁺ ²⁾⁄₍₋₁ ₊ ₂₎ = ⁽ˣ ⁻ ⁵⁾⁄₍₄ ₋ ₅₎

⁽ʸ ⁺ ²⁾⁄₁ = ⁽ˣ ⁻ ⁵⁾⁄₍₋₁₎

y + 2 = -x + 5

y = -x + 3

Substitute the third point (1, 2).

2 = -1 + 3

2 = 2

1 + 2 - 3 = 0

0 = 0

Therefore, the third point (1, 2) satisfies the equation. 

Hence, the given three points are collinear.

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