Collinear Points Free Online Calculator

More that two points are collinear if they are on the same line.
Given three points \( A \), \( B \) and \( C \), an online calculator to calculate the slopes of the line through \( A \) and \( B \), and the line through \( B \) and \( C \) and hence decide whether the three points are collinear or not.

Collinear points

Formulas Used in Calculator

The three points \( A(x_A,y_A) \), \( B(x_B,y_B)\) and \( C(x_C,y_C) \) are collinear if the slopes of the lines through any two points are equal.

The slope \( m_{AB} \) of line through \( A \) and \( B \) is given by
   \( m_{AB} = \dfrac{y_B - y_A}{x_B-x_A} \)

The slope \( m_{AC} \) of line through \( A \) and \( C \) is given by
   \( m_{AC} = \dfrac{y_C - y_A}{x_C-x_A} \)

The slope \( m_{BC} \) of line through \( B \) and \( C \) is given by
   \( m_{BC} = \dfrac{y_C - y_B}{x_C-x_B} \)

The equation of the line through the points \( A \) and \( B \) may be written as
\( y - y_B = m_{AB}(x - x_B ) \)

For point \( C \) to be on the line through the points \( A \) and \( B \), the following equation may be satisfied
\( y_C - y_B = m_{AB}(x_C - x_B ) \)

which may be written as
\( m_{AB} = \dfrac{y_C - y_B}{x_C - x_B} = m_{BC}\)

Conclusion: For the three points to be collinear, we need to satify the following
\( m_{AB} = m_{BC} \)


Example
Are the points \( A(-1,5) \) , \( B(1,1) \) and \( C(3,-3) \) collinear?
Solution
The slope \( m_{AB} \) of line through \( A \) and \( B \) is given by
   \( m_{AB} = \dfrac{y_B - y_A}{x_B-x_A} = \dfrac{1 - 5}{1-(-1)} = - 2 \)

The slope \( m_{BC} \) of line through \( B \) and \( C \) is given by
   \( m_{BC} = \dfrac{y_C - y_B}{x_C-x_B} =\dfrac{-3 - 1}{3-1} = -2 \)
Hence \( m_{BC} = m_{AB} \) and therefore the three points are collinear


Use of Online Calculator to Verify that three Given Points Are Collinear

Enter the coordintes of the three points \( A \),\( B\) and \( C \) as real numbers and press "Calculate".
The results are: the slopes \( m_{AB} \) and \( m_{BC} \) and the conclusion whether the three points are collinear or not.


Point \( A: \quad \) \( x_A \) = , \( y_A \) =
Point \( B: \quad \) \( x_B \) = , \( y_B \) =
Point \( C: \quad \) \( x_C \) = , \( y_C \) =
Decimal Places =

Results


    

    


Activities

Use the calculator to find slopes \( m_{AB} \), \( m_{BC} \) and verify that the three points are collinear. Then calculate the slopes \( m_{AB} \), \( m_{BC} \) and \( m_{AC} \) analytcally and verify that they are all equal.
a) \( A(-5,-2) \),   \( B(-2,1) \) ,   \( C(2,5) \).
b) \( A(-5,7) \),   \( B(-1,-1) \) ,   \( C(1,-5) \).
c) \( A(0,3) \),   \( B(2,2) \) ,   \( C(6,0) \).


More References and Links

Slope
General Equation of a Line: ax + by = c .
Equations of Lines in Different Forms .
Online Geometry Calculators and Solvers .

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