Collinear Points Free Online Calculator - Free Mathematics Tutorials

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.

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 satisfy the following equality \[ m_{AB} = m_{BC} \]

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 \( A: \quad \) \( x_A \) = 0 , \( y_A \) = 3 Point \( B: \quad \) \( x_B \) = 1.5 , \( y_B \) = 0 Point \( C: \quad \) \( x_C \) = 3 , \( y_C \) = -3 Decimal Places = 2

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.