More than two points are collinear if they lie on the same straight line. This calculator checks whether three given points \( A(x_A,y_A) \), \( B(x_B,y_B) \) and \( C(x_C,y_C) \) are collinear by comparing the slopes of lines \( AB \) and \( BC \).
More than two points are collinear if they lie on the same straight line. This calculator checks whether three given points \( A(x_A,y_A) \), \( B(x_B,y_B) \) and \( C(x_C,y_C) \) are collinear by comparing the slopes of lines \( AB \) and \( BC \).
The three points \( A(x_A,y_A) \), \( B(x_B,y_B) \) and \( C(x_C,y_C) \) are collinear if the slopes through any two pairs of points are equal.
| Slope of line through \( A \) and \( B \): | \[ m_{AB} = \frac{y_B - y_A}{x_B - x_A} \] |
| Slope of line through \( B \) and \( C \): | \[ m_{BC} = \frac{y_C - y_B}{x_C - x_B} \] |
| Condition for collinearity: | \[ m_{AB} = m_{BC} \] |
Note: For vertical lines (where \( x_B = x_A \) or \( x_C = x_B \)), the slope is undefined. Special handling is applied in such cases.
Problem: Are the points \( A(-1,5) \), \( B(1,1) \) and \( C(3,-3) \) collinear?
Solution:
Since \( m_{AB} = m_{BC} = -2 \), the three points are collinear.
Enter the coordinates of points A, B, and C to determine if they are collinear
Activity 1: Verify that the following three points are collinear by calculating slopes \( m_{AB} \) and \( m_{BC} \).
a) \( A(-5,-2) \), \( B(-2,1) \), \( C(2,5) \)
b) \( A(-5,7) \), \( B(-1,-1) \), \( C(1,-5) \)
c) \( A(0,3) \), \( B(2,2) \), \( C(6,0) \)
Activity 2: Find the value of \( k \) such that points \( A(1,2) \), \( B(3,4) \), and \( C(5,k) \) are collinear.
Hint: Use the condition \( m_{AB} = m_{BC} \)