Question

Determine whether the points \( A(1, -3), B(2, -5), C(-4, 7) \) are collinear.

Hint:

To check for collinearity, verify if the slopes of the line segments between the points are equal.

Answer 1

Step 1: To determine collinearity, check if the slope of \( AB \) is equal to the slope of \( BC \).
Step 2: Slope of \( AB \): \[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - (-3)}{2 - 1} = \frac{-5 + 3}{1} = -2. \] Step 3: Slope of \( BC \): \[ m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - (-5)}{-4 - 2} = \frac{7 + 5}{-6} = \frac{12}{-6} = -2. \] Step 4: Since \( m_{AB} = m_{BC} \), the points are collinear.