Determine if AB and CD are parallel, perpendicular or neither if: A(-1,3),B(0,5),C(2,1), and D(6,-1)

To determine if the lines AB and CD are parallel, perpendicular or neither, we need to find the slopes of AB and CD.

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

m = \frac{y_2 - y_1}{x_2 - x_1}

First, calculate the slope of AB:

A(-1,3), B(0,5)

m_{AB} = \frac{5 - 3}{0 + 1} = \frac{2}{1} = 2

Next, calculate the slope of CD:

C(2,1), D(6,-1)

m_{CD} = \frac{-1 - 1}{6 - 2} = \frac{-2}{4} = -\frac{1}{2}

To check if they are parallel, we compare their slopes. Lines are parallel if their slopes are equal:

m_{AB} \neq m_{CD}

To check if they are perpendicular, we multiply their slopes. Lines are perpendicular if the product of their slopes is -1:

m_{AB} \cdot m_{CD} = 2 \cdot -\frac{1}{2} = -1

Since the product is -1, the lines AB and CD are perpendicular.