Slope and Equations of Lines
Example 1
Determine whether \( \overleftrightarrow{A B} \) and \( \overleftrightarrow{C D} \) are parallel, perpendicular, or neither. Graph each line to verify your answer.
1. \( A(-2,2), B(4,4), C(-1,4), D(1,-2) \)
2. \( A(-6,8), B(-4,-2), C(6,4), D(8,-6) \)
3. \( A(-1,-3), B(1,-4), C(4,-2), D(2,-6) \)
4. \( A(-4,-4), B(-6,-6), C(10,-8), D(-2,8) \)

Answer

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Answer

\(m_{AB_4} = \frac{-6 - (-4)}{-6 - (-4)} = \frac{-2}{-2} = 1\), \(m_{CD_4} = \frac{8 - (-8)}{-2 - 10} = \frac{16}{-12} = -\frac{4}{3}\)

Steps

Step 1 ：\(m_{AB} = \frac{4 - 2}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}\), \(m_{CD} = \frac{-2 - 4}{1 - (-1)} = \frac{-6}{2} = -3\)

Step 2 ：\(m_{AB_2} = \frac{-2 - 8}{-4 - (-6)} = \frac{-10}{2} = -5\), \(m_{CD_2} = \frac{-6 - 4}{8 - 6} = \frac{-10}{2} = -5\)

Step 3 ：\(m_{AB_3} = \frac{-4 - (-3)}{1 - (-1)} = \frac{-1}{2}\), \(m_{CD_3} = \frac{-6 - (-2)}{2 - 4} = \frac{-4}{-2} = 2\)

Step 4 ：\(m_{AB_4} = \frac{-6 - (-4)}{-6 - (-4)} = \frac{-2}{-2} = 1\), \(m_{CD_4} = \frac{8 - (-8)}{-2 - 10} = \frac{16}{-12} = -\frac{4}{3}\)