Slope and Equations of Lines
Example 1
Determine whether $\overset{\leftrightarrow}{A B}$ and $\overset{\leftrightarrow}{C D}$ 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. $A (- 4, - 4), B (- 6, - 6), C (10, - 8), D (- 2, 8)$

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Answer

$m_{A B_{4}} = \frac{- 6 - (- 4)}{- 6 - (- 4)} = \frac{- 2}{- 2} = 1$ , $m_{C D_{4}} = \frac{8 - (- 8)}{- 2 - 10} = \frac{16}{- 12} = - \frac{4}{3}$

Steps

Step 1 ： $m_{A B} = \frac{4 - 2}{4 - (- 2)} = \frac{2}{6} = \frac{1}{3}$ , $m_{C D} = \frac{- 2 - 4}{1 - (- 1)} = \frac{- 6}{2} = - 3$

Step 2 ： $m_{A B_{2}} = \frac{- 2 - 8}{- 4 - (- 6)} = \frac{- 10}{2} = - 5$ , $m_{C D_{2}} = \frac{- 6 - 4}{8 - 6} = \frac{- 10}{2} = - 5$

Step 3 ： $m_{A B_{3}} = \frac{- 4 - (- 3)}{1 - (- 1)} = \frac{- 1}{2}$ , $m_{C D_{3}} = \frac{- 6 - (- 2)}{2 - 4} = \frac{- 4}{- 2} = 2$

Step 4 ： $m_{A B_{4}} = \frac{- 6 - (- 4)}{- 6 - (- 4)} = \frac{- 2}{- 2} = 1$ , $m_{C D_{4}} = \frac{8 - (- 8)}{- 2 - 10} = \frac{16}{- 12} = - \frac{4}{3}$

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