Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither.
$$\begin{array} { l }
L _ { 1 } : ( 7 , - 4 ) , ( - 9 , - 1 ) \
L _ { 2 } : ( 4 , - 6 ) , ( - 3,9 )
\end{array}$$
A) parallel
B) perpendicular
C) neither

Question

Determine whether lines $L _ { 1 }$ and $L _ { 2 }$ passing through the pairs of points are parallel, perpendicular, or neither.
$$\begin{array} { l } 
L _ { 1 } : ( 7 , - 4 ) , ( - 9 , - 1 ) \
L _ { 2 } : ( 4 , - 6 ) , ( - 3,9 )
\end{array}$$
A) parallel
B) perpendicular
C) neither

Quizplus · Accepted Answer

To determine if two lines are parallel, perpendicular, or neither, we need to calculate their slopes. The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

For $L_1$, we have:

$$m_{L_1} = \frac{-1 - (-4)}{-9 - 7} = \frac{3}{16}$$

For $L_2$, we have:

$$m_{L_2} = \frac{9 - (-6)}{-3 - 4} = -\frac{3}{7}$$

Two lines are parallel if and only if their slopes are equal. Two lines are perpendicular if and only if the product of their slopes is $-1$. If the slopes are neither equal nor opposite reciprocals, then the lines are neither parallel nor perpendicular.

In this case, the slopes are neither equal nor opposite reciprocals, so the lines are neither parallel nor perpendicular. Therefore, the answer is $\boxed{	ext{(C)}}$.

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