Question

Determine if JK and LM are parallel, perpendicular, or neither. J(1, 9), K(7, 4), L(8, 13), M(-2, 1).

Answer 1

First of all, we need to remember two statements:

1) two lines are perpendicular when the product of its slopes is equal to -1.

2) two lines are parallel when they have the same slope.

Now, we need to find the slope for both lines, we start with the segment of line JK

$\begin{gathered} m=\text{slope}=\frac{(y_2-y_1)}{(x_2-x_1)};\text{ } \\ J(1,9)=J(x_1,y_1);x_1=1;y_1=9 \\ K(7,4)=J(x_2,y_2);x_2=7;y_2=4 \\ m=\text{slope}=\frac{(y_2-y_1)}{(x_2-x_1)}=\frac{(4-9)}{(7-1)}=\frac{-5}{6} \\ m=\frac{-5}{6} \end{gathered}$

In the same way, we need to find the value of the slope to the segment LM

$\begin{gathered} m=\text{slope}=\frac{(y_2-y_1)}{(x_2-x_1)}; \\ L(8,13)=L(x_1,y_1);x_1=8;y_1=13 \\ M(-2,1)=M(x_2,y_2);x_2=-2;y_2=1 \\ m=\text{slope}=\frac{(y_2-y_1)}{(x_2-x_1)}=\frac{(1-13)}{(-2-8)}=\frac{-12}{-10}=\frac{6}{5} \\ m=\frac{6}{5} \end{gathered}$

From that we can conclude that both segments are NOT parallel,

Now, we verify if they are perpendicular multiplying the slopes, like this:

$\frac{-5}{6}.\frac{6}{5}=-1$

Finally, we conclude that both segments are parallel because the product of its slopes are -1.