Question

Line 1 goes through the points (9,7) and (10,1). Line 2 passes through (4,4) and (10,5). Are the lines parallel or perpendicular?

Answer 1

perpendicular

Explanation

to solve this we need to find the slopes of the lines, and then compare the slopes

Step 1

find the slope of line 1

the slope is given by:

$\begin{gathered} \text{slope}=\frac{change\text{ in y}}{\text{change in x}}=\frac{y_2-y_1}{x_2-x_1} \\ \text{where} \\ P1(x_1,y_1) \\ P2(x_2,y_2) \end{gathered}$

P1 and P2 are 2 known points of the line,

so

Let

P1(9,7)

P2(10,1)

now, replace to find slope 1

$\begin{gathered} \text{slope}=\frac{y_2-y_1}{x_2-x_1} \\ slope=\frac{1-7}{10-9} \\ \text{slope}=\frac{-6}{1} \\ \text{slope}_1=-6 \end{gathered}$

Step 2

now, slope of line 2

Let

P1(4,4)

P2(10,5)

replace to find slope 2

$\begin{gathered} \text{slope}=\frac{y_2-y_1}{x_2-x_1} \\ slope_2=\frac{5-4}{10-4}=\frac{1}{6} \\ slope_2=\frac{1}{6} \end{gathered}$

Step 3

remember:

when 2 lines are parallel , the slope is the same,hence

$\begin{gathered} \text{slope}_1=-6 \\ \text{slope}_2=\frac{1}{6} \\ \text{slope}_1\ne slope_2\rightarrow the\text{ lines are not parellel} \end{gathered}$

now, 2 lines are perpendicular if

$slope_1\cdot slope_2=-1$

replace to check

$\begin{gathered} slope_1\cdot slope_2=-1 \\ -6\cdot\frac{1}{6}=-1 \\ -1=-1\rightarrow true,\text{ so the lines are perpendicular} \end{gathered}$

I hope this helps you