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

Step-by-step 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}}=(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}=(y_2-y_1)/(x_2-x_1) \\ slope=(1-7)/(10-9) \\ \text{slope}=(-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}=(y_2-y_1)/(x_2-x_1) \\ slope_2=(5-4)/(10-4)=(1)/(6) \\ slope_2=(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=(1)/(6) \\ \text{slope}_1\\e 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(1)/(6)=-1 \\ -1=-1\rightarrow true,\text{ so the lines are perpendicular} \end{gathered}`

I hope this helps you