Question

3.5 equations of parallel and perpendicular lines
geometry - classwork
determine which of the lines are parallel and which of the lines are perpendicular. (see example 1.)
1.
2.
write an equation of the line passing through point p that is (a) parallel to the given line and (b) perpendicular to the given line. (see example 2.)

1. p(0, 1), y = - 2x + 3

parallel: ____________
perpendicular: ____________

1. p(4, 0), - x + 2y = 12

parallel: ____________
perpendicular: ____________
recall: all vertical lines are parallel to each other, all horizontal lines are parallel to each other
vertical lines are parallel to horizontal lines
to graph a vertical line, type “x =#”
(where # is any number of your choosing)
this is the line x = 8
to graph a horizontal line, type “y = #”
(where # is any number of your choosing)
this is the line y = -2

1. p(- 2, 6), x = - 5

parallel: ____________
perpendicular: ____________

1. p(2, 3), 2 = - 2x

parallel: ____________
perpendicular: ____________

Explanation:

Step1: Q3 Parallel Line

Given line: $y=-2x+3$ (slope $m=-2$). Parallel slope $m=-2$. Point $P(0,1)$. Using $y - y_1 = m(x - x_1)$: $y - 1 = -2(x - 0) \Rightarrow y=-2x+1$.

Step2: Q3 Perpendicular Line

Perpendicular slope $m=\frac{1}{2}$. Using point-slope: $y - 1 = \frac{1}{2}(x - 0) \Rightarrow y=\frac{1}{2}x+1$.

Step3: Q4 Parallel Line

Given line: $-x + 2y=12 \Rightarrow y=\frac{1}{2}x+6$ (slope $m=\frac{1}{2}$). Parallel slope $m=\frac{1}{2}$. Point $P(4,0)$. $y - 0 = \frac{1}{2}(x - 4) \Rightarrow y=\frac{1}{2}x-2$.

Step4: Q4 Perpendicular Line

Perpendicular slope $m=-2$. $y - 0 = -2(x - 4) \Rightarrow y=-2x+8$.

Step5: Q5 Parallel Line

Given line $x=-5$ (vertical). Parallel vertical line through $(-2,6)$: $x=-2$.

Step6: Q5 Perpendicular Line

Perpendicular to vertical is horizontal line through $(-2,6)$: $y=6$.

Step7: Q6 Parallel Line

Given line $2=-2x \Rightarrow x=-1$ (vertical). Parallel vertical line through $(2,3)$: $x=2$.

Step8: Q6 Perpendicular Line

Perpendicular horizontal line through $(2,3)$: $y=3$.

Answer:

1. Parallel: $y=-2x+1$, Perpendicular: $y=\frac{1}{2}x+1$
2. Parallel: $y=\frac{1}{2}x-2$, Perpendicular: $y=-2x+8$
3. Parallel: $x=-2$, Perpendicular: $y=6$
4. Parallel: $x=2$, Perpendicular: $y=3$