Parallel & Perpendicular Lines

The student is able to (I can):

• Use slopes to identify parallel and perpendicular lines.

• Write equations of line parallel or perpendicular to a given line through a given point.

Parallel Lines Theorem

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope.

Any two vertical lines are parallel

x

y

ms = mt ⇒ s � t

s t

s

1 3 4m 2

2 0 2

− − −= = =− − −

t

3 1 4m 2

1 1 2

− − −= = =− − −

Perpendicular Lines Theorem

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is —1 (negative reciprocals).

Vertical and horizontal lines are perpendicular.

x

y

p

q

p

2 4 6m 3

2 0 2

+= = = −− − −

q

0 1 1 1m

3 0 3 3

− −= = =− − −

p qm m 1= − ⇒ ⊥i p q

Practice

Given A(—3, —1), B(3, 3), C(—4, 4), and D(0, —2), is AB parallel or perpendicular to CD?

Practice

Given A(—3, —1), B(3, 3), C(—4, 4), and D(0, —2), is AB parallel or perpendicular to CD?

The two slopes are not equal, so the lines are not parallel. The product of the slopes is —1, so the lines are perpendicularperpendicularperpendicularperpendicular.

3 ( 1) 4 2AB : m

3 ( 3) 6 3

− −= = =

− −

���

2 4 6 3CD : m

0 ( 4) 4 2

− − −= = = −

− −

���

Pairs of Lines

Two lines will do one of three things:

• Not intersect (parallel)

• Intersect at one point

• Intersect at all points (coincide)

• To determine which of these possibilities is true, look at the slope and y-intercept:

• To compare slopes and y-intercepts, put both equations in slope-intercept form (y=mx+b). If we do that to the last equation, we can see why the two coincide:

y — 5 = 3(x — 1)

y = 3x — 3 + 5

y = 3x + 2

Parallel LinesParallel LinesParallel LinesParallel Lines Intersecting LinesIntersecting LinesIntersecting LinesIntersecting Lines Coinciding LinesCoinciding LinesCoinciding LinesCoinciding Lines

y = 2x — 9

y = 2x + 7

y = 3x + 5

y = —4x — 1

y = 3x + 2

y — 5 = 3(x — 1)

same slope, different intercept

different slopessame slope, same

intercept

To write the equation of a line that is parallel (or perpendicular) to a given line through a given point:

• Determine the slope of the given line

• Determine the slope of the new line

— Parallel lines have the same slope

— Perpendicular lines have slopes that are the negative reciprocal

• Write the new equation in point-slope form

• Solve for y if necessary

Example: Write the equation of the line that is parallel to x − 3y = 15 through the point (−3, 2).

Example: Write the equation of the line that is parallel to x − 3y = 15 through the point (−3, 2) in slope-intercept form.

So, our slope is .

− =

− = − +

= −

x 3y 15

3y x 15

1y x 5

31

3

( )− = +

= + +

= +

1y 2 x 3

31

y x 1 231

y x 33

Example: Write an equation of the line that is perpendicular to that goes through the point (8, −3), in point-slope form.

= −y 4x 3

Example: Write an equation of the line that is perpendicular to that goes through the point (8, −3), in point-slope form.

orig. slope = ⊥ slope =

= −y 4x 3

−1

4

4

1

( )+ = − −1

y 3 x 84