1.4.4 parallel and perpendicular line equations
Post on 13-Apr-2017
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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
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