# 4.5 Equations of Parallel and Perpendicular Lines

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Name
Class
4.5
Date
Equations of Parallel
and Perpendicular Lines
Essential Question: How can you find the equation of a line that is parallel
or perpendicular to a given line?
Explore
Resource
Locker
Exploring Slopes of Lines
Recall that the slope of a straight line in a coordinate plane is
the
of the rise to the run. In the figure, the slope of
_ ratio
rise
1
4
__
__
AB is ___
run = 8 = 2 .
3
y
B(4, 1)
1
-5
-3
A(-4, -3)
x
-1 0
Run = 4-(-4) = 8
-5
A
Graph the equations y = 2(x + 1) and y = 2x - 3.
B
What do you notice about the graphs of the two lines?
About the slopes of the lines?
The lines are parallel. The slopes are equal.
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Use a protractor. What is the measure of the angle formed
by the intersection of the lines. What does that tell you
90°; the lines are perpendicular.
D
2
x
-4
-2
What are the slopes of the two lines? How are they related?
1
-__
and 3; the slopes are opposite reciprocals.
Complete the statements: If two nonvertical lines
are
0
2
4
-2
-4
y
8
6
3
E
y
4
The graphs of x + 3y = 22 and y = 3x - 14 are shown.
C
Rise = 1-(-3)
=4
3
4
2
x
0
2
4
6
8
parallel
, then they have equal slopes. If two nonvertical lines are perpendicular,
–1
then the product of their slopes is
.
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Correcti
Reflect
1.
Your friend says that if two lines have opposite slopes, they are perpendicular.
He uses the slopes 1 and –1 as examples. Do you agree with your friend? Explain.
No; although lines with slopes of 1 and -1 are perpendicular, it’s because the product
of the slopes is -1. Slopes of 2 and -2 are opposites, but the corresponding lines are not
perpendicular.
2.
The frets on a guitar are all perpendicular to one of the strings. Explain why the
frets must be parallel to each other.
The frets are lines that are perpendicular to the same line (the string), so the frets must
be parallel to each other.
Writing Equations of Parallel Lines
Explain 1
You can use slope relationships to write an equation of a line parallel to a given line.
Example 1 Write the equation of each line in slope-intercept form.
 The line parallel to
y = 5x + 1 that passes through (-1, 2)
Parallel lines have equal slopes. So the slope of the required line is 5.
y - y 1 = m(x - x 1)
Use point-slope form.
y - 2 = 5(x - (-1))
m, x 1,y 1.
Substitute for
y - 2 = 5x + 5
Simplify.
Solve for
y = 5x + 7
y.
The equation of the line is
y = -3x + 4 that passes through (9, -6)
equal
Parallel lines have
Use point-slope form.
m, x 1,y 1.
Substitute for
slopes. So the slope of the required line is -3 .
y - y 1 = m(x - x 1)
(
y - -6 = -3 x - 9
y + 6 = -3 x + 27
Simplify.
Solve for
The equation of the line is
GE_MNLESE385795_U2M04L5.indd 206
)
y = -3 x + 21
y.
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 The line parallel to
y = 5x + 7.
y = -3x + 21 .
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Reflect
3.
What is the equation of the line through a given point and parallel
to the x-axis? Why?
The equation is y = y 1 , wherey 1 is they-coordinate of the given point. This is because the
x-axis is a horizontal line with equation y = 0.
Write the equation of each line in slope-intercept form.
4.
The line parallel to y = -x that passes
through (5, 2.5)
5.
The line parallel to y = __32 x + 4 tha t passes
through (-4, 0)
_3 (x - (-4))
2
3
y = _x + 6
y - (0) =
y - 2.5 = -1(x - 5)
y - 2.5 = -x + 5
2
y = -x + 7.5
Writing Equations of Perpendicular Lines
Explain 2
You can use slope relationships to write an equation of a line perpendicular to a given line.
Example 2 Write the equation of each line in slope-intercept form.
 The line perpendicular to
y = 4x - 2 that passes through (3, -1)
Perpendicular lines have slopes that are opposite reciprocals, which means that
the product of the slopes will be -1. So the slope of the required line is -__14 .
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y - y 1 = m(x - x 1) Use point-slope form.
1 (x - 3)Substitutefor
y - (-1) = -_
4
3Simplify.
1x + _
y + 1 = -_
4
4
1x - _
1Solvefor
y = -_
y.
4
4
The equation of the line is
y = -__14 x - __14 .
 The line perpendicular to
m, x 1,y 1.
y = -__25 x + 12 that passes through (-6, -8)
The product of the slopes of perpendicular lines is
5
.
-1 . So the slope of the required line is __
2
y - y 1 = m(x - x 1) Use point-slope form.
5
__
y - -8 = 2
y+8=
y=
5
__
2
5
__
2
(x - -6 )
Substitute for m, x 1,y 1.
x + 15
Simplify.
x+
Solve for y.
7
5
x+ 7 .
The equation of the line is y y = __
2
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Correcti
Reflect
6.
A carpenter’s square forms a right angle. A carpenter places the
square so that one side is parallel to an edge of a board, and then
draws a line along the other side of the square. Then he slides the
square to the right and draws a second line. Why must the two
lines be parallel?
Both lines are perpendicular to the edge of the
board. If two coplanar lines are perpendicular to
the same line, then the two lines are parallel to each
other, so the lines must be parallel to each other.
Write the equation of each line in slope-intercept form.
7.
3
The line perpendicular to y = __
x + 2 that
2
passes through (3, –1)
8.
The line perpendicular to y = -4x that
passes through (0, 0)
1(
x - 0)
y - 0 = __
4
2 (x - 3)
y − (-1) = -__
3
2
__
y = -3x + 1
1
y = __
x
4
Elaborate
9.
intersects it or it is the same line.
10. Would it make sense to find the equation of a line perpendicular to a given line,
and through a point on the given line? Explain.
Yes; the line will be perpendicular to the given line at the point.
11. Essential Question Check-In How are the slopes of parallel lines and
perpendicular lines related? Assume the lines are not vertical.
Parallel lines have the same slope; perpendicular lines have slopes whose product is -1.
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Discussion Would it make sense to find the equation of a line parallel to a
given line, and through a point on the given line? Explain.
No; if the point is on the line, the line can’t be parallel to the line, because it either
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