# 6.7 Prll _ Prpndclr Lines

Chapter 6 Coordinate Geometry
6.7
MATHPOWERTM 10, WESTERN EDITION 6.7.1
Parallel Lines
m AB 
50

0 3
m AB 
5
3
m CD 
m CD 
0 5
B(0, 5)
30
5
3
D(3, 0)
A(-3, 0)
If the slopes of two lines are
equal, the lines are parallel.
If two lines are parallel,
their slopes are equal.
C(0, -5)
AB is parallel to CD.
6.7.2
Verifying Parallel Lines
Show that the line segment AB with endpoints A(2, 3)
and B(6, 5) is parallel to the line segment CD with
endpoints C(-1, 4) and D(3, 6).
y 2  y1
m 
x2  x1
m AB 
m AB 
5 3
6 2
1
2
m CD 
m CD 
64

3 1
1
2
Since the slopes are equal, the line segments are parallel.
6.7.3
Using Parallel Slopes to Find k
The following are slopes of parallel lines.
Find the value of k.
a)
2
3
2
4
,
k

3
c)
-k
5
k
5
b)
4
k
,

3
2
-2k = 15
k =
15
2
2
,
k

5
d)
2
5
1
2k = 12
k=6
3
-1
k
-k
3
k
3
-1k = 10
k = -10
2
,

-2
7
2
-7k = -6
7
k=
6
7
6.7.4
Perpendicular Lines
m AB 
m AB 
2  2

4 2
2
3
m CD 
4  2
1  3
m CD  
D(-1, 4)
B(4, 2)
3
2
If the slopes of two lines
are negative reciprocals,
the lines are perpendicular.
A(-2, -2)
C(3, -2)
AB is perpendicular to CD.
If two lines are perpendicular,
their slopes are negative reciprocals.
6.7.5
Perpendicular Line Segments
Show that the line segment AB with endpoints A(0, 2)
and B(-3, -4) is perpendicular to the line segment CD
with endpoints C(2, -4) and D(-8, 1).
m 
m AB 
4  2
3  0
m AB  2
y 2  y1
x2  x1
m CD 
1  4
8  2
m CD  
1
2
The slopes are equal so line segments are perpendicular.
6.7.6
Using Perpendicular Slopes to Find k
The following are slopes of perpendicular lines.
Find the value of k.
a)
2
,
3
2
4
b)
k

c)
k
-3k = 8
1
4
k =
5
3
2
k
2

,

k
3
k=
10
3
k
3
k=

2
5
-2
,
3
-3k = -10
-5k = -2
k
2
-k
d)
5
5
8
3
,
2
5
3
-k
-1
7
7
2
-2k = 21
k =
21
2
6.7.7
Parallel and Perpendicular Lines
Given the following equations of lines, determine
which are parallel and which are perpendicular.
A) 3x + 4y - 24 = 0
B) 3x - 4y + 10 = 0
-4y = -3x - 10
4y = -3x + 24
Slope = 
3
y =
3
4
x+6
Slope =
4
y=
3
3
4
x + 5/2
4
C) 4x + 3y - 16 = 0
D) 6x + 8y + 15 = 0
8y = -6x - 15
3y = -4x + 16
Slope = 
4
3
y 
4
3
x
16
3
y 
Slope = 
3
3
4
x
15
8
4
Lines A and D have the same slope, so they are parallel.
Lines B and C have negative reciprocal slopes, so they are
6.7.8
perpendicular.
Writing the Equation of a Line
Find the equation of the line through the point
A(-1, 5) and parallel to 3x - 4y + 16 = 0.
Find the slope.
3x - 4y + 16 = 0
-4y = - 3x - 16
3
y= x+4
4
Slope =
3
y - y1 = m(x - x1)
y-5=
3
4
(x - -1)
4y - 20 = 3(x + 1)
4y - 20 = 3x + 3
0 = 3x - 4y + 23
4
3x - 4y + 23 = 0
6.7.9
Writing the Equation of a Line
Find the equation of the line through the point
A(-1, 5) and perpendicular to 3x - 4y + 16 = 0.
Find the slope.
3x - 4y + 16 = 0
-4y = -3x - 16
3
y= x+4
4
Slope =
3
4
Therefore, use
the slope  4 .
y - y1 = m(x - x1)
y-5=
4
3
(x - -1)
3y - 15 = -4(x + 1)
3y - 15 = -4x - 4
4x + 3y - 11 = 0
4x + 3y - 11 = 0
3
6.7.10
Writing the Equation of a Line
Determine the equation of the line parallel to 3x + 6y - 9 = 0
and with the same y-intercept as 4x + 4y - 16 = 0.
3x + 6y - 9 = 0
6y = -3x + 9
y 
1
2
x
3
2
The slope
. is
1
2
.
4x + 4y - 16 = 0
For the y-intercept, x = 0:
4(0) + 4y - 16 = 0
4y = 16
y=4
A point is (0, 4).
y - y1 = m(x - x1)
y-4=
1
2
(x - 0)
2y - 8 = -1x
x + 2y - 8 = 0
6.7.11
Writing the Equation of a Line
Determine the equation of the line that is
perpendicular to 3x + 6y - 9 = 0 and has the
same x-intercept as 4x + 4y - 16 = 0.
3x + 6y - 9 = 0
4x + 4y - 16 = 0
6y = -3x + 9
For the x-intercept, y = 0:
y 
1
2
x
3
2
The slope is 2.
y - y1 = m(x - x1)
y - 0 = 2(x - 4)
y = 2x - 8
0 = 2x - y - 8
4x + 4(0)- 16 = 0
4x = 16
x=4
A point is (4, 0).
The equation of the
line is 2x - y - 8 = 0.
6.7.12
Writing the Equation of a Line
Determine the equation of each of the following lines.
A) perpendicular to 5x - y - 1 = 0 and passing through (4, -2)
x + 5y + 6 = 0
B) perpendicular to 2x - y - 3 = 0 and intersects the y-axis at -2
x + 2y + 4 = 0
C) parallel to 2x + 5y + 10 = 0 and same x-intercept as 4x + 8 = 0
2x + 5y + 4 = 0
D) passing through the point (3, 6) and parallel to the x-axis
y = 6 or y - 6 = 0
E) passing through the y-intercept of 6x + 5y + 25 = 0 and
parallel to 4x - 3y + 9 = 0
4x - 3y - 15 = 0
F) passing through the x-intercept of 6x + 5y + 30 = 0 and
perpendicular to 4x - 3y + 9 = 0
3x + 4y + 15 = 0
6.7.13
Suggested Questions:
Pages 294 and 295
1 - 25 odd,
27ace,
28 - 42 even,
44 - 50
6.7.14