1. Parallel and Perpendicular Lines
Chapter 6 Coordinate Geometry
6.7
Parallel and
Perpendicular
Lines
MATHPOWERTM 10, WESTERN EDITION
6.7.1

2. If the slopes of two lines are equal, the lines are parallel.
Parallel Lines
B(0, 5)
D(3, 0)
A(-3, 0)
If the slopes of two lines are
equal, the lines are parallel.
C(0, -5)
If two lines are parallel,
their slopes are equal.
AB is parallel to CD.
6.7.2

3. 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).
Since the slopes are equal, the line segments are parallel.
6.7.3

4. Using Parallel Slopes to Find k
The following are slopes of parallel lines.
Find the value of k.
2k = 12
k = 6
-1k = 10
k = -10
-7k = -6
-2k = 15
k =
k =
6.7.4

5. If the slopes of two lines are negative reciprocals,
Perpendicular Lines
D(-1, 4)
B(4, 2)
If the slopes of two lines
are negative reciprocals,
the lines are perpendicular.
C(3, -2)
A(-2, -2)
AB is perpendicular to CD.
If two lines are perpendicular,
their slopes are negative reciprocals.
6.7.5

6. 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).
The slopes are equal so line segments are perpendicular.
6.7.6

7. Using Perpendicular Slopes to Find k
The following are slopes of perpendicular lines.
Find the value of k.
-5k = -2
-3k = 8
k =
k =
-3k = -10
-2k = 21
k =
k =
6.7.7

8. Parallel and Perpendicular Lines
Given the following equations of lines, determine
which are parallel and which are perpendicular.
A) 3x + 4y - 24 = B) 3x - 4y + 10 = 0
C) 4x + 3y - 16 = D) 6x + 8y + 15 = 0
4y = -3x + 24
y = x + 6
-4y = -3x - 10
y = x + 5/2
Slope =
Slope =
8y = -6x - 15
3y = -4x + 16
Slope =
Slope =
Lines A and D have the same slope, so they are parallel.
Lines B and C have negative reciprocal slopes, so they are
perpendicular.
6.7.8

9. 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.
y - y1 = m(x - x1)
3x - 4y + 16 = 0
-4y = - 3x - 16
y - 5 = (x - -1)
y = x + 4
4y - 20 = 3(x + 1)
4y - 20 = 3x + 3
0 = 3x - 4y + 23
Slope =
3x - 4y + 23 = 0
6.7.9

10. 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
y - y1 = m(x - x1)
y - 5 = (x - -1)
y = x + 4
3y - 15 = -4(x + 1)
3y - 15 = -4x - 4
4x + 3y - 11 = 0
Slope =
Therefore, use
the slope
4x + 3y - 11 = 0
6.7.10

11. 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
4x + 4y - 16 = 0
For the y-intercept, x = 0:
4(0) + 4y - 16 = 0
4y = 16
y = 4 .
The slope is .
A point is (0, 4).
y - y1 = m(x - x1)
y - 4 = (x - 0)
2y - 8 = -1x
x + 2y - 8 = 0
6.7.11

12. 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
6y = -3x + 9
4x + 4y - 16 = 0
For the x-intercept, y = 0:
4x + 4(0)- 16 = 0
4x = 16
x = 4
The slope is 2.
A point is (4, 0).
y - y1 = m(x - x1)
y - 0 = 2(x - 4)
y = 2x - 8
0 = 2x - y - 8
The equation of the
line is 2x - y - 8 = 0.
6.7.12

13. 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

14. Assignment Suggested Questions: Pages 294 and 295 1 - 25 odd, 27ace,
even,
6.7.14