Parallel and Perpendicular
Lines
6-6
6-6
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To determine whether lines
What is the reciprocal of each fraction?
are parallel
• To determine whether lines
are perpendicular
. . . And Why
To use parallel and
perpendicular lines to plan a
bike path, as in Example 4
Lessons 2-3 and 6-2
Objectives
1
1. 12 21
2. 43 34
3. 2 25 –52
What are the slope and y-intercept of each equation?
4. 2 75 –57
2
5. y = 53 x + 4 53 ; 4 6. y = 53 x - 8 53 ; –8 7. y = 6x 6; 0
8. y = 6x + 2
6; 2
Examples
New Vocabulary • parallel lines • perpendicular lines
• negative reciprocal
1
2
3
61
To determine whether lines
are parallel
To determine whether lines
are perpendicular
4
Determining Whether Lines
Are Parallel
Writing Equations of Parallel
Lines
Writing Equations for
Perpendicular Lines
Real-World Problem Solving
Parallel Lines
In the graph at the right, the red and blue lines are parallel.
Parallel lines are lines in the same plane that never
intersect. The equation of the red line is y = 12 x + 32 .
The equation of the blue line is y = 12 x - 1.
y
Math Background
2
⫺2 O
x
2
⫺2
Key Concepts
Property
Slopes of Parallel Lines
Nonvertical lines are parallel if they have the same slope and different
y-intercepts. Any two vertical lines are parallel.
More Math Background: p. 306D
Example The equations y = 23 x + 1 and y = 23 x - 3 have the same slope, 23 ,
and different y-intercepts. The graphs of the two equations are parallel.
EXAMPLE
Determining Whether Lines Are Parallel
PowerPoint
Are the graphs of y = 2 13 x + 5 and 2x + 6y = 12 parallel? Explain.
Write 2x + 6y = 12 in slope-intercept form. Then compare with y =
6y = -2x + 12
Real-World
Connection
The lanes for competitive
swimming are parallel.
6y
22x 1 12
6 =
6
y = 2 13 x + 2
Bell Ringer Practice
2 13 x
+ 5.
For intervention, direct students to:
Divide each side by 6.
Multiplying and Dividing
Lesson 2-3: Example 7
Extra Skills and Word
Problem Practice, Ch. 2
Simplify.
Slope-Intercept Form
Lesson 6-2: Example 1
Extra Skills and Word
Problem Practice, Ch. 6
1 Are the graphs of -6x + 8y = -24 and y = 3 x - 7 parallel? Explain.
4
yes; same slope, different y-intercepts
Lesson 6-6 Parallel and Perpendicular Lines
Special Needs
Check Skills You’ll Need
Subtract 2x from each side.
The lines are parallel. The equations have the same slope, 2 13 , and
different y-intercepts.
Quick Check
Lesson Planning and
Resources
See p. 306E for a list of the
resources that support this lesson.
You can use slope-intercept form to determine whether the lines are parallel.
1
Vertical lines have an undefined,
or infinite, slope because the
denominator in the ratio is zero.
This is why the Property of Slopes
of Parallel Lines includes the word
nonvertical.
Below Level
L1
343
L2
Some students may confuse the terms parallel and
perpendicular. Have them underline the ll in the word
parallel. Ask: Do the 2 l’s in the word parallel look
parallel or perpendicular?
To demonstrate that the signs of the slopes of parallel
lines are the same, have students graph y = 12 x - 1.
Then have them compare their graphs with the one
on p. 343.
learning style: visual
learning style: visual
343
2. Teach
You can use the fact that the slopes of parallel lines are the same to write the
equation of a line parallel to a given line. To write the equation, you use the slope
of the given line and the point-slope form of a linear equation.
nline
Guided Instruction
1
EXAMPLE
2
Writing Equations of Parallel Lines
Write an equation for the line that contains (5, 1) and is parallel to y = 35 x - 4.
Step 1 Identify the slope of the given line.
Alternative Method
Write y = 2x + 4 and y = 2x - 1
on the board. Ask: What number
do the equations have in
common? 2 What does this
number represent in each
equation? the slope Have
students find the y-coordinates
of the point on each line when
x = 4. Then find the difference
between the y-coordinates. Tell
students that the difference is
the vertical distance between
the two lines. Repeat for
x = 100, x = 50,000 and
x = 100,000. Compare the
vertical distance between the
lines for each pair of points. The
vertical distances are all the
same. Ask: What geometric term
describes lines that are always the
same distance apart? parallel
What might you conclude about
lines that have the same slope?
Lines that have the same slope
are parallel.
EXAMPLE
y = 35 x - 4
c
slope
Visit: PHSchool.com
Web Code: ate-0775
Step 2 Write the equation of the line through (5, 1) using slope-intercept form.
y - y1 = m(x - x1)
y-1=
y-1=
y-1=
y=
Quick Check
2
1
3 (x 2 5)
5
3 x - 3 (5)
5
5
3x - 3
5
3x - 2
5
point-slope form
Substitute (5, 1) for (x1, y1) and 35 for m.
Use the Distributive Property.
Simplify.
Add 1 to each side.
2 Write an equation for the line that contains (2, -6) and is parallel to y = 3x + 9.
y ≠ 3x – 12
Perpendicular Lines
The lines at the right are perpendicular.
Perpendicular lines are lines that intersect
to form right angles. The equation of the red line
is y = 2 14 x - 1. The equation of the blue line is
y = 4x + 2.
y
2
x
O
⫺6 ⫺4
2
⫺2
PowerPoint
Additional Examples
Key Concepts
Property
1 Are the graphs of y = -2x - 1
Slopes of Perpendicular Lines
and 4x + 2y = 6 parallel? yes
Two lines are perpendicular if the product of their slopes is -1. A vertical and
a horizontal line are also perpendicular.
2 Write an equation for the line
that contains (-2, 3) and is parallel
to y = 52 x - 4. y ≠ 52 x ± 8
Example The slope of y 5 2 14x 2 1 is 2 14. The slope of y = 4x + 2 is 4.
Since 2 14 ? 4 = -1, the graphs of the two equations are perpendicular.
The product of two numbers is -1 if one number is the negative reciprocal of the
other. Here is how to find the negative reciprocal of a number.
Start with
a fraction: 2 35. S
Find its
reciprocal: 2 53.
S
Start with
an integer: 4.
Find its
reciprocal: 41.
S
Write the
negative reciprocal: 35.
Since 2 35 ? 35 = -1, 53 is the negative reciprocal of 2 35.
S
Write its
negative reciprocal: 2 14.
Since 4 Q 2 14 R = -1, 2 14 is the negative reciprocal of 4.
344
Chapter 6 Linear Equations and Their Graphs
Advanced Learners
English Language Learners ELL
L4
Draw a square and its diagonals on a coordinate grid.
Challenge students to prove that the diagonals are
perpendicular.
344
learning style: verbal
In Example 2, have students “undo” negative
reciprocal one word at a time. Ask: What do you
multiply by to make a fraction negative? –1 Discuss
the meaning of reciprocal.
learning style: verbal
3
You can use the negative reciprocal of the slope of a given line to write an
equation of a line perpendicular to that line.
3
B
A
2
4
5
A
A
B
B
C
C
D
D
E
E
y = 15x 2 2
y = 5x - 2
E
D
C
B
E
D
C
B
A
3
E
D
C
Multiple Choice Which equation describes a line that contains (0, -2) and is
perpendicular to y = 5x + 3?
E
D
C
B
A
1
Writing Equations for Perpendicular Lines
EXAMPLE
Test-Taking Tip
Step 1 Identify the slope of
the given line.
After finding the
slopes of the
perpendicular lines,
multiply the two
slopes as a check.
y = 2 15x 2 2
Step 2 Find the negative reciprocal
of the slope.
PowerPoint
Additional Examples
3 Find an equation of the
line that contains (6, 2) and is
perpendicular to y = -2x + 7.
y ≠ 12 x – 1
Step 3 Use the slope-intercept form to write an equation.
y = mx + b
y = 2 15 x + (-2)
y = 2 15 x - 2
Substitute 2 15 for m, and –2 for b.
Simplify.
The equation is y = 2 15 x - 2. So D is the correct answer.
Quick Check
Students may think that a
negative reciprocal is always
negative. Tell students that
the phrase negative reciprocal
actually means the negative of
the reciprocal. So they are really
just finding the opposite of the
reciprocal.
The negative reciprocal of 5 is 2 15.
y = 5x + 3
c
slope
5(2 1
5 ) 5 21
y = 15x 1 2
3 Write an equation of the line that contains (1, 8) and is perpendicular to y = 3 x + 1.
4
y ≠ –43 x ± 9 13
4 The line in the graph represents
the street in front of a new house.
The point is the front door. The
sidewalk from the front door will
be perpendicular to the street.
Write an equation representing
the sidewalk. y ≠ 32 x – 3
You can use equations of parallel and perpendicular lines to solve some
real-world problems.
4
Real-World
EXAMPLE
4
y park
2
oa
kR
Par
2
⫺2 O
2
x
new path
perpendicular
to Park Road
2
str
ee
O
2
x
6
front
door
4
(2, 1)
x
⫺2 O
4
2
Resources
• Daily Notetaking Guide 6-6 L3
• Daily Notetaking Guide 6-6—
L1
Adapted Instruction
(4, 5)
4
d
4
y
y-intercept
entrance
y
t
Problem Solving
Urban Planning A bike path for a new city park will connect the park entrance to
Park Road. The path will be perpendicular to Park Road. Write an equation for the
line representing the bike path.
Park
Error Prevention
EXAMPLE
2
4
Closure
Step 1 Find the slope m of Park Road.
Real-World
Connection
Careers Urban planners use
mathematics to make
decisions on community
problems such as traffic
congestion and air pollution.
y 2y
21=4=2
m = x2 2 x1 = 45 2
2
2
2
1
Points (2, 1) and (4, 5) are on Park Road.
Step 2 Find the negative reciprocal of the slope.
The negative reciprocal of 2 is 2 12 . So the slope of the bike path is 2 12 .
The y-intercept is 4.
The equation for the bike path is y = 2 12 x + 4.
Quick Check
4 A second bike path is planned. It will be parallel to Park Road and will also
contain the park entrance. Write an equation for the line representing this
bike path. y ≠ 2x ± 4
Lesson 6-6 Parallel and Perpendicular Lines
Ask: Compare the equations
of non-vertical parallel lines.
Parallel lines have the same
slope, but different y-intercepts.
Compare the equations of
perpendicular lines.
Perpendicular lines have slopes
that are negative reciprocals of
each other. They will have the
same y-intercept only if that is
where the two lines intersect.
345
345
3. Practice
EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practice
Practiceand
andProblem
ProblemSolving
Solving
Assignment Guide
A
1 A B 1-18, 40, 41, 44, 45,
Practice by Example
Example 1
51-60
2 A B
19-39, 42, 43, 46-50,
61-63
C Challenge
64-70
Test Prep
Mixed Review
GO for
Help
(page 343)
Find the slope of a line parallel to the graph of each equation.
1. y = 12 x + 2.3 12
4. y = 6 0
3. y = x 1
6. 7x - y = 5 7
Are the graphs of the lines in each pair parallel? Explain. 7–12. See margin.
7. y = 4x + 12
-4x + 3y = 21
71-76
77-89
10. y = 2 12 x + 32
5x - 10y = 15
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 18, 26, 44, 47-49, 54.
2. y = 2 23 x - 1 –23
5. 3x + 4y = 12 –34
Example 2
(page 344)
9. y = 13 x + 3
x - 3y = 6
12. y = 34 x - 2
-3x + 4y = 8
11. y = -3x
21x + 7y = 14
Write an equation for the line that is parallel to the given line and that passes
through the given point. 13–18. See margin.
13. y = 6x - 2; (0, 0)
Exercise 4 Suggest to students
8. y = 2 32 x + 2
3x + 2y = 8
14. y = -3x; (3, 0)
15. y = -2x + 3; (-3, 5)
16. y = 2 72 x + 6; (-4, -6) 17. y = 0.5x - 8; (8, -5)
that they graph the line y = 6.
Example 3
(page 345)
18. y = 2 23 x + 12; (5, -3)
Find the slope of a line perpendicular to the graph of each equation.
19. y = 2x –12
20. y = -3x 13
22. y = 2 x5 - 7 5
23. 2x + 3y = 5 32
21. y = 75 x - 2 –57
24. y = -8 undefined
Write an equation for the line that is perpendicular to the given line and that
passes through the given point.
GPS Guided Problem Solving
L3
L4
Enrichment
L1
Adapted Practice
Practice
Name
Class
Practice 6-5
Example 4
L3
Date
(page 345)
Parallel and Perpendicular Lines
Find the slope of a line parallel to the graph of each equation.
2. y = 27 x + 1
1. y = 4x + 2
5. 6x + 2y = 4
6. y - 3 = 0
9. -x + 3y = 6
4. y = 2 21 x + 1
3. y = -9x - 13
7. -5x + 5y = 4
10. 6x - 7y = 10
8. 9x - 5y = 4
11. x = -4
12. -3x - 5y = 6
Write an equation for the line that is perpendicular to the given line and that
passes through the given point.
13. (6, 4); y = 3x - 2
14. (-5, 5); y = -5x + 9
15. (-1, -4); y = 16 x + 1
16. (1, 1) ; y = 2 14 x + 7
17. (12, -6); y = 4x + 1
18. (0, -3); y = 2 43 x - 7
19.
20.
21.
3
2
1
y
5
4
3
2
1
x
⫺3⫺2⫺1O 1 2 3
⫺2
⫺3
y
2
1
x
O
⫺4⫺3⫺2⫺1
1 2
B
y
Apply Your Skills
x
O
⫺2⫺1
⫺2
⫺3
⫺4
1 2 3 4
© Pearson Education, Inc. All rights reserved.
25. (4,0); y = 23 x + 9
28.
y
⫺2⫺1
⫺1
⫺2
⫺3
⫺4
⫺5
⫺6
O
1 2
23. (1, 3); y = -4x + 5
24. (4, -1); y = x - 3
26. (-8, -4); y = 2 43 x + 5
27. (9, -7); -7x - 3y = 3
29.
30.
5
4
3
2
1
x
4
⫺3⫺2⫺1
y
2
1
⫺2⫺1
⫺2
⫺3
⫺4
x
O
2 3
y
x
O
1 2 3 4
3x - y = -1
34. 9x + 3y = 6
3x + 9y = 6
32. 3x + 2y = -5
y = 2x + 6
3
33. y = 2 25 x + 11
-5x + 2y = 20
35. y = -4
y=4
36. x = 10
y = -2
28. 3x + 5y = 7; (-1, 2) y ≠ 35 x ± 11
3
31. Maps A city’s civil engineer is planning a new parking
garage and a new street. The new street will go from the
entrance of the parking garage to Handel St. It will be
perpendicular to Handel St. What is the equation of the
line representing the new street? y ≠ 54 x ± 1
Exercises
Ha
nd
el
St.
Entrance
2
4
O
2
2 x - 6, y = 2 x + 6 parallel
3
3
5x, y = -5x + 7 neither
2, y = 9 perpendicular
39. 3x - 5y = 3, -5x + 3y = 8 neither
346
y = 3x - 1
y = 35 x + 8
Chapter 6 Linear Equations and Their Graphs
7. no, different slopes
10. no, different slopes
13. y ≠ 6x
16. y ≠ –72 x – 20
8. yes, same slopes and
different y-intercepts
11. yes, same slopes and
different y-intercepts
14. y ≠ –3x ± 9
17. y ≠ 0.5x – 9
9. yes, same slopes and
different y-intercepts
12. yes, same slopes and
different y-intercepts
15. y ≠ –2x – 1
18. y ≠ –32 x ± 13
346
x
40. Multiple Choice Which pair of equations graph as perpendicular lines? C
y = 4x + 12
y = 23 x + 1
y = 235 x - 9
y = 235 x - 9
y = -4x + 3
pages 346–349
y
Tell whether the lines for each pair of equations are
parallel, perpendicular, or neither.
38. 2x + y = 2, 2x + y = 5 parallel
Tell whether the lines for each pair of equations are parallel, perpendicular,
or neither.
31. y = 3x - 8
27. y = 2 13 x + 2; (4, 2) y ≠ 3x – 10
32. y = 4x + 34 , y = 2 14 x + 4
33. y =
perpendicular
34. y = -x + 5, y = x + 5
35. y =
perpendicular
x
1
36. y = 3 - 4, y = 3 x + 2 parallel
37. x =
Write an equation for the line that is parallel to the given line and that
passes through the given point.
22. (3, 4); y = 2x - 7
26. y = x - 3; (4, 6) y ≠ –x ± 10
29. -10x + 8y = 3; (15, 12)
30. 4x - 2y = 9; (8, -2) y ≠ –12 x ± 2
y ≠ –45 x ± 24
L2
Reteaching
25. y = 2x + 7; (0, 0) y ≠ –12 x
y = 35 x + 4
Teaching Tip
Find the equation for each line.
41.
y≠
42.
y
1
3x
4
2
⫺6
⫺2 O
y ≠ –45 x – 19
5 ;
x
y≠
⫺2
Real-World
–45 x
±
±
y
4
3;
y ≠ –3x ± 7
Exercises 32– 39 Ask students
to describe equations of lines
that are neither parallel nor
perpendicular. The slopes are
neither the same nor negative
reciprocals of each other.
2
3
5
⫺2 O
2
x
4
Error Prevention!
⫺2
Connection
43.
A skier’s fastest speed occurs
when the skis are parallel.
4
44.
y
⫺4
2
–12 x;
⫺2 O
2
y≠
x y ≠ 2x
4
Exercise 64 Suggest that students
y
O
x
2
y ≠ 25 x ± 53 ;
y ≠ 25 x – 21
5
⫺2
first sketch a graph of the
triangle to determine which two
sides may be perpendicular.
⫺5
45.
46.
y
4
y
3
⫺4 ⫺2 O
x
2
2
y ≠ 4;
y≠2
1
⫺2
O
4
x
y ≠ x;
y ≠ –x ± 1
Maps Use the map below for Exercises 47–49.
GPS
Av
e.
Washington, D.C.
Ma
Ne
w
Ha
mp
sh
ire
ssa
Pen
nsy
chu
sett
s Av
e.
lvan
ia
Ave
.
47. What is the slope of New Hampshire Avenue? about 54
48. Show that the parts of Pennsylvania Avenue and Massachusetts Avenue near
New Hampshire Avenue are parallel. Answers may vary.
1
Sample: same slope of –2
49. Show that New Hampshire Avenue is not perpendicular to Pennsylvania
Avenue. Answers may vary. Sample: 54 ? Q 212 R u –1
GO
nline
Homework Help
Visit: PHSchool.com
Web Code: ate-0606
50. a. The graphs of y = x and y = -x are shown
on the standard screen at the right. The product
of the slopes is -1. Explain why the lines do not
appear to be perpendicular. a–b. See margin.
b. Graphing Calculator Graph y = x and y = -x
on a graphing calculator. In the
feature,
choose the square screen. What do you notice?
51. Open-Ended Write an equation for a line parallel to the graph of 4x - y = 1.
Answers may vary. Sample: y ≠ 4x ± 1
Lesson 6-6 Parallel and Perpendicular Lines
347
50a. The units on the axes
are different, so the
screen is not square.
b. The lines appear
perpendicular.
347
4. Assess & Reteach
PowerPoint
Lesson Quiz
52. No; the slopes are
not equal.
52. Are the graphs of 2x + 7y = 6 and 7y = 2x + 6 parallel? Explain. See left.
53. No; the slopes are
not neg.
reciprocals.
54. Writing Are all horizontal lines parallel? Explain. See margin.
Problem Solving Hint
1. Find the slope of a line parallel
to 3x - 2y = 1. 32
For Exercises 55–57,
sketch a graph to help
you understand the
statement in each
exercise.
2. Find the slope of a line
perpendicular to
4x + 5y = 7. 54
Tell whether the lines for each
pair of equations are parallel,
perpendicular, or neither.
3. y = 3x - 1, y = - 31 x + 2
perpendicular
4. y = 2x + 5, 2x + y = -4
neither
5. y = -32 x - 1, 2x + 3y = 6
parallel
6. Write an equation of the line
that contains (2, 1)
and is perpendicular to
y = - 12 x + 3. y ≠ 2x – 3
55. False; the product
of two positive
numbers can’t be
–1.
53. Are the graphs of 8x + 3y = 6 and 8x - 3y = 6 perpendicular? Explain. See left.
Tell whether each statement is true or false. Explain your choice.
55–57. See below left.
55. Two lines with positive slopes can be perpendicular.
56. Two lines with positive slopes can be parallel.
57. The graphs of two different direct variations can be parallel.
58–60. See
Geometry A quadrilateral with both pairs of opposite sides parallel is a margin.
parallelogram. Use slopes to determine whether each figure is a parallelogram.
58.
2
56. True; y ≠ x ± 2 and
y ≠ x ± 3 are
parallel.
57. False; all direct
variations go
through the point
(0, 0). If they have
the same slope, they
are the same line,
not parallel lines.
4
y
A
O
348
⫺3
C
6 x
4
4
y
Q
O
1
3
x
L
S
⫺2 ⫺1 O
x
2
R
Geometry A quadrilateral with four right angles is a rectangle. Use slopes to
determine whether each figure is a rectangle. 61–63. See margin.
61.
2
62.
y B
4
K
x
A
⫺2
O
1
63.
y
4
L
2
3
y
Q
2
P
⫺2
C
N
O
1
3
x
O
⫺2
1
R
x
S
M
D
Challenge
60. P
K
M
D
⫺4
C
y
1
⫺2
Write an equation in slopeintercept form on the board.
Have students first write an
equation of a line that is parallel
to the given line, and then write
an equation of a line that is
perpendicular to the given
equation. Let students compare
their answers. Repeat.
3
J
Alternative Assessment
54. yes; same slopes and
different y-intercepts
4
4
58. The slopes of AD and BC
are both undefined, so
they are parallel. The
4
4
slopes of AB and CD are
2
both 5, so they are
parallel. The quadrilateral
is a parallelogram.
4
59. The slope of JK is 15. The
4
slope of KL is –2. The
4
slope of LM is 16. The
4
slope of JM is –4. The
quadrilateral is a
parallelogram.
59.
B
64. Geometry A quadrilateral with two pairs of parallel sides and with diagonals
that are perpendicular is a rhombus. Quadrilateral ABCD has vertices
A(-2, 2), B(1, 6), C(6, 6), and D(3, 2). Show that ABCD is a rhombus. See margin.
65. Geometry A triangle with two sides that are perpendicular to each other is a
right triangle. Triangle PQR has vertices P(3, 3), Q(2, -2), and R(0, 1).
Determine whether PQR is a right triangle. Explain. See margin.
Tell whether the lines in each pair are parallel, perpendicular, or neither.
68. y ≠ –38 x – 17
8 ;
8
y ≠ 3x ± 7
69. y ≠ 12 x – 3;
y ≠ –2x ± 7
66. ax - by = 5; -ax + by = 2
67. ax + by = 8; bx - ay = 1
perpendicular
parallel
Assume the two lines are perpendicular. Find an equation for each line. 68–69.
See left.
68.
69.
y
y
1
1
x
⫺2 O
2
4 x
⫺4 ⫺2 O
2
⫺2
⫺4
⫺4
70. For what value of k are the graphs of 3x + 12y = 8 and 6y = kx - 5
parallel? Perpendicular? –1.5; 24
348
Chapter 6 Linear Equations and Their Graphs
4
4
60. The slopes of PQ and RS
are both –12 . The slopes
4
4
of QR and SP are both
–32 . The quadrilateral is
a parallelogram.
4
4
61. The slopes of AB and CD
are both 25 . The slopes of
4
4
BC and AD are both –52 .
The product is –1, so
the quadrilateral is a
rectangle.
4
4
62. The slopes of KL and MN
are both –16 . The slopes
4
4
of LM and KN are both 5.
The product is not –1, so
the quadrilateral is not a
rectangle.
Test Prep
Test Prep
Standardized Test Prep
Resources
Multiple Choice
71. Which equation has as its graph a line perpendicular to a line that has a
slope of 23 ? D
A. y = 2
B. y = 3
C. y = 3x - 2
D. 3x + 2y = 8
3x + 5
2x - 1
72. Which equation has as its graph a line parallel to the graph of -2x - 4y = 3? F
F. y = 2 1
G. y = 2x - 6
H. y = -2x + 4
J. y = 1
2x + 5
2x - 2
73. A parallelogram has vertices A(0, 2), B(2, -1), C(6, 3), and D(p, q). Which of
the following ordered pairs has possible values for (p, q)? C
A. (0, 6)
B. (6, 1)
C. (4, 6)
D. (6, 4)
74. Which equation has as its graph a line perpendicular to the graph of the line
that passes through the points (0, -1) and (8, 9)? F
F. (y 2 2) 5 2 4
5 (x 1 1)
H. 4x 2 5y 5 0
G. y 5 4
5x 1 10
J. y 5 2x 1 7
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 369
• Test-Taking Strategies, p. 364
• Test-Taking Strategies with
Transparencies
Connection to Geometry
Exercise 73 Tell students that
they can eliminate some possible
answers by first sketching the
three points in the exercise on a
coordinate plane.
75. Which equation has as its graph a line parallel to the graph of the line that
passes through the points (-8, 2) and (-3, 3)? C
A. (y 2 2) 5 2(x 1 8)
B. y 5 25x 1 10
D. 2y 5 2 1
5x 1 7
C. x 2 5y 5 26
Short Response
76. Suppose the line through points (x, 6) and (1, 2) is parallel to the graph of
2x + y = 3. Find the value of x. Show your work. See margin.
A-headMixed Review
GO for
Help
Lesson 6-4
Write an equation for the line through the given point with the given slope.
77. (0, 4); m = 3 y ≠ 3x ± 4
78. (-2, 0); m = -4 y ≠ –4x – 8
y ± 9 ≠ –23(x ± 1)
79. (5, -3); m = 34 y ± 3 ≠ 34(x – 5)
80. (-1, -9); m = 2 23
81. (-6, 4); m = 2 35 y – 4 ≠ –35(x ± 6) 82. (7, 11); m = 12 y – 11 ≠ 12(x – 7)
Lesson 6-1
21
5
Find the slope of each line.
83.
1
y
O
⫺6 ⫺4 ⫺2
2
4
85.
y
1
2
y
4
O
x
2
⫺2
x
⫺4
⫺3
Lesson 5-2
4
84.
⫺6
⫺4
O
2
x
⫺2
Determine whether each relation is a function.
86. {(1, 1), (2, 2), (3, 3)} yes
87. {(1, 3), (2, 5), (3, 5)} yes
88. {(5, 1), (5, 2), (4, 3)} no
89. {(1, 3), (2, 2), (3, 1)} yes
lesson quiz, PHSchool.com, Web Code: ata-0606
4
4
63. The slopes of PQ and RS
are both 12 . The slopes of
4
4
PS and QR are both –2.
The product is –1, so
the quadrilateral is a
rectangle.
Lesson 6-6 Parallel and Perpendicular Lines
4
4
64. BC and AD both have a
4
slope of zero. BC and
4
4
AD are parallel. AB and
4
CD both have a slope of
4
4 4
3 . AB and CD are
parallel. The diagonal
349
4
BD has a slope of –2.
4
The diagonal AC has a
1
slope of 2. The diagonals
are perpendicular.
~ABCD is a rhombus.
4
4
65. RP has a slope of 23 . RQ
4
has a slope of –32 . RP is
the neg. reciprocal of
4
RQ, so kPQR is a right
triangle.
76. [2] Find slope of
2x ± y ≠ 3:
y ≠ –2x ± 3,
therefore slope is –2.
6
Find x: 21 2
2 x ≠ –2,
24
1 2 x ≠ –2,
–4 ≠ –2 ± 2x,
x ≠ –1
(OR equivalent
explanation)
[1] correct value of x but
no work OR minor
computation error in
work
349