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84 Chapter 2 Linear Equations and Functions
PARALLEL AND PERPENDICULAR LINES Recall that two lines in a plane are
parallel if they do not intersect. Two lines in a plane are perpendicular if they
intersect to form a right angle.
Slope can be used to determine whether two different nonvertical lines are
parallel or perpendicular.
KEY CONCEPT For Your Notebook
Slopes of Parallel and Perpendicular Lines
Consider two different nonvertical lines l 1 and l 2 with slopes m 1 and m 2 .
Parallel Lines The lines are parallel if and only if
they have the same slope.
E XAMPLE 4 Classify parallel and perpendicular lines
Tell whether the lines are parallel, perpendicular, or neither.
a. Line 1: through (22, 2) and (0, 21) b. Line 1: through (1, 2) and (4, 23)
Line 2: through (24, 21) and (2, 3) Line 2: through (24, 3) and (21, 22)
Solution
a. Find the slopes of the two lines.
m 1 5
m 2 5
21 2 2
} 5
0 2 (22) 23
} 52
2 3 }
2
3 2 (21)
} 2 2 (24) 5 4 } 6 5 2 } 3
c Because m 1 m 2 52 3 } 2 p 2 } 3 521, m 1 and m 2
are negative reciprocals of each other. So, the
lines are perpendicular.
b. Find the slopes of the two lines.
23 2 2
m 5} 5 1 4 2 1 25
} 52
3 5 }
3
m 2 5
m 1 5 m 2
Perpendicular Lines The lines are perpendicular if
and only if their slopes are negative reciprocals of
each other.
m 52 1 1 } , or m m 521
m2 1 2
22 2 3
} 21 2 (24) 5 25
} 3 52 5 } 3
c Because m 1 5 m 2 (and the lines are different),
you can conclude that the lines are parallel.
l 1
(22, 2)
2
21
(24, 21)
(24, 3)
Line 2
(21, 22)
1
y l 1 l 2
y
y
1
y
(2, 3)
Line 2
(0, 21)
(1, 2)
(4, 23)
l 2
x
x
Line 1
Line 1
x
x