4–7
Geometry: Parallel and Perpendicular Lines
BUILD YOUR VOCABULARY (pages 80–81)
MAIN IDEAS
• Write an equation of
the line that passes
through a given point,
parallel to a given line.
• Write an equation of
the line that passes
through a given point,
perpendicular to a
given line.
Lines in the same plane that do not
parallel lines.
Parallel Lines in a
Coordinate Plane Two
nonvertical lines are
parallel if they have the
same slope. All vertical
lines are parallel.
are called
Lines that intersect at
perpendicular lines.
EXAMPLE
KEY CONCEPT
are called
Parallel Line Through a Given Point
Write the slope-intercept form of an equation for the line
that passes through (4, -2) and is parallel to the graph
1
of y = _
x - 7.
2
1
The line parallel to y = x - 7 has the same slope, _
.
2
1
_
Replace m with and (x, y) with (4, -2) in the
2
point-slope form.
y - y 1 = m(x - x 1)
Point-slope form
1
, y with -2,
Replace m with _
2
and x with 4.
2
1
=_
(x - 4)
Simplify.
2
y+2=
y+2-
Distributive Property
1
=_
x-2-
Subtract
2
y=
from each side.
Write the equation in slopeintercept form.
Check Your Progress Write the slope-intercept form of
an equation for the line that passes through (2, 3) and is
1
x - 1.
parallel to the graph of y = _
2
98
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
y - (-2) = _
(x - 4)
4–7
KEY CONCEPT
Perpendicular Lines in a
Coordinate Plane Two
nonvertical lines are
perpendicular if the
product of their slopes is
-1. That is, the slopes are
opposite reciprocals of
each other. Vertical lines
and horizontal lines are
also perpendicular.
EXAMPLE
Determine Whether Lines are Perpendicular
GEOMETRY The height of a
trapezoid is measured on a
segment that is perpendicular
to a base. In a trapezoid ARTP,
−−
−−
−−
RT and AP are bases. Can EZ
be used to measure the height
of the trapezoid? Explain.
y
T
P
E
x
O
R
Z
Find the slope of each segment.
−−
Slope of RT: m =
A
-3 - 1
__
or
-5 - (-1)
−−
Slope of AP: m =
or
−−
Slope of EZ: m =
or
−−
−−
The slope of RT and AP is
−−
and the slope of EZ is
-7 · 1 ≠
−−
. EZ is not
.
−−
−−
to RT and AP, so
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
it cannot be used to measure height.
Check Your Progress The graph
shows the diagonals of a rectangle.
−−
Determine whether JL is
−−−
perpendicular to KM.
y
J(2, 6)
K(6, 6)
M(2, 1)
L(6, 1)
O
x
Glencoe Algebra 1
99
4–7
EXAMPLE
Perpendicular Line Through a Given Point
Write the slope-intercept form for an equation of a line
that passes through (4, -1) and is perpendicular to the
graph of 7x - 2y = 3.
Step 1 Find the slope of the given line.
7x - 2y = 3
Original equation
7x - 2y - 7x = 3 - 7x
Subtract
from
each side.
= -7 + 3
-2y
-7x + 3
_
= __
-2
-2
y=
Simplify.
Divide each side by
.
Simplify.
Step 2 The slope of the given line is
. So, the slope of the
line perpendicular to this line is the opposite reciprocal
7
2
, or - _
.
of _
2
7
Step 3 Use the point-slope form to find the equation.
y - y 1 = m(x - x 1)
7
2
= -_
(x - 4)
7
8
_
y + 1 = -2x + _
7
7
y+1
2
8
= -_
x+_
7
y=
HOMEWORK
ASSIGNMENT
Page(s):
Exercises:
100
Glencoe Algebra 1
7
(x 1, y 1) = (4, -1), m =
Simplify.
Distributive Property
Subtract.
Simplify.
Check Your Progress Write the slope-intercept form
for an equation of a line that passes through (-3, 6) and is
perpendicular to the graph of 3x + 2y = 6.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
y - (-1) = - _
(x - 4)
Point-slope form