# Parallel And Perpendicular Lines Worksheet Page 38

Distance Between Parallel Lines
The distance between two parallel lines is the distance between one of the lines
and any point on the other line.
Recall that a locus is the set of all points that satisfy
Study Tip
a given condition. Parallel lines can be described as the
d
Look Back
locus of points in a plane equidistant from a given line.
d
To review locus, see
Lesson 1-1.
Theorem 3.9
Theorem 3.9
In a plane, if two lines are equidistant from a third line, then the two lines are
parallel to each other.
Distance Between Lines
Example
Example
3
3
Find the distance between the parallel lines and m whose equations are
1
1
1
y
x
3 and y
x
, respectively.
3
3
3
You will need to solve a system of equations to find the endpoints of a segment
1
that is perpendicular to both and m. The slope of lines
and m is
.
3
• First, write an equation of a line p perpendicular
y
to and m. The slope of p is the opposite reciprocal
p
1
, or 3. Use the y-intercept of line , (0,
of
3), as
3
one of the endpoints of the perpendicular segment.
O
x
y
y
m(x
x
)
Point-slope form
1
1
m
y
( 3)
3(x
0)
x
0, y
3, m
3
1
1
( 0,
3 )
y
3
3x
Simplify.
y
3x
3
Subtract 3 from each side.
• Next, use a system of equations to determine the
point of intersection of line m and p.
1
1
1
1
1
1
Substitute
x
for y in the
m: y
x
x
3x
3
3
3
3
3
3
3
second equation.
p: y
3x
3
1
1
x
3x
3
Group like terms on each side.
3
3
1
0
1
0
x
Simplify on each side.
3
3
1
0
x
1
Divide each side by
.
3
y
3(1)
3
Substitute 1 for x in the equation for p.
y
0
Simplify.
The point of intersection is (1, 0).
• Then, use the Distance Formula to determine the distance between (0,
3)
and (1, 0).
2
2
d
(x
x
)
(y
y
)
Distance Formula
2
1
2
1
2
2
(0
1)
( 3
0)
x
= 0, x
= 1, y
=
3, y
= 0
2
1
2
1
10
Simplify.
The distance between the lines is
Lesson 3-6 Perpendiculars and Distance 161