Parallel And Perpendicular Lines Worksheet Page 38
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50Distance 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
10 or about 3.16 units.
Lesson 3-6 Perpendiculars and Distance 161
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