Distance Between Parallel
Lines
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: November 15, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. Distance Between Parallel Lines
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Distance Between Parallel
Lines
Here you’ll learn that the shortest distance between two parallel lines is the length of a perpendicular line between
them.
What if you were given two parallel lines? How could you find how far apart these two lines are? After completing
this Concept, you’ll be able to find the distance between two vertical lines, two horizontal lines, and two non-vertical,
non-horizontal parallel lines using the perpendicular slope.
Watch This
MEDIA
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CK-12 Finding The Distance Between Parallel Lines
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James Sousa: Determining the Distance Between Two Parallel Lines
Guidance
All vertical lines are in the form x = a, where a is the x−intercept. To find the distance between two vertical lines,
count the squares between the two lines. You can use this method for horizontal lines as well. All horizontal lines
are in the form y = b, where b is the y− intercept.
In general, the shortest distance between two parallel lines is the length of a perpendicular segment between them.
There are infinitely many perpendicular segments between two parallel lines, but they will all be the same length.
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Remember that distances are always positive!
Example A
Find the distance between x = 3 and x = −5.
The two lines are 3 – (-5) units apart, or 8 units apart.
Example B
Find the distance between y = 5 and y = −8.
The two lines are 5 – (-8) units apart, or 13 units apart.
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Chapter 1. Distance Between Parallel Lines
Example C
Find the distance between y = x + 6 and y = x − 2.
Step 1: Find the perpendicular slope.
m = 1, so m⊥ = −1
Step 2: Find the y−intercept of the top line, y = x + 6. (0, 6)
Step 3: Use the slope and count down 1 and to the right 1 until you hit y = x − 2.
Always rise/run the same amount for m = 1 or -1.
Step 4: Use these two points in the distance formula to determine how far apart the lines are.
q
(0 − 4)2 + (6 − 2)2
q
= (−4)2 + (4)2
√
= 16 + 16
√
= 32 = 5.66 units
d=
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CK-12 Finding The Distance Between Parallel Lines
–>
Guided Practice
1. Find the distance between y = −x − 1 and y = −x − 3.
2. Find the distance between y = 2 and y = −4.
3. Find the distance between x = −5 and x = −10.
Answers:
1. Step 1: Find the perpendicular slope.
m = −1, so m⊥ = 1
Step 2: Find the y−intercept of the top line, y = −x − 1. (0, -1)
Step 3: Use the slope and count down 1 and to the left 1 until you hit y = x − 3.
Step 4: Use these two points in the distance formula to determine how far apart the lines are.
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Chapter 1. Distance Between Parallel Lines
q
(0 − (−1))2 + (−1 − (−2))2
q
= (1)2 + (1)2
√
= 1+1
√
= 2 = 1.41 units
d=
2. The two lines are 2 – (-4) units apart, or 6 units apart.
3. The two lines are -5 – (-10) units apart, or 5 units apart.
Explore More
Use each graph below to determine how far apart each pair of parallel lines is.
1.
2.
3.
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4.
Determine the shortest distance between the each pair of parallel lines. Round your answer to the nearest hundredth.
5.
6.
7.
8.
9.
10.
x = 5, x = 1
y = −6, y = 4
y = 3, y = 15
x = −10, x = −1
x = 8, x = 0
y = 7, y = −12
Find the distance between the given parallel lines.
11.
12.
13.
14.
y = x − 3, y = x + 11
y = −x + 4, y = −x
y = −x − 5, y = −x + 1
y = x + 12, y = x − 6
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 3.11.
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