Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
K12 LibreTexts

4.38: Distance Between Parallel Lines

( \newcommand{\kernel}{\mathrm{null}\,}\)

Length of a perpendicular segment between parallel lines.

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.

f-d_a5654a10f6ef82f03b3a166e4c290d7fa1582402ea756fec0200c16f+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.1

Remember that distances are always positive!

Example 4.38.1

Find the distance between x=3 and x=5.

f-d_73a5eedfb5a83e601a9684a35c3e4d9135d085876a977eb14bf1b0ea+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.2

Solution

The two lines are 3(5) units apart, or 8 units apart.

Example 4.38.2

Find the distance between x=5 and x=10.

Solution

The two lines are 5(10) units apart, or 5 units apart.

Example 4.38.3

Find the distance between y=5 and y=8.

f-d_01841c05d72ef3c9c43e45aa95cc0ebbea7ee4391d30b9916842b204+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.3

Solution

The two lines are 5(8) units apart, or 13 units apart.

Example 4.38.4

Find the distance between y=x+6 and y=x2.

f-d_dda43ce8031bd9d34ec5c4fe9a68b6b40c180cf987fdf4a0477a0d3a+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.4

Solution

Step 1: Find the perpendicular slope.

m=1, so m=1.

Step 2: Find the y-intercept of the top line, y=x+6.

The intercept is (0,6).

Step 3: Use the slope and count down 1 and to the right 1 until you hit y=x2.

Always rise/run the same amount for m=1 or m=1.

f-d_66ac1f309ccdb9bb4a8d56722dcecdd0f5d4ea2c94f5a4051d72b775+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.5

Step 4: Use these two points in the distance formula to determine how far apart the lines are.

d=(04)2+(62)2=(4)2+(4)2=16+16=32=5.66units

Example 4.38.5

Find the distance between y=x1, and y=x3.

f-d_17a69cb952c6d5e771a1bf8579fb3acd295e0436375da06b960fdcce+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.6

Solution

Step 1: Find the perpendicular slope.

m=1, so m=1.

Step 2: Find the y-intercept of the top line, y=x1.

The intercept is (0,1).

Step 3: Use the slope and count down 1 and to the left 1 until you hit y=x3.

f-d_e7cba0e239341827ec5dff12a53279e6985239990663bc2e3e41e002+IMAGE_TINY+IMAGE_TINY.png
Figure 4.38.7

Step 4: Use these two points in the distance formula to determine how far apart the lines are.

d=(0(1))2+(1(2))2=(1)2+(1)2=1+1=2=1.41units

Review

Use each graph below to determine how far apart each pair of parallel lines is.

  1. f-d_dab9056943a53623e49490753babd8daa99f435a513472203e9940f7+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.38.8
  2. f-d_b2673221e7c7724d90b28f2d79bb2ce8ffd2f5f044daff5e57a9fff5+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.38.9
  3. f-d_9b5821a7afce2599e667652e43432834528c7f64825c167497be6775+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.38.10
  4. f-d_f9050a2af91200bfe2b576821698434792c868c89dc9f9ba3a1b276c+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.38.11

Determine the shortest distance between the each pair of parallel lines. Round your answer to the nearest hundredth.

  1. x=5,x=1
  2. y=6,y=4
  3. y=3,y=15
  4. x=10,x=1
  5. x=8,x=0
  6. y=7,y=12

Find the distance between the given parallel lines.

  1. y=x3,y=x+11
  2. y=x+4, y=x
  3. y=x5, y=x+1
  4. y=x+12, y=x6

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.11.

Resources

Vocabulary

Term Definition
Distance Formula The distance between two points (x1,y1) and (x2,y2) can be defined as d=(x2x1)2+(y2y1)2.
Perpendicular Perpendicular lines are lines that intersect at a 90 angle. The product of the slopes of two perpendicular lines is -1.

Additional Resources

Interactive Element

Video: Finding The Distance Between Parallel Lines Principles - Basic

Activities: Distance Between Parallel Lines Discussion Questions

Study Aids: Lines in the Coordinate Plane

Practice: Distance Between Parallel Lines

Real World: Flipping Out


This page titled 4.38: Distance Between Parallel Lines is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.

CK-12 Foundation
LICENSED UNDER
CK-12 Foundation is licensed under CK-12 Curriculum Materials License

Support Center

How can we help?