# How to Find the Distance Between Parallel Lines

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• 0:03 Finding the…
• 1:49 Finding the Perpendicular Line
• 3:07 Finding the Crossing Points
• 6:23 Concise Expressions…
• 8:12 Looking at Some Special Cases
• 9:37 Lesson Summary

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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Parallel lines are very common features in architecture and construction. We see them in ladder rungs, railroad tracks and city streets. In this lesson, we'll use algebra and geometry to find the distance between parallel lines.

## Finding the Distance

You're driving to Elm Street Bakery, the best bakery in town. On Oak Street, a street that runs parallel to Elm Street, you notice that your car is almost out of gas. You could take a street that runs perpendicular to both Oak and Elm Street, but you're not sure if your car will make it. Knowing the exact distance between these two parallel streets would be helpful.

When finding the distance between two parallel lines, we determine the distance along a perpendicular line crossing both parallel lines. This calculation uses many concepts and skills from geometry and algebra: expressing the equation of a line, using the value of the slope, solving two equations with two unknowns and using distance formulas.

## Plotting Parallel Lines

Let's say we have two lines where the equation for line 1 is 4x - 3y = 5 while for line 2, we have 4x - 3y = -15. We'll organize each equation into its slope-intercept form, which looks like y = mx + b.

For line 1: 4x - 3y = 5 becomes -3y = -4x + 5. Dividing by -3 gives y = (4/3) x - 5/3. The slope m is 4/3 and the intercept b is -5/3.

Similarly for line 2: 4x - 3y = -15 becomes y = (4/3)x + 5. Both lines have the same slope of m = 4/3, meaning the lines are parallel to each other and never cross.

## Finding the Perpendicular Line

With parallel streets, like Elm Street and Oak Street, a perpendicular street lets us cross over from one parallel street to the other. Streets have a finite length, so there's a finite number of perpendicular streets between any two parallel streets. Lines, however, are infinitely long, so there's an infinite number of perpendicular lines, or perp lines, crossing line 1 and line 2. Since we have to choose one perp line, we might as well choose one that simplifies the calculations.

The simplest is the perp line passing through origin. This line has the form y = (-3/4)x. Note the slope is the negative reciprocal of 4/3. All perp lines have a negative reciprocal, sometimes called opposite reciprocal, slope to the line they are perpendicular to. We are choosing the perp line through the origin because it has an intercept of zero. Let's plot the perp line along with the two parallel lines:

See how this perp line passes through the origin at (0,0)? See the point where the perp line crosses line 2? This point is (x2, y2). The crossing at line 1 is (x1, y1). We are getting closer to the bakery. Can we go the distance?

## Finding the Crossing Points

When we solve for two unknowns in two equations, we are finding the point where two lines cross. To find the crossing point (x2, y2) we substitute y = (-3/4)x into y = (4/3)x + 5 and solve for x. You can label your solution as x2 at the end or at the beginning. Labeling x as x2 and y as y2:

x2 is -12/5, which equals -2.4. To find y2: y2 = (-3/4)x2 = (-3/4)(-2.4) = 1.8.

We find the point (x1, y1) by solving y = (-3/4)x with y = (4/3)x - 5/3. We substitute (-3/4)x1 for y1 and solve for x1:

x1 is 4/5, which equals .8. Then, y1 = (-3/4)x1 = (-3/4) * .8 = -.6.

So far we have x2 = -2.4, x1 = .8, y2 = 1.8 and y1 = -.6. The distance between two points is calculated by finding the square root of the sum of the squared differences of the individual coordinates:

We have a distance of 4 between the parallel lines.

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