How to Find the Distance Between Perpendicular Lines

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will walk us through the steps of finding the distance between two perpendicular lines given two points on the lines. We will use multiple examples to illustrate these steps and solidify our understanding of this process.

Distance Between Two Perpendicular Lines

Imagine that your and your friend Sally's houses are situated on two roads that create a 90-degree angle where they intersect. Your house is 30 yards from the intersection of the roads, and Sally's house is 40 yards from the intersection of the roads. However, because you two visit each other so often, you've stomped down a trail through the grass, creating a shortcut trail that runs directly from your house to her house. You're curious how long this trail is.


distperp1


It turns out that finding the distance between your and Sally's houses is an example of finding the distance between two perpendicular lines. Two lines are perpendicular if they create a 90-degree angle where they intersect. In this scenario, the two roads are perpendicular lines, your and Sally's houses are points on the lines, and the length of the trail that you and Sally created is the distance between the two lines at those points.

Of course, the distance between the lines is dependent on the two points on the line because any two points will give a different distance. For instance, your friend, Micha, lives a little farther up Sally's road. If there were a trail between your and Micha's houses, it would have a different length from the trail between your and Sally's houses, but it would still represent the distance between the roads (or two parallel lines).


distperp2


This is all really neat, but ultimately, we want to know the length of the trail between your and Sally's houses. In other words, we want to know how to find the distance between two perpendicular lines, given two points on the lines. Let's see how to do this!

How to Find the Distance Between Two Perpendicular Lines

As we mentioned, perpendicular lines create a 90-degree angle where they intersect. Because of this, if we connect any two points on the lines with a line, we create a right triangle.


distperp3


This is extremely helpful when it comes to finding the distance between two perpendicular lines, because we have a really nice theorem for finding the length of the sides of a right triangle, and this is called the Pythagorean Theorem. The Pythagorean Theorem states that if a right triangle has side lengths a, b, and c, where c is the longest side called the hypotenuse, then it must be the case that

  • a2 + b2 = c2


distperp4


This is becoming clearer! Basically, to find the distance between two perpendicular lines given two points, all we have to do is find the distance from the points to the intersection of the lines, then we can use the Pythagorean Theorem to find the distance between the two points. That is, to find the distance between two perpendicular lines, we use these steps:

  1. Find the distance between the intersection of the lines and each of the given points on the lines; call them a and b.
  2. To find the distance between the two points, call it c, use the Pythagorean Theorem, a2 + b2 = c2, and solve for c. This will give you the distance between the two lines, given two points.

Okay, let's use these steps to figure out how long that trail is! Step one is done for us because we know that your house is 30 yards from the intersection and Sally's house is 40 yards from the intersection. All we have to do is step two, which is to use the Pythagorean Theorem to find the distance of the trail.

a2 + b2 = c2 Plug in a = 30 and b = 40
302 + 402 = c2 Simplify
2500 = c2 Take the square root of both sides
50 = c This is the length of the trail

We get that the length of the trail, or the distance between the perpendicular roads, from your house to Sally's house, is 50 yards.

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