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How to Find the Distance Between a Point & a Line

How to Find the Distance Between a Point & a Line
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  • 0:02 Choices
  • 0:41 The Point and the Line
  • 1:10 Using a Formula to Get…
  • 3:23 Distance Without the Formula
  • 6:38 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.

Finding the distance from a point to a line is useful in many science and engineering applications. In this lesson, we calculate this distance using a formula, as well as with some fundamental concepts from geometry and algebra.

Choices

Choices! Buy the cake ready-made, or bake it from fundamental ingredients like flour, eggs, and milk? The issue might be time or availability. Maybe the store is closed, or maybe there isn't time to bake a cake from scratch.

Similar choices often exist in math: use a formula or get the answer from simpler ideas. In this lesson, we will calculate the distance from a point to a line using a formula, as well as determining this distance using fundamental ideas and equations. As with a cake, will the results be the same?

The Point and the Line

Imagine a line and a point in two-dimensional space.


The point and the line
the point and the line


We could be calculating the distance from a road to an address or the distance from the end-zone line to the location of a player on the field. In all these cases, we have the equation of a line and the coordinates of a point.

For our example, the equation for the line is y = (1/2)x - 1 and the point (x1,y1) is located at the point (2, 3).

Using a Formula to Get the Distance

Having the line equation in the form ax + by + c = 0 and knowing the coordinates of the point gives us the distance from the point to the line using the formula:


formula


What if the equation of the line is in some other form? This is the situation in our example where the line y = (1/2)x - 1 is in slope-intercept form. Let's convert to the ax + by + c = 0 form. First, multiply both sides of the equation by 2 to give 2y = x - 2.

Then, transfer all the terms to the same side and organize the equation so the x term is first: x - 2y - 2 = 0.

Comparing this equation to ax + by + c = 0, we identify: a = 1, b = -2 and c = -2. The location of the point says x1 is 2 and y1 is 3. Substituting into the formula for the distance:


formula


What if this distance formula was not available? Could we determine the distance using some ideas from geometry and algebra? It's like baking from fundamental ingredients as an alternative to buying the ready-made store product. For practice, let's do this. We'll take it one ingredient at a time.

Distance Without the Formula

This will be fun! First, visualize a line perpendicular to our line. The perpendicular line makes a right-angle (90o) with our line. Also, we want the perpendicular line to pass through the point (x1, y1).


The perperdicular line
the perp line


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