Distance Between a Point and a Line Proof (Algebraic Method)
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In this video, we will show how to derive the distance between a point and a line using only algebra. There are faster ways to obtain this formula, however, some may be required to derive the formula using algebraic methods.
Here are the steps:
1) Place the line ax + by + c = 0 in slope intercept form
2) Obtain the slope intercept form of the perpendicular line
3) Find the point of intersection by solving a linear system
4) Plug the two points into the distance formula and simplify
d = (|ax_1 + by_1 + c|) / (sqrt(a^2 + b^2))
00:00 Introduction
01:00 Sketch of setting up a perpendicular line segment
02:33 Placing the line in slope-intercept form
03:27 Finding the perpendicular line
05:04 Finding x_2
12:12 Finding y_2
18:44 Setting up the distance formula
19:36 Simplifying (x_2 - x_1)^2
24:09 Simplifying (y_2 - y_1)^2
28:57 Final steps
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