# Point-To-Line Distance Formula: Algebraic Proof

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• Published on Jan 13, 2021

In this video I go over deriving the formula for the shortest distance between a point and a line. There are several different ways of deriving this, and in this video I use an algebraic derivation. This method involves using the fact that the shortest distance between a point and a line is the line that is perpendicular to the other line. Thus both lines are negative reciprocals of each other. Combined with the Pythagorean theorem to obtain the square of the distance in determines of the squares of the differences in x and y, we can then play around with some algebra to obtain our final formulation.

Note the general proof used in this video involves a derivation which is not valid for vertical or horizontal lines BUT the final result still holds true nonetheless! This is actually a very interesting result and illustrates how we must always use mathematical rigor regardless of whether the final formula is valid for cases that weren't valid in the proof methodology; so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhv8AcV6RCgPi8zuO4g

View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-point-to-line-distance-formula-algebraic-proof

Related Videos:

Negative Reciprocals and Perpendicular Lines: http://youtu.be/Ue7FmrfmuX4
Foil Method - Simple Proof and Quick Alternative Method: http://youtu.be/tmj_r94D6wQ
Simple Proof of the Pythagorean Theorem: http://youtu.be/yt-EJlbJQp8 .

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