Theorem 1.11: The perpendicular distance from the point

(x1, y1)

to the line

y = mx + b

is

Proof: We need only use the distance formula to find the distance from the point   (x1, y1)   to its foot in the line   y = mx + b,   which, by Theorem 1.10, is

So the distance is

We will need to find common denominators.

or

We can factor an   m   out of all the terms on the top of the first square.

Incredibly, the terms on the top of the second square are all negatives of the terms of the top of the first square, but that does not matter because they are being squared. So we can factor out the common square factors, and take their square roots outside the radical.

It will prove to be more useful to irrationalize the numerator.

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