Theorem 1.10: Given a line whose equation is

y = mx + b

and a point whose coordinates are

(x1, y1)

the coordinates of the foot of the point in the line are

Proof: By Theorem 1.6, the coordinates of the point of intersection of the perpendicular line through   (x1, y1)   with the line   y = mx + b   are

and

where

m1 = -1/m

by Theorem 1.9,

m2 = m

and

b2 = b

We substitute these in and get

by Theorem 1.6. Multiply top and bottom by   m   to clear denominators

If we substitute this into   y = mx + b,   we get

We need common denominators.

Remove parentheses and add the fractions

or

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next theorem (1.11)