Theorem 4.7: The reflection of the point   (x1, y1)   in the line   y = mx + b   is

The reflection of the point   (x1, y1)   in the vertical line   x = a   is

R(x1, y1) = (2a - x1, y1)

Proof: Let   A = (x1, y1),   and let   F   be the foot of the point   A in the line. By Theorem 4.6,

R(A) = 2F - A

If the line is the vertical line   x = a,   then

F = (a, y1)

and, for the case of the vertical line, the result follows.

If the line is not vertical it will have an equation of the form

y = mx + b

by Theorem 1.2, and by Theorem 1.10

So

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