Theorem 1.9: Given a line whose equation is

y = mx + b

and a point whose coordinates are

(x1, y1)

the equation of the unique line through the point, perpendicular to the line, is

Proof: By Theorem 1.8, there will be a unique line through the point perpendicular to the line whose equation is   y = mx + b.   Since this line is to be perpendicular to the line whose equation is   y = mx + b,   its slope will be the negative reciprocal of   m   or   -1/m.   If we substitute   -1/m   for the slope in the formula for the   y   intercept in Theorem 1.3, we get

If we substitute this into the   y = mx + b   form of the equation for the perpendicular line, we get the desired result.

next theorem (1.10)