If you have already learned about systems of linear equations, then you have probably discussed that the product of the slope of perpendicular lines is . The proof of this theorem comes from the fact that any point rotated 90 degrees at about the origin becomes . One example of this is shown below. The point , when rotated degrees counterclockwise becomes .
With this fact, we prove this theorem.
Slope of Perpendicular Lines Theorem
If two lines with slopes and are perpendicular, then .
Let and be points on line passing through the origin.
If we let be the slope of and be the slope of , then
Multiplying and , we have
That proves the theorem above.