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 .

**Proof**

Let and be points on line passing through the origin.

If we rotate the points on the origin, then, the new coordinates of the points will be

and .

If we let be the slope of and be the slope of , then

Multiplying and , we have

.

That proves the theorem above.