Slope Criterion For Perpendicular Lines

Remember the slope of a line is

Prove the slope criterion for perpendicular lines.

The Slope Criterion for Perpendicular Lines states:

The non-vertical lines are perpendicular if and only if the product of their slopes is -1

Lets start out with a simple picture of a line.

But this isn't the exact mathematical formula. When you're dealing with the negative numbers, we need the exact formula.

In this picture, we made a slope triangle to make it easier. We can see that y₂ - y₁ = r, while x₂ - x₁ = q. This means

For our new line, we can see y₂ - y₁ = q, but this time x₂ - x₁ = -r. Why negative? Because this time, x₂ is smaller than x₁. (you can even see that if we use the actual numbers on our picture, -3 - (-1) = -2

So this means

Now we're going to rotate everything 90°. This will give us a perpendicular line, and its slope triangle.

simplifying

So our original theorem was that multiplying the two slopes will give us -1. Let's see if this is true

Cross canceling

X = -1

And thus we have proved the criterion (not really, but good enough).

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