Proof: perpendicular lines have negative reciprocal slope | High School Math | Khan Academy

What I'd like to do in this video is use some geometric arguments to prove that the slopes of perpendicular lines are negative reciprocals of each other.

So, just to start off, we have lines L and M, and we're going to assume that they are perpendicular, so they intersect at a right angle. We see that depicted right over here.

Now, I'm going to construct some other lines here to help us make our geometric argument. So, let me draw a horizontal line that intersects at this point right over here. Let's call that point A. And so, let me see if I can do that. There you go. So that's a horizontal line that intersects at A.

Now I'm going to drop some verticals from that. So I'm going to drop a vertical line right over here, and I'm going to drop a vertical line right over here. And so that is 90° and that is 90°. I've constructed it that way. This top line is perfectly horizontal, and then I've dropped two vertical things, so they're 90° angles.

Let me now set up some points. So that already said point, that's point A. Let's call this point B, let's call this point C, let's call this point D, and let's call this point E right over here.

Now let's think about what the slope of line L is. So, slope of... Let me move this over a little bit. So, slope of L is going to be what? Well, you could view line L as line the line that connects point C to A. So, it's the slope of CA. You could say this is the same thing as the slope of line CA. L is line CA, and so to find the slope, that's change in Y over change in X.

So, our change in Y is going to be CB. So, it's going to be the length of segment CB; that is our change in Y. So, it is CB over our change in X, which is the length of segment BA, which is the length of segment BA right over here. So that is BA.

Now, what is the slope of line M? So, slope of M... and we could also say slope of... we could call line M line AE, line AE like that. Well, if we're going to go between points A and point E, once again, it's just change in Y over change in X. Well, what's our change in Y going to be? Well, we're going to go from this level down to this level as we go from A to E.

We could have done it over here as well. We're going to go from A to E—that is our change in Y. So we might be tempted to say, well, that's going to be the length of segment DE, but remember our Y is decreasing, so we're going to subtract that length as we go from this Y level to that Y level over there.

And what is our change in X? So our change in X, we're going to go... as we go from A to E, our change in X is going to be the length of segment AD. So, AD.

So, our slope of M is going to be negative DE; it's going to be the negative of this length because we're dropping by that much. That's our change in Y over segment AD.

So, some of you might already be quite inspired by what we've already written because now we just have to establish that these two triangles, triangle CB and triangle AED, are similar. Then we're going to be able to show that these are the negative reciprocals of each other.

So, let's show that these two triangles are similar. Let's say that we have this angle right over here, and let's say that angle has measure X, just for kicks. And let's say that we have... let me do another color for... let's say we have this angle right over here, and let's say that the measure that that has, measure Y.

Well, we know X + Y + 90 is equal to 180 because together they are supplementary. So, I could write that X + 90 + Y is going to be equal to 180°. If you want, you could subtract 90 from both sides of that, and you could say, look, X + Y is going to be equal to 90°; DE is going to be equal to 90°.

These are algebraically equivalent statements. So, how can we use this to fill out some of the other angles in these triangles? Well, let's see. X plus this angle down here has to be equal to 90°, or you could say X + 90 + what is going to be equal to 180? I'm looking at triangle CBA right over here; the interior angles of a triangle add up to 180°.

So, X + 90 + what is equal to 180? Well, X + 90 + Y is equal to 180; we already established that. Similarly, over here, Y + 90 + what is going to be equal to 180? Well, same argument; we already know Y + 90 + X is equal to 180.

So, Y + 90 + X is equal to 180°. And so, notice we have now established that triangle ABC and triangle EDA, they all have their interior angles—their corresponding interior angles are the same.

So, they have three different angle measures; they correspond to each other. They both have an angle of X; they both have a measure of X; they both have an angle of measure Y, and they're both right triangles.

So, just by Angle-Angle-Angle (AAA), one of our similarity postulates, we know that triangle EDA is similar to triangle ABC. This tells us that the ratio of corresponding sides are going to be the same.

So, for example, we know—let's find the ratio of corresponding sides. We know that the ratio of, let's say, CB to BA. So, let's write this down. We know that the ratio—this tells us that the ratio of corresponding sides are going to be the same.

So, the ratio of CB over BA is going to be equal to... well, the corresponding side to CB is the side opposite it, the X degree angle right over here. So, the corresponding side to CB is side AD. So, that's going to be equal to AD over—what's the corresponding side to BA? Well, BA is opposite the Y degree angle, so over here the corresponding side is DE.

Let me do that in the same color—over DE. And so, this right over here, we saw from the beginning, this is the slope; this is the slope of L.

So, slope of L... and how does this relate to the slope of M? Notice the slope of M is the negative reciprocal of this. You take the reciprocal, you're going to get DE over AD, and then you have to take this negative right over here.

So, we could write this as the negative reciprocal of the slope of M. Negative reciprocal of M's slope. And there you have it. We've just shown that if we start with—if we assume these lines L and M are perpendicular, and we set up these similar triangles, we were able to show that the slope of L is the negative reciprocal of the slope of M.