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Proof: perpendicular lines have negative reciprocal slope

CCSS Math: HSG.GPE.B.5

Video transcript

- [Voiceover] What I'd like to do with this video is use some geometric arguments to prove that the slopes of perpendicular lines are negative reciprocals of each other. And so, just to start off. We have lines L and M and we are going to assume that they are perpendicular. So, they intersect at a right angle. We see that depicted right over here. And so, I'm gonna now construct some other lines here to help us make our geometric argument. So, let me draw a horizontal line that intersects 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. And now, I'm gonna drop some verticals in that. So, I'm gonna drop a vertical line right over here, and I'm gonna drop a vertical line right over here. And so, that is 90 degrees, and that is 90 degrees. And I've constructed it that way. This top line is perpendicular horizontal. And then, I've dropped to vertical things. So, there are at 90 degree angles. And let me know setup some points. So, that I already said, 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 slop of line L is. So, slope of L is going to be what? Well, you view line L as the line that connects point CA, so it's the slope of CA, you could say. This is the same thing as 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 gonna be the length of CB. That is our change in Y. So, it is CB over our change in X, which is the length of segment of BA, which is the length of 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 point 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, our change in Y. Well, we're gonna 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 gonna go from A to E. That is our change in Y. So, we might be tempted to say, well that's just gonna be the length of segment DE. But remember, our Y is decreasing. So, we're gonna 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, as we go form 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. That's going to be the negative of this length, cause we're dropping by that much. That's our change in Y, over segment A, over segment AD. So, some of you might already be quite inspired by what we've already written, because now we just have to established that these two triangles, triangle CBA and triangle ADE are similar, and then we're going to be able to show that these are the negative reciprocal of each other. So, let's show that these two triangles are similar. So, 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 degrees. If you want, you can subtract 90 from both sides of that, and you could say look, X + Y is going to be equal to 90 degrees. These are algebraically equivalent statements. So, is equal to 90 degrees. And how can we use this to fill out some of the other angles in these triangles. Well, let's see X + this angle down here has to be equal to 90 degrees. 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, all of their interior angles, their corresponding interior angles are the same, or that their three different angle measures all 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, so we could say by angle, angle, angle, one of our similarity postulates. We know that triangle EDA is similar to triangle ABC. And so, that tells us that the ratio of corresponding sides are going to be the same. And so, for example, we know. Let's find the ration 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, so 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, it's the side opposite 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. AD over DE. Let me do that same color. Over DE. And so, this right over, we saw from the beginning, 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 slope of M. Negative reciprocal of M's slope. And there you have it. We've just shown that if we assume L and M are perpendicular, and we setup these similar triangles, and we're able to show, that the slope of L is the negative reciprocal of the slope of M.