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# Proof: perpendicular lines have negative reciprocal slope

## Video transcript

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 and so just to start off we have lines ln 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 and so I'm going to now 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 and 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 degrees and that is 90 degrees and I've constructed it that way this top line is perfectly horizontal and then I've dropped two vertical things so there are 90-degree angles and let me now set up some points so that I 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 me move this over a little bit so slope slope of L is going to be what well that's at you could view line L is line the line that connects Point C a so it's the slope of of CA you could say this is the same thing as slope of line see a 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 C B so it's going to be the length of segment CB that is our change in Y so it is CV over our change in X which is the length of segment B a to the length of segment ba right over here so that is B a now what is the slope of line M so slope 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 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 that's just going to be the length of segment D e but remember our Y is decreasing so we're going to subtract that length as we go from this wide level to that Y level over there and what is our change in X so our change in X we're going to go for 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 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 establish that these two that these two are these two triangles triangles CBA and triangle a de 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 let's let's show that these two triangles are similar so let's 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 why plus 90 is equal to 180 because together they are supplementary so I could write I could write that X plus 90 plus 90 plus y plus y is going to be equal to is going to be equal to 180 degrees if you want you could subtract 90 from both sides of that and you could say look X plus y is going to be equal to 90 degrees 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 plus this angle down here has to be equal to 90 degrees or you could say X plus 90 plus what is going to be equal to 180 I'm looking at triangle CBA right if you're the interior angles of a triangle add up to 180 so X plus 90 plus what is equal to 180 well X plus 90 plus y is equal to 180 we already established that similarly over here y plus 90 plus what is going to be equal to 180 well same argument we already know y plus 90 plus X is equal to 180 so y plus 90 plus X is equal to is equal to 180 and so notice we have now established that triangle triangle ABC and triangle e da they're all of their interior angles their corresponding interior angles are the same or they and so or that there are three different angle measures all correspond to each other they both have an angle of X they both have a measure X they both have an angle of measure Y and they're both right angle 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 II triangle e da e da is similar to triangle two triangle a B C to triangle a b c 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 type find the ratio of corresponding sides we know that the ratio of let's say c b to be a 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 C B over B a over b a is going to be equal to is going to be equal to well the corresponding side to C B it's the side opposite the x degree angle right over here so the corresponding side to C B is side ad so that's going to be equal to a D over what's the corresponding side to be a 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 here this right over here we saw from the beginning this is the slope this is the slope of L so the slope 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 can 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 reciprocal of of em's of M's slope and there you have it we've just shown that if we start with purpose if we assume these lot L and M are perpendicular and we use and we set up these similar triangles and we were able to show that the slope of L is the negative reciprocal of the slope of M