Proving slope is constant using similarity

we tend to be told in algebra class that if we have a line our line will have a constant rate of change of Y with respect to X or another way of thinking about that a line will have a constant inclination or that our line will have a constant slope and our slope is literally defined as your change in Y this triangle is the Greek letter Delta it's a shorthand for change in it means change in Y Delta Y means change in Y over change in X and if you're dealing with a line this right over here is constant constant for a line what I want to do in this video is to actually prove that using similar triangles from geometry so let's think about two two sets of two points so let's say that's a point then we do in a different color let me start at this point and let me end up at that point so what is our change in X between these two points so this points X value is right over here this points X value is right over here so our change in X our change in X is going to be that right over there and what's our change in Y well this point to Y value is right over here this points Y value is right over here so this height or this height is our change in Y so that is our change in Y now let's look at two other points let's say I have this point and this point right over here and let's do the same exercise what's the change in X well let's see if we're going this Y this points X values here this points X values here so if we start here and we go this far this would be this would be the change in X between this point and this point and this and this is going to be the change let me do that same green color so this is going to be the change in X between those two points and our change in Y well this Y value is here this Y value is up here so our change in Y is going to be that right over here so what I need to show I'm just picking two arbitrary points I need to show that the ratio of this change in Y to this change of X is going to be the same as the ratio of this change in Y to this change of X or the ratio of this purple side to this green side is going to be the same as the ratio of this purple side to this green side remember I'm just picking two sets of arbitrary points here and the way that I will show it is through similarity if I can show that this triangle is similar to this triangle then we are all set up and just as a reminder of what similarity is two triangles are similar and there's multiple ways of thinking about it so you're similar if and only if all corresponding or I should say all three angles are the same or are congruent so all three and let me be carefully they don't have to be the same exact exact angle the corresponding angles have to be the same so corresponding so core this corresponding always spell it corresponding corresponding angles angles are going to be equal or we could say they are congruent so for example if I have this triangle right over here and this is 30 this is 60 and this is 90 and then I have this triangle right over here I'll try to draw it so this triangle where this is 30 degrees this is 60 degrees and this is 90 degrees even though their side lengths are different these are going to be similar triangles or essentially scaled up versions of each other all the corresponding angles 60 is corresponds to this 60 30 corresponds to this 30 and 90 corresponds to this one so these two triangles are similar and what's neat about similar triangles if you can establish that two triangles are similar then the ratio between corresponding size is going to be the same so if these two are similar then the ratio of this side to this side is going to be the same as the ratio of this let me do that pink color of this side of this side to this side and so you can see why that will be useful in proving that the slope is constant here because all we have to do is look if these two triangles are similar then the ratio between corresponding sides is always going to be the same we've picked two arbitrary sets of points then this would be true really for any two arbitrary sets of points across the line would be true for the entire line so let's try to prove similarity so the first thing we know is that both of these are right triangles these green lines are perfectly horizontal these purple lines are purple lines literally go along the exit or go in the horizontal direction the purple lines go in the vertical direction so let me make sure that we mark that so we know that these are both right-angles so we have one corresponding angle that is congruent now we have to show that the other ones are and we can show that the other ones are using our knowledge of parallel lines and transversals let's look at these two green lines so and I'll continue them these are line segments but if we view them as lines and we just continue them on and on and on so let me do that just like here so this line is clearly parallel to that one they essentially are perfectly horizontal and now you can view our orange line as a transversal and if you view it as a transversal then you know that this angle corresponds to this angle and we know from transversals of parallel lines that corresponding angles are congruent so this angle is going to be congruent to that angle right over there now we make a very similar argument for this angle but now we use the two vertical lines we know that this segment we could continue it as a line so we could continue it if we want it as a line so just like that a vertical line and we can continue this one as a vertical line we know that these are both vertical they're just measuring they're exactly in the Y direction in the vertical direction so this line this line is parallel to this line right over here once again our orange line is a transversal is a transversal of it and this angle corresponds to this angle right over here and there we have it they're congruent corresponding angles the transversal of two parallel lines are congruent we learn that in geometry class and there you have it all of the corresponding this angle is congruent to this angle this angle is congruent to that angle and then both of these are 90 degrees so both of these are similar triangles so both of these are similar so let me write that down so we know that these are both similar similar similar triangles and now we can use the common ratio of both sides so for example if we called if we call this side length a and we said that this has the side has length B and we said this side has length C and this side has length D we know for a fact that the ratio because these are similar triangles the ratio between corresponding sides the ratio of A to B the ratio of A to B is going to be equal to the ratio of C to D is equal to the ratio of C to D and so and that ratio is literally the definition of slope your change in Y over your change in X and this is constant because any sum any triangle that you generate this right triangles that you generate between these two points we've just shown that they are going to be similar and if they are similar than the ratio of this vertical line the length of this vertical line segment to this horizontal line segment is constant that is the definition of slope so the slope is constant for a line