High school geometry
- Pythagorean theorem proof using similarity
- Exploring medial triangles
- Proof: Parallel lines divide triangle sides proportionally
- Prove theorems using similarity
- Proving slope is constant using similarity
- Proof: parallel lines have the same slope
- Proof: perpendicular lines have opposite reciprocal slopes
Sal uses a clever proof involving similar triangles to show that slope is constant for a line. Created by Sal Khan.
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 it, that our 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 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 2 sets of 2 points. So let's say that's a point there. Let me do it 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 2 points? So this point's x value is right over here. This point's x value is right over here. So our change in x is going to be that right over there. And what's our change in y? Well, this point's y value is right over here. This point's y value's 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 2 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 point's x value's here. This point's x value's here. So if we start here and we go this far, this would be the change in x between this point and this point. And this is going to be the change-- let me do that in the 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's 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 2 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 2 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, 2 triangles are similar-- and there's multiple ways of thinking about it-- if and only if all corresponding-- or I should say, all three angles are the same, or are congruent. And let me be careful here. They don't have to be the same exact angle. The corresponding angles have to be the same. So corresponding-- I always misspell it-- 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 I have 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. They're essentially scaled up versions of each other. All the corresponding angles-- 60 corresponds to this 60, 30 corresponds to this 30, and 90 corresponds to this 1. So these 2 triangles are similar. And what's neat about similar triangles, if you can establish that 2 triangles are similar, then the ratio between corresponding sides is going to be the same. So if these 2 are similar, then the ratio of this side to this side is going to be the same as the ratio of-- let me do that pink color-- 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 2 triangles are similar, then the ratio between corresponding sides is always going to be the same. We've picked 2 arbitrary sets of points. Then this would be true, really, for any 2 arbitrary set of points across the line. It 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 perfectly vertical because the green lines literally 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 1 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 2 green lines. So 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 1. 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 2 vertical lines. We know that this segment, we could continue it as a line. So we could continue it, if we wanted, as a line, so just like that, a vertical line. And we could 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, the vertical direction. So this line is parallel to this line right over here. Once again, our orange line 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 of the transversal of 2 parallel lines are congruent. We learned 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. Just let me write that down so we know that these are both similar triangles. And now we can use the common ratio of both sides. So for example, if we called this side length a. And we said that this 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, between corresponding sides, the ratio of a to b is going to be equal to the ratio of c to d. And that ratio is literally the definition of slope, your change in y over your change in x. And this is constant because any 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, then the ratio of 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.