Using similar triangles to reason about slope | Grade 8 (TX) | Khan Academy

So you have likely already learned about the notion of the slope of a line and what we define that is. The change in y over the change in x as we go from any one point on the line to another point on the line. Some of you, when you first saw this, might be wondering, “Hey, will that come up with the same slope no matter what two points I pick?” And that's what we're going to talk about in this video.

So let's start off with this line on the left and let's make ourselves feel good that no matter which two points on this line we pick, if we calculate the slope correctly, we will always get the same value. For a line, the slope is always going to be the same.

Let's start with these two points that they have given us. We can think about what our change in x is going to be; I'll do that in purple. That feels like a good color for change in x. Our change in x over here, that's supposed to be looks like an “a.” That's a Delta, a triangle right over Delta x. Our change in x over here, we're going from x = 8 to x = 4, so it takes us 8 to get to the y-axis and then four more. Our change in x is 12.

What is our change in y? Well, let's go from where our y started, which is 4, and let's go all the way up over here. So our y is now going to be 5. Our change in y is equal to, it takes us 4 to get to the x-axis and 5 more. Our change in y is 9.

So if we only used these two points and we wanted to calculate the slope, our change in y over change in x would be 9 over 12. 9 over 12. So it looks like, at least just using those two points, the slope of this line is equal to 9 over 12. But let's see if that's true if we pick some other points on this line.

Let's say I were to pick—and I'm going to pick some points where I can clearly see the coordinates—so let's say that point right over there and that the coordinates there. Let's see, x is -4 and then y is -1 there. Then let me pick another point that clearly seems to be on the line. So let's see, this one right over here, this is the point x = 0 and y is equal to 2.

So let's look at the slope between these two points. To do that, I can construct another triangle here where I can say my change in x is that right over there, and that's 1, 2, 3, 4. So my change in x is equal to 4. And then what's my change in y? My change in y, I'm going 3 up. My change in y is equal to 3.

So what's my slope over here? It's change in y over change in x or 3 over 4. So I am getting a different number here, but notice these are equivalent. 3 over 4 is the same as 9 over 12. Those are equivalent fractions, so the slope is the same for these two that I picked.

Now you might say, “Hey, maybe you just got lucky; you just picked points that happen to work out.” But now let's think a little bit more generally. Look at these triangles—these purple and red triangles that I've drawn over here. Some of you might be familiar with the notion of similar triangles because the base of these is parallel to the x-axis and then our height is parallel to the y-axis.

We know that these are right triangles because the x and y axes are also perpendicular to each other. So these are both right triangles. I won't go into all of the details here, but we also know that the corresponding angles are going to be the same. If you don't know what that is just now, don't worry about it, but these are similar triangles.

In fact, any triangle that you draw between any two points on this line in a similar way are going to be similar. So that's similar to that and to that. The ratio of the height to the base in this situation is always going to be the same. You're always going to get the same ratio—whether it's 3 over 4 or 9 over 12.

Now what about a scenario where we're dealing with a negative slope? Well, let's figure that out as well. We could draw multiple triangles here, and we have to think about what's happening with directions here.

Here, we only deal with positive values, but obviously, the direction that we're moving in matters a lot. So let's say as we go from this point to this point, what's going on? We could do change in x first. So our change in x, we're going from x = -5 to x = -2, so our change in x is equal to positive 3.

But then when our change in x is positive 3, what is our change in y? Well, our y is actually going down in this scenario. It's going down by how many? It's going from y = 9 to y = 3.

It's going down by 1, 2, 3, 4, 5, 6. So change in y is equal to -6. If we wanted to know our slope, our change in y over our change in x, this is going to be equal to -6. Change in y over our change in x over 3, which is equal to -2.

Let's see if that's true somewhere else, and I encourage you—you can pause this video and you can try this anywhere you want on this line. Just make sure you get your signs right. Let's try it. I'm going to find some other point where it clearly intersects.

Let's go between this point and that point where we clearly are intersecting the grid, so to speak. Here, our change in x. Change in x, we're going two in the positive x direction, which is 2. And then our change in y, we are going down again. Our change in y is equal to -4.

Well, what is -4 over 2? Well, -4 divided by 2 is -2 again. You can do this anywhere you want. You're just going to keep generating these similar triangles.

It's going to be the same no matter where we go on this line. The slope on this line, no matter which two points we use, is always going to be the same. The slope of a line, by definition, is going to be constant.