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Triangle medians and centroids (2D proof)

Showing that the centroid is 2/3 of the way along a median. Created by Sal Khan.

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Video transcript

- [Voiceover] In the video on triangle medians and centroids I did, essentially, the proof that the centroid is 2/3 along the way of a median. I did it using a two-dimensional triangle in three dimensions, and I mentioned that I thought, at least it made the math a little simpler, but someone mentioned that they'd be interested in seeing the two-dimensional version of the proof, so why not do that? So let's just draw an arbitrary triangle, and I'll make one side of the triangle right along the x-axis just to make, I think it'll make the math a little bit easier, maybe there's easier ways of doing it. So that's one side of the triangle, that's the other side of the triangle, and let's put the height of the triangle, let's put it on the y-axis, so this triangle right here, this is the y-axis that is the y-axis, and then this right here, is our x-axis. Now, we said that this is an arbitrary triangle, so let's just make this distance equal to a, so this point right here, this vertex would be the point a comma zero, so that distance is a. Let's make this distance right here be equal to, I don't know, let's make it equal to b, so this is the point negative b comma zero, and then let's make this distance, this is the point y is equal to c, so this is a point zero comma c, so it's an arbitrary triangle, any triangle can be represented this way. Now, let's think about its medians. Let's think about its medians and the centroid. I'm only going to do it for two of the medians, because we know that the third median will also intersect at the same centroid, so we have a median on this, we have a midpoint, I should say, on this side of the triangle, and its coordinates are just going to be the midpoint of those two points, or zero plus a over two, which would just be a over two, and then c plus zero over two, which would just be c over two. Let's do the same thing over here. The midpoint of this side right over here is going to be negative b plus zero over two, so it's negative b over two, and then zero plus c over two, so it's just going to be c over two. Now that we know all the coordinates for the vertices and the midpoints, at least of these two sides, we can find the equations for the lines that the medians are part of, so we can find the equation for this line, we could find the equation for this line in purple, then we can find the equation for the other line that connects to these two dots, find their intersection, and we'll essentially have the coordinates of our centroid, of the intersection of the medians, and we can do it for this one, too, but that would be redundant, because it's going to intersect in the same point. So what would be the equation for this line right over here? Well, our slope is going to be change and y, so c over two minus zero, so that's just c over two, over change and x, so a over two minus minus b, so minus negative b, I should say, so it's plus b, I could write a plus b over here, but just to makes the maths involved right, plus 2b over two. I just subtracted a negative b, which is the same thing as adding a b, and adding a b is the same thing as adding a 2b over two, and I did that so these have the same denominator. I could add them, or I could just multiply the numerator and the denominator both by two and I'll get the slope is equal to, c over a plus 2b, so that's the slope of this line right over here, and then we know some points here. We could just use the point slope formula for the equation of a line. We know the point negative b comma zero is on the line, so we know that the equation of this line in purple, let me stay in the purple, is going to be y minus zero, is equal to x minus negative b, or I could say x plus b, times our slope, times c over a plus 2b. And obviously, this would just simplify to y is equal to all of this business over here, so this is essentially the equation of this median right here if the line just kept going, or the median is a line segment that is part of this line. Let us do the same thing for this line right over here, this median right over here. So once again, its slope is change and y over change and x, so the change and y, c over two minus zero, so the slope here is equal to c over two minus zero, which is just c over two, over negative b over two, minus a, or I could say minus 2a over two, that's the same thing as minus a, and do the same thing, multiply the numerator and the denominator by two. We get c over negative b minus 2a, so the equation of this line right here, the other median, the equation of this other median right over here, well we have this point over here, so we could use the point slope again. We get y minus zero is equal to x minus a times the slope, times c over negative b minus 2a, so we have the two equations for these lines. We can now use that information to find their intersection, which is going to be the centroid. Well, to do that, we have both of these in terms of y, right? Y minus zero is the same thing as y, so we could just set this as being equal to that, so let's do that. We get x plus b times c, over a plus 2b is equal to x minus a times c over negative b minus 2a. Well, both sides are divisible by c, we can assume that c is non-zero, so this triangle actually exists, it actually exists in two dimensions, so we can divide both sides by c and we get that. Let's see, now we can cross-multiply. We can multiply x plus b times this quantity right here, and that's going to be equal to a plus 2b times this quantity over here. We can just distribute it, so it's going to be x times all of this business, it's going to be x times negative b minus 2a, plus b times all of this, so that's going to be minus b squared minus 2ab, so I just multiply that times that, is going to equal to this times that, so it's going to be x times all of this business. X times a plus 2b, minus a times that, so distribute the a, minus a squared minus 2ab. Let's see if we can simplify this. We have a negative 2ab on both sides, so let's just subtract that out, so we cancelled things out, let's subtract this from both sides of the equation, so we'll have a minus x times a plus 2b, and I'll subtract that from here, but I'll write it a little different, it'll simplify things, so this will become x times, I'll distribute the negative inside the a plus 2b, so negative a minus 2b, and let's add a b squared to both sides, so plus b squared, plus b squared. I want to collect all the x terms on one side and all of the constants on the other. This becomes on the left-hand side, the coefficient on x, I have negative b minus 2b is negative 3b negative 2a minus a is negative 3a, times x is equal to, these guys cancel out, and these guys cancel out, is equal to b squared minus a squared. Let's see if we can factor, this already looks a little suspicious in a good way, something that we should be able to solve. This can be factored. We can factor out a negative three. We can actually factor out a three. We'll get three times b minus a, actually, let's factor out a negative three, so negative three times b plus a times x is equal to, now this is the same thing as b plus a times b minus a. So we can divide both sides by b plus a, we have negative three times x is equal to b minus a, let's just divide both sides by negative three. We get x is equal to b minus a over negative three, which is the same thing as a minus b over three, so we have the x-coordinate so far for our centroid. It is a minus b over three. You kind of see hints of the solution over here. A minus b is this entire distance. It's one third of this entire distance, although I don't want to jump the gun too much because it complicates just the x-coordinate. But let's figure out the y-coordinate. To do that, we can just substitute back into one of these equations up here. This was the equation of this median, the grey median that we had done, so lets substitute back in. We have, I'll do it over here, y is equal to x minus a times c. X is this thing over here, so it's a minus b over three minus a, so I'm going to subtract, instead of just writing a minus a over here, I'm going to write a negative three over three, so minus three a over three. That's this minus a right over here. I just multiplied and divided it by three. It's going to be all of that times c over negative b minus 2a, so this is going to be equal to c and the numerator up here, we have an a minus 3a, so this is equal to negative b, we have a negative b there and then we have a minus 3a is minus 2a, so this becomes negative b minus 2a, we can factor out the 1/3, times c over three, so that's what the numerator is. All of that over negative b minus 2a, well, that's going to cancel with that, and so our y-coordinate is going to be equal to c over three, so our y-coordinate over here is c over three. Now, we could just use the distance formula now. We know all the coordinates, and feel free to do it if you like, but there's a slight simplification, or at least in my mind a slight simplification. We know that the height, we know that the y-coordinate of the centroid is c over three, so this right here, this point over here, is c over three. We could actually use, maybe an easier argument, because remember, our whole point was to show that this point is 2/3 along the median away from the vertex. Let me draw this vertex now here. Our whole point of this video as we said this is the midpoint of this side over here is to show that this right here is 2/3 along the way, or that this length is twice this length, or that this length is 2/3 of the entire length of the median. So how can we do that? Well, the simplest way I can do that, I mean you could actually just use the coordinates and use the distance formula, but a simpler way, just the way that we've arranged this, is just to use an argument of similar triangles. We know that this entire height right here in blue is c. We know that this height right over here is c over three. Or you could do that as 1/3 times c, so we know that this height over here we know that that distance is 2/3 c, so we can use a similar triangle argument. Let's say that this entire thing right here is the length of the median, and let's just call that, I don't know, let's call that l, and let's say we want to figure out how far along the median away from the vertex that is and let's just call that, let me use another, let me just call that distance, let me do this in a different color. Let's call this distance, right over here, let's see, I've already used a, b, and c, let's call that d. So if we can show that the ratio of d to l is 2/3, that this is 2/3 of the distance, then we're done. We can do that by similar triangles. We have two similar triangles. Let me use a new color. We have this triangle, which is kind of embedded inside of this larger triangle right over here. They both share the same vertex, this angle over here, they both have right angles over here, and since they have two of the same angles, their third angle has to be the same, so they're definitely similar triangles. So we can just set up a ratio here. 2/3 c, which is this length right over here, that's just kind of the vertical side of these smaller triangles, we could write 2/3 c over this entire length, over the length of the larger, this corresponding side of the larger similar triangle, over c is equal to the hypotenuse of the smaller similar triangle, d over the hypotenuse of the longer similar triangle, or the bigger similar triangle. D over l. This is clearly just dividing the numerator and denominator by c, this clearly becomes 2/3, so the ratio of this length to this larger length is 2/3, or this is 2/3 along the median away from the vertex. So there you go, that's the two-dimensional proof that the centroid is 2/3 along the way of any median from the vertex, or 1/3 along the median, or 1/3 the distance of the median from the opposite midpoint. Anyway, hopefully you enjoyed that.