Geometry (all content)
Centroid & median proof
Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median). Created by Sal Khan.
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- Is this really a textbook "proof"? It seems that my textbook doesn't want proofs to take the form of a set of equations. I see that this does "prove" the point, but my book wants a list of theorems and postulates and such. HELP!!!(4 votes)
- I know. In a real test, sadly, you will probably have to list all those theorems and postulates, even completely obvious ones, (such as the fact that there is a postulate for x=x)! Sal, is doing this proof to make it easier to understand this topic.(16 votes)
- Doesn't he show this in a previous video as well?(9 votes)
- Yes, Triangle Medians and Centroid. He solve it with a three dimensional plot (x,y,z) axes. The video following that showed how to solve it in 2D which is more difficult.(5 votes)
- at0:01sal says " i have drawn an Arbitrary triangle" what's an Arbitrary Triangle?(2 votes)
- an Arbitrary triangle is a triangle that has no definite side lengths, no definite angles, and the vertices have no definite position. In other words, it is equally likely to be ANY POSSIBLE TRIANGLE.(6 votes)
- Why is the centroid known as the center of gravity ? Why isn't it the circumcenter, incenter, or any other point of concurrency? Why the centroid?(2 votes)
- Because the centroid is the physical center of gravity. If you had a paper triangle, you could balance it on a pencil if you put the pencil under the centroid.(2 votes)
- Is there a relationship between the Circumcenter, the Inradius and the Centroid of a triangle? Would these three points all be the same in an Equilateral Triangle?(1 vote)
- Yes, all three points would be the same in an equilateral triangle. If the triangle was not equilateral, then the points would fall on the same line, known as the Euler line.(3 votes)
- How do we know for sure that the area for all 6 triangles inside the bigger triangle are equal?(2 votes)
- At3:59, how do we know that the two blue triangles together have twice the area of the orange triangle? We don't know the relationship between them other than the height, but the height doesn't even matter when you split of the blue triangles.(2 votes)
- In this video, http://www.khanacademy.org/math/geometry/triangle-properties/medians_centroids/v/medians-divide-into-smaller-triangles-of-equal-area, Sal proves that the 6 smaller triangles formed by the 3 medians all have equal area.(1 vote)
- what would be the reasons for each step(1 vote)
- Well most of the work in this proof was in showing the areas of the sub-divided triangles were all equal. That's done in this earlier video: https://www.khanacademy.org/math/geometry/triangle-properties/medians_centroids/v/medians-divide-into-smaller-triangles-of-equal-area
In this video the steps are just "formula for the area of a triangle" and then some basic algebra.(2 votes)
- so what's the difference between a median and a perpendicular bisector?(1 vote)
- Median - A line segment that joins the vertice of a triangle to the midpoint of opposite side.
Angle bisector - A line segment that divides an angle of a triangle into two equal angles.
Perpendicular bisector - A line segment that makes an angle of 90 deg (right angle) with the side of a triangle.
The common point where the medians intersect is the centroid.
The common point where the angle bisectors intersect is the incenter.
The common point where the perpendicular bisectors intersects is the circumcenter.(1 vote)
- But why is triangle AGE is twice the size of triangle ABG? Is there a way to prove it, and not speculate?(1 vote)
I've drawn an arbitrary triangle right over here, and I've also drawn its three medians: median EB, median FC, and median AD. And we know that where the three medians intersect at point G right over here, we call that the centroid. What I want to do in this video is prove to you that the centroid is exactly 2/3 along the way of each median. Or another way to think about it, we can pick any one of these medians, and let's say let's pick EB. What I want to do is I want to prove that EG is equal to 2 times GB. So whatever distance this is it's twice this distance there. Or another way to think about it is EG is 2/3 along the way of EB. And the logic that I'm using to prove this you can use for any of the medians to show that the centroid is exactly 2/3 along the way of any median, or divides it into a segment that's twice as long as the other segment. And to do that, let's focus-- I want to focus on triangle ABE right over here. And I'm going to draw this median as essentially the base. So let me draw it that way. I'm going to try to color code it similarly. So we draw it a little bit flatter than that. So it's like that. And then we have the two yellow sides, so it looks something like this. It looks something like that. And then we have the centroid, right over here at G. That is our centroid, and then we have this magenta line going to A. Let me draw it a little bit neater than that. We have that line going to A, and then we have this blue line going to F right over here. And let me label all the points. Go back to the orange color. So this is going to be E, this is going to be B, this is going to be A, this is going to be F right over here. And just to make sure we have all the same markings, that little marking there is that marking, these two markings, these two markings are on this side right over there. And the whole way that I'm going to prove that EG is twice as long as GB is just refer to the result that we did, I think, a couple of videos ago that the medians divide this triangle into six smaller triangles that all have equal area. So another way to think about it is each of these three small triangles have equal area. These are three of the total of six smaller triangles. So these three all have equal area. So let's think about this triangle right over here. Let's think about this triangle, triangle AGB. This is triangle AGB right over there. Those are the same triangles. And let's compare that to triangle EAG right over here. Let's compare it to this triangle, which is this triangle right over here on the original drawing. Now, they both have the exact same height. If you view EG as their base-- or I the guess their shared base, they don't have the exact same base. The smaller triangle has the base E-- sorry, the smaller orange triangle has GB as it's base. The larger blue triangle has EG as its base, but they definitely both have the same height, or altitude, when you draw it this way. So their height, in both cases, is this right over here. Now the other thing that we do know is that this blue triangle EAG has twice the area of the orange triangle. How do we know that? Because it's got two of these triangles in it. So one way to think about it is if this orange triangle has area x, actually let me call it a-- well I already used a, so I'm going to call it area x-- then each of these blue triangles have area x. Or you could say this entire blue region has area 2x. So if you look at this blue triangle right over here, we know that 1/2 times base times height is equal to area. So we get 1/2-- the base is EG. 1/2-- I'll do that in the green color-- 1/2 EG times height times this yellow height is going to be equal to 2x. I'm just applying the formula for area of a triangle. 1/2 base times height is equal to area. This is our area. Now, let's do the same thing for this orange triangle. 1/2-- let me scroll over a little bit to the right-- we have 1/2 GB times the yellow height is going to be equal to x. Well we can substitute it. If this is equal to x, we can place this entire expression right over here for x. So let's do that. We get 1/2-- and you might already see where this is going, but I won't skip any steps here-- we get 1/2 times EG times h is equal to 2 times x. But instead of x, I'm going to write this here. Is equal to 2 times 1/2 times GB times this length-- times the base of the smaller triangle times h. And now we can just simplify this. We have 2 times 1/2 is just going to be 1. You can divide both sides by h, and we are left with 1/2 EG is equal to GB. Or we could write EG over 2 is equal to-- let me do it in the same color since I've gone this far with the same colors. So we can write 1/2 EG is equal to GB. And we're done. This is essentially saying that GB is half of EG. So for example, if EG is 2, this is going to be 1. If EG is 4, this is going to be 2. So we've actually proven our result. Well actually, let's go back to-- this is the result we wanted to prove. To get to there we just multiply both sides of this equation by 2. You multiply this, the left-hand side by 2, you get EG. You multiply the right-hand side by 2, you get GB. So we've proven that EG is twice GB . And you can apply the same logic to any of the medians to show that the centroid is exactly 2/3 along the way of the median.