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# Triangle medians & centroids

## Video transcript

I want to do a quick refresher on medians of triangles and also explore an interesting property of them that will be useful I think in future problems so let me just draw an arbitrary and arbitrary triangle over here that's good enough now a median of the triangle and we'll see that me a triangle has three of them is just a line that connects a vertex of the triangle with the midpoint of the opposite side so the opposite sides midpoint looks right about there this length is equal to that length and so this is a median close enough and of course we have three vertices so we'll have three medians if we start at this vertex we want to go to the midpoint of the opposite side and it looks right about there so this blue line right over here is another median it's not a completely straight line but I think you get the idea then we could also do it from this point right over here draw a line from this vertex to the midpoint of the opposite side let's say the midpoint of the opposite side is there and we draw a line each of these I could draw a straighter line than that let me draw it my there well I think you get the idea these are all medians of this triangle now what's neat about medians is that they all all three medians always intersect in one point and that by itself is a pretty neat property and at one point that they intersect in is called a centroid it's called the centroid and if this was actually a physical triangle let's say you made it out of iron and if you were if you were to toss it well even before you toss it the centroid would actually be the center of mass so let's say this is an iron triangle let's say that this right here is an iron triangle that has its centroid right over here then this irons triangles center of mass would be where the centroid is assuming it has a uniform density and if you were to throw that iron triangle it would rotate it would rotate around this point around around a lot of Sumanth at you it had some rotational motion it would rotate around that centroid around the center of mass but anyway the point of this video is not to focus on physics and throwing iron triangles the point here is I want to show you need property of medians and the property is is that the distance from if you pick any median the distance from the centroid to the midpoint of the opposite side so this distance is going to be half of this distance so if this distance right here is a then this distance right here is 2a or another way to think about it is this distance is 2/3 of the length of the entire median and this distance right here is 1/3 of the length of the entire median and let's just prove it for ourselves just so you don't have to take things take things on faith and to do that I'll draw an arbitrary triangle I'll do a two dimensional triangle I'll do it in three dimensions because at least in my mind it makes the math it makes the math a little bit easier in general whenever you take a if whenever you have an n-dimensional figure and you embed it in n plus 1 dimensions it makes the math a little bit easier that the actual tetrahedron problem that we did you could actually embed it in 4 dimensions and make the math easier just much harder to visualize so I didn't do it that way but let's just have an arbitrary triangle and let's so let's say it has a vertex there a vertex there and a vertex there so I'm not making any assumptions about the triangle I'm not saying it's isosceles or equilateral or anything it's just an arbitrary triangle and so let's say it's let's say it's coordinate let's say this coordinate right over here is I'll you I'll call this the x-axis so this is the x-axis the y-axis and the z-axis I know some of y'all are used to swapping these two axes but it doesn't make a difference so let's call this coordinate right here a 0 0 so it's a along the x-axis let's call this coordinate 0 B 0 and let's call this coordinate up here 0 0 0 0 C and if you connect the points you're going to have a triangle you're going to have a triangle or just like that now the centroid of a triangle especially in three dimensions the centroid of a triangle is just going to be the average of the average of the coordinates of the vertices or the coordinate of the centroid the coordinate of the centroid here is just going to be the average of the coordinates of the vertices so the this coordinate right over here is going to be so for the x-coordinate we have zero plus zero plus a so we have three coordinates they add up to a and we have to divide by three so it's a over three the y-coordinate is going to be b plus zero plus zero they add up to B but we have three of them so the average is B over three and then same thing we do it for the Z coordinate the average is going to be C is C over three and I'm not proving it to you right here you could verify it for yourself that it's going to be the average that if you were to figure out what this line is this line is and this line is this centroid or the center of mass of this triangle if it had some mass is just the average of these coordinates now what we want to do is use this information let's just use this coordinate right here and then compare just using the distance formula let's compare this distance let's compare this distance up here in orange to this distance down here down here in yellow and remember this point right over here is the point this is the median of this of this bottom side right over here so the median of that bottom side is just going to be the average of these two points and so the x-coordinate zero plus a over two is going to be a over to be plus zero over two is going to be B over two and then it has no z coordinate so it's just going to be zero zero plus zero over two is zero so we know the Corden's for this point that point and that point so we can calculate the yellow distance and we can calculate the orange distance so let's calculate the orange distance so that that is going to be equal to the square root the square root the orange distance is going to be equal to the square root of and we just take the difference of each of these points squared so it's a over three minus zero squared so that's going to be a squared over nine plus B over three minus zero squared so that's B squared over nine plus C over C C over three minus C which is negative two thirds that's negative two thirds and we want to square that so we're going to have positive positive four over nine e squared did I do that right C over 3 so 1/3 minus 1 is negative 2/3 negative this is negative 2/3 see you square it you're going to get 4/9 C squared so that's the orange distance now let's calculate and if we want to do it we can actually we can express this let me express it a little bit simpler than this this is the same thing as the square root of a squared plus B squared plus 4c squared over the square root of 9 which is just equal to that's just equal to 3 now let's do the same thing with the yellow distance so it's going to be equal to the square root of so if we have a over 2 minus a over 3 so 1/2 minus 1/3 that's the same thing as 3 6-2 6-1 6-1 6 squared is a squared over 36 B over 2 minus B over 3 is B over 6 you square it you get plus B squared over 36 and then finally you have 0 minus C over 3 squared that's going to be C squared over 9 but just so we get a common denominator C squared over 9 is the same thing as plus 4 C squared over 36 and we can rewrite this as the square root of a squared plus B squared plus 4c squared over 6 so you can see this distance right here if you multiply this orange distance by 1/2 you're going to get so if you multiply the orange distance by 1/2 or if you divide it by 2 so if you multiply by 1/2 you get the yellow distance so this is always going to be twice the distance as this because we did this in the most general possible way we assumed nothing about this triangle so remember that little property that the centroid the intersection of the medians if the intersection happens 2/3 along every medians 2/3 away from the vertex or 1/3 away of the length of the median a from the the midpoint of the opposite side and we can use that property in what we'll probably use it in a bunch of problems but anyway hopefully you found that interesting