Medians and centroids
Triangle Medians and Centroids Seeing that the centroid is 2/3 of the way along every median
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- I wanna do a quick refresher on medians of triangles
- also explore interesting property of them that would be useful,
- think,to a future problem so let me just draw an arbitrary,
- an arbitrary triangle over here, now that's good enough
- Now a median of the triangle, and we'll see the median,
- a triangle has 3 of them, it's just a line that connects
- a vertex of a triangle with the midpoint of the opposite sides,
- so the opposite sides midpoint looks like right about there,
- this length is equal to the length, and so this is a median,
- close enough
- And of course we have 3 vertexes, we'll have 3 medians
- if start at this vertex, we wanna go to the midpoint
- of the opposite side, it looks like it's right about there
- so this blue line over here is another median
- It's not a completely straight line but you get the idea
- And we can also do it from this point, right over here,
- draw a line from this vertex to the mid point of the opposite side
- let's see the midpoint of the opposite side is there
- And we draw a line, each of these;
- I can draw a straighter "line" than that
- We draw, well think you get the idea
- These are all medians of this triangle
- And what's next about medians,
- is that all 3 medians always intersect in one point
- And that by itself is a pretty neat property
- That one point that they intersect is a called a centroid
- It's called a centroid, it's called the centroid
- And if this was a physical triangle, let's say you made it out of iron,
- and if you were to toss it, or even before you toss it,
- the centroid would actually be the center of mass
- Let's say this is an iron triangle,
- lets say right here this is an iron triangle,
- that has it's centroid right over here,
- then this iron's triangle center of mass would be the centroid is,
- assuming the uniform density, and if you would have thrown,
- it would have rotate, it would have rotate around this points,
- around, around, assuming it had some a rotational motion
- around that centroid, around the center mass
- But anyway the point of this video is not to focus on physics
- and throwing iron, triangles
- The point here is, I wanna show you a neat properties of medians
- and the property is the distance from, if you pick any median
- the distance from the centroid to the midpoint of the opposite side
- so this distance is gonna be half of this distance
- So this distance right here is A, then this distance right here is 2A
- Or another way to think about this,
- 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 2 dimensional triangle, I'll even do a 3 dimensions,
- at least in my mind it makes the math,
- it makes the math a little bit easier
- In general whenever you take a,
- Whenever you have N dimensional
- that you embed on an N +1 dimension,
- it makes the math a little bit easier
- That actually tetrahedron problem we did,
- you can actually embed a four dimension
- that would make the math easier,
- it's much harder to visualize because I didn't do it that way
- So lets just have an arbitrary triangle,
- let's say it has a vertex there, a vertex there and a vertex there
- I'm not making any assumptions about the triangle
- I'm not saying its an isosceles and equilateral, anything,
- its just an arbitrary triangle
- 8And so lets say its coordinate, lets say its coordinate right over here is,
- I call this the X-axis, this is the X-axis, this the Y-axis and the Z-axis,
- I know some of you are already used to swapping these two axis
- but it doesn't make a different so lets call this coordinator right here
- A, 0, 0 so it's A along the X-axis
- Let's call this coordinate 0, B, 0
- Let's call this coordinate up here 0, 0, 0, C
- So if you connect the points, you're gonna have a triangle
- You're going to have a triangle just like that
- Now the centroid of a triangle especially in 3 dimensions
- the centroid of is going to be the average,
- the average of the coordinates of the vertexes
- or the coordinate of the centroid, the coordinate of the centorid here
- its just gonna be the average of the coordinates of the vertexes
- So the coordinate right this over here is gonna be,
- so for the x coordinate we have 0, 0, 0 + A
- so we have 3 coordinates they add up to A so we have to divide up by 3 so its
- A over 3
- The Y coordinate is gonna be B+0+B+0 they add to B, but we have 3 off them
- And the same thing we do it for the for the Z coordinate,
- the average is gonna C, it's C over 3 and
- I'm not proving anything to right here, you can verify it for yourself,
- but its going to be the average, figuring out what this line is,
- this ine is and this line is
- This centroid or the center of this mass of this triangle,
- if it had some
- mass is just the average of these coordinates
- Now what we wanna do is use this information,
- let's just use this coordinate
- and compare just using the distance formula,
- lets compare this distance, lets compare this distance
- up here in orange to this distance down here, down here in yellow
- And remember this point down here, this is the point
- the median of this, of this bottom side right over here, so the median of this bottom side
- Is just gonna be the average of the these two points
- and so the X coordinates 0 plus A
- B + 0 over 2 is gonna be B over 2
- And then it has no Z coordinates so it's just gonna be 0
- 0 + 0 over 2 = 0
- So we know the coordinates for this point, that point and that point
- so we can calculate the yellow distance
- and we can calculate the orange distance
- So lets calculate the orange distance
- So that is gonna be equal to the square root of, square root,
- The orange disntace is going to be the square root of,
- which we'll take the difference of each of these point square
- so it's A over 3 minus 0 squared,
- that's gonna be A square over nine over 9
- so that's gonna plus B over three minus 0 square over 9,
- plus C over 3, minus C which is negative 2/3, that's negative 2/3
- and we want to square that, so we're gonna have positive
- 4 over 9 positive C square, did I do that right?
- C over 3, so 1/3 minus 1 is negative 2/3, so this is negative 2/3 C,
- you square it, you're gonna get 4 9 C squar
- So that's the orange distance, now let's calculate,
- and if we wanna do it we can actually,
- we can express it a little simpler than this ,
- this is the same thing as a square root
- of A square + B square + 4 C square over the square root of 9,
- which is just, that's just equal to that's just, the equal to 3
- So lets do the same thing with the yellow distance
- so if gona be equal to have A over 2 - A over 3
- so 1/2 minuses 1/3 that's the same thing
- as 3/6 minuses 2/6 so 1/6 A square over 36
- B over 2 minus B over 3 is B over 6, you square it,
- you got plus B square over 36
- An then finally you have 0 minus C over 3
- over square, that's gonna be a C square over nine
- that's when we get a common denominator
- C square over nine is the same things as plus 4 C square over 36
- and we can rewrite this as a square root of
- A + B square + Four C square over 9
- 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 by 1/2
- or you divide it by 2, so if you multiply by 1/2
- you get the yellow distance
- So this is always gonna be twice the disntsaces
- So we do this in the most general possible way,
- we assume nothing about this triangle
- Remember that little property the centroid,
- the if the intersection of the medians,
- if the intersection happens 2/3s along every median
- 2/3s away from the vertex or 1/3 away
- of the length of median away from the midpoint of the opposite side
- We can use this property in,
- well we can use it in a bunch of problems
- Anyway, I hope you found that interesting
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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