Medians and centroids
Medians divide into smaller triangles of equal area Showing that the three medians of a triangle divide it into six smaller triangles of equal area. Brief discussion of the centroid as well
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- But what I think about now is the if we're given some triangle,
- and here we have triangle ABC
- What are the medians of the triangle
- and how do they relate to each other
- and do they have some interesting properties,
- and you might guess that they do
- So the median is, if we start at one of the vertices
- so let's start at this one right over here
- And then we bisect the opposite sides
- So this right here would be a median
- And so it started there, and it bisects this side right over here
- So the length from B to let's call this D
- is equal to the length from D to C
- Now, let's do that with every side
- I can draw a median over here, like that
- So, let's call this point E
- So the length from A to E
- is going to be equal to the length from E to C
- Although it's kind of a little lopsided,
- but that gets it pretty close
- And then we're gonna draw another median
- And I'm not gonna prove it in this video,
- but all the medians, and this is another neat thing, that, you know,
- when you have three lines that will always intersect at one point,
- that all the medians do intersect at one point
- They all are concurrent
- They all hit one common point in the center
- So, let's draw it that way
- I'm not gonna prove it in this video, so this length,
- let's call this point right over here F
- So this length right over here,
- this length right over there is equal to that length over there
- And the point at which these medians intersect
- is called the centroid, the centroid
- And when you start studying physics,
- if you're actually and this was a uniform triangle
- and you were to throw it,
- it would rotate around the centroid right over here
- But we'll study it geometrically for now
- So let's call this ten centroid well, we've already gone up to F
- So let's call this centroid G right over there
- Now, what I wanna that by itself is neat that you have a centroid,
- that if you were to throw it, if it was a uniform mass,
- it would rotate around the centroid
- But what's even neater about this,
- is that we can see that we've divided this triangle
- into six smaller triangles
- What's really cool,
- even though these aren't congruent triangles necessarily,
- but they do have the same area
- And that's what we're going to prove in this video
- That these six triangles all have the same area
- Now, to start off, I'm just going to look at two
- I'm gonna look at different pairs of triangles
- So let's look at let's look at these two triangles right over here
- Let's look at those two triangles over here
- And to show that those two have the same area,
- we're just going to invoke a very simple principle
- So, imagine rotating, just those two triangles over
- So just rotating, just those two triangles over
- It would look something like this
- It would look something like this
- Let me try my best to draw it
- Where this would be point G I'll even try to color code it the same
- that is point G
- That is that side right over there
- This is point C
- This is point B, that is point B
- And then this right over here
- would be the second part of that median right over there
- That over there would be point D
- Now, we know and I didn't draw it I should, at nicely over here
- We know that this length is equal to this length right over here
- And these two triangles, if we're starting to think about the area,
- they have the same base
- And we know area
- Area is equal to ? base * height
- So, they definitely have the same base
- What about their heights?
- Well, they also have the same heights
- Both of their heights, both of their height is exactly that tall
- They both have the same height
- So both of them have the same base
- They both have the same height
- On an ob on an obtuse triangle right here, the altitude sits outside of it
- So that might be a little kind *unclear 00:03:26 7*
- So if you have an obtuse triangle like that
- I say obtuse cause this is more than 90 degrees
- Your altitude, your height *unclear 00:03:33 4*
- sit outside of the triangle, but that's okay
- Both of these triangles have the same base, and the same height,
- so they must have the same area
- So if this one right here has area x,
- this one right here will have area x as well
- And you can use the exact same logic to say, well, look
- This guy and this guy, they have the same base,
- and they both have the same height
- So if this one right over here is of area y,
- then this one over here is also going to be of area y
- They will have the same area
- And then finally,
- we could do the same thing for these two characters up here
- They both have the same base
- This was BF is equal to FA, and they both have the same height
- We drop an altitude like that
- And so if we call this area, if we call this area right over here Z,
- you could call that area Z, as well
- So, so far, we've shown that we can divide this
- into three pairs of triangles that have the same area
- But we wanna now show that they all have the same area
- And to do that, we can invoke this same principle,
- but we'll do it with different sets of triangles
- So now, let's look at triangle let's look at triangle BAE
- Look at triangle BAE, triangle BAE
- So the area of triangle BAE,
- area of BAE, is going to be equal to z + z + y
- z + z + y
- And let's look at the area of triangle BEC, BEC right over there
- That's going to be,
- this triangle right over here, is going to be x + x + y
- So the area of BEC is going to be x + x + y, but the same principle
- They both have the same base, and they both have the same height
- We could drop an altitude like this
- This one is obtuse, so that the altitude sits outside of it,
- but they have the exact same height
- So these two areas need to be equal to each other
- So you have Z, well, let me just add that up
- Now you have 2z + y is going to be equal to,
- is going to be equal to 2x + y
- 2x + y
- Subtract y from both sides
- You get 2z is equal to 2x
- Divide both sides by 2
- You get z is equal to x
- And so we could say, we could write an x here, and an x there
- So we know that all of these will have the same area,
- but we still have to worry about these y's here
- And to do that,
- we just have to kind of rotate the way that we look at it
- And now look at triangle ADC
- Let me do that in a different color
- Triangle ADC, which I'm highlighting right over here
- Triangle ADC, whose area, so the area of ADC is going to be 2y + x,
- is equal to 2y + x
- And then look at triangle let's see, what color have I not used yet
- Let me use this green
- Triangle ADB, triangle ADB
- Triangle ADB is going to be equal to,
- well, you could say it's 2z + x,
- but we know that the z's are equal to x
- It's really just x + x + x
- It's really equal to ADB is equal to 3x
- And we have the same idea here
- ADB has this base, which is the same as ADC's base,
- and they both have the same height
- We can draw up an altitude like this,
- we can draw up an altitude like that
- They have the same height
- We're just invoking this principle over and over and over again
- So these things have to equal each other
- So we get 2y + x is equal to 3x, is equal to 3x
- Subtract x from both sides
- You get 2y is equal to 2x, or, or y is equal to x
- So that's a really neat result
- You go from each of the vertices of a triangle to the opposite side,
- and you bisect that opposite side, and you do that three times,
- you have three medians
- These lines are called medians, where they intersect as a centroid
- You know what's really cool is,
- it divides the triangle into six smaller triangles of equal area
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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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