Medians and centroids
Proving that the Centroid is 2-3rds along the Median Showing that the centroid divides each median into segments with a 2:1 ratio (or that the centroid is 2/3 along the median)
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- I've drawn an arbitrary triangle right over here
- And I've also drawn is three medians
- Median EB median FC and median AD
- And we know where the three median intersect
- at point G right over here we call that the centroid
- What I wanna do in this video is prove to you
- That centroid is exactly two thirds along the way of each median
- Or another way to think about it we can pick any one of
- these medians then let's say let's pick EB
- What I wanna do is I wanna prove that EG,
- EG is equal to two times GB two times GB
- So whatever distance this is it's twice this distance there
- Or other way to think about it is EG is two thirds along
- the way of EB and that logic that I'm using to prove this
- You can use for any of the medians to show
- That the centroid is exactly two thirds along the way of any median
- Or divides it to a segment that's twice as long as the other segment
- And to do that let's focus I wanna focus on triangle A B E
- Triangle A B E right over here
- And I'm gonna draw this median as essentially the way
- So let me draw that way I'm gonna try to color code it similarly
- So we draw it I'm gonna 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 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
- We have this blue line going to F
- 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 are on this side right over there
- And the whole way that I'm gonna prove
- That EG is twice as long as GB is
- Just to refer the results that we did I think a couple of videos ago
- That the medians divide in this triangle to just
- Into six smaller triangles that all have equal area
- So another way to think about it
- Is each of these three smaller triangles have equal layer
- These are three of the total of six smaller triangle
- So these three all have equal area
- So let's think about of this triangle right over here
- Let's think about of this triangle, triangle AGB
- This is triangle AGB right over there those are same triangles
- And let's compare that to triangle
- Let's compare it 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 as their base or not I guess their shared base
- They don't have the exact same base
- This smaller triangle has the base E oh sorry
- The smaller orange triangle has GB as its base
- The larger blue triangle has EG as its base
- But the definitely 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
- 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 this orange triangle has area X
- You want me to call it A well I already used A
- So I'm gonna call it area X
- Then each of these blue triangles have area X
- Or you could say this entire blue region
- This entire blue region has area two X area two X
- So if you look at this two blue triangles
- We know that one half times base time height is equal to area
- So we get one half the base is EG one half
- I'll do that in green color
- One half EG times height times this yellow height
- Is going to be equal to two X
- Is going to be equal to two X
- I'm just way I'm just applying the formula for the area of a triangle
- One half base times height is equals to area this is our area
- Now let's do the same thing for this orange triangle
- One half let me scroll a little bit to the right
- We have one half GB GB times the yellow height
- Times height is going to be equal to X
- Is going to be equal to X
- Well we can substitute if this is equal to X we can
- We can place this entire expression right over here for X
- So let's do that we get one half
- And you might already see where this is going
- But I won't I won't skip any steps here
- We get one half time EG times H times H
- Is equals to two is equals to two time X
- But instead of X I'm gonna write this here
- As you can look so two times two times one half times GB
- Times this length times the base of the smaller triangle
- Times H times H
- And now we can just simplify this
- We have two times one half is just going to be
- It's just going to be one you can divide both sides by H
- And we are left with one half EG is equals to GB
- Or we could write EG over two is equal
- Let me do this in same color since I've gone this far
- With the same colors
- So we can write we can write on half EG, EG is equal to GB
- Is equal to GB and we're done
- This is essentially saying that GB is half of EG
- GB is half of EG so for example if EG is two
- This is going to be one if EG is four this is going to be two
- So we've actually proven our result
- We'll 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 two
- You multiply thisthe left hand side by two you get EG
- You multiply the right hand side by two you get GB
- So we've proven the EG is twice GB
- And you can apply the same logic to any of the medians
- To show that the centroid
- Is exactly two thirds along the way of the median.
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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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