Medians and centroids
Exploring Medial Triangles What a medial triangle is and its properties
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- So I've got an arbitrary triangle here, we'll call it triangle ABC
- What I wanna do is look at the midpoints of each of the sides of ABC,
- So this is the midpoint of one of the sides of side BC
- Let's call that point D
- Let's call this midpoint E
- And let's call this midpoint right over here F
- And so this is the midpoint,
- we know that the distance from BD
- is equal to the distance from D to C
- So this distance is equal to this distance
- We know that AE is equal to EC,
- So this distance is equal to that distance
- And we know that AF is equal to FB,
- so this distance is equal to this distance
- Instead of drawing medians going from this midpoints to the vertices,
- What I wanna do is I wanna connect
- these midpoints and see what happens
- So if I connect them I clearly have three points
- So if you connect three nonlinear points
- like this you will get another triangle,
- And this triangle that's formed from the midpoints
- of the sides of this larger triangle,
- we call this a medial triangle
- Medial triangle and that's all nice and cute by itself
- But what we're gonna see in this video,
- is that the medial triangle actually have some very neat properties
- What we're actually going to show is that divides
- Any triangle into four smaller triangles
- They are congruent to each other
- That all four of these triangles are, are identical to each other
- And they are similar to their larger triangle
- And you could think of them
- Each has having one fourth of the area of the larger triangle
- So let's go about proving it,
- So first let's focus on this triangle down here
- Triangle CDE and it looks it looks similar to the larger triangle
- The triangle CBA, but let's prove it to ourselves
- So one thing we could say is well look
- Both of them share this angle right over here,
- Both the larger triangle, triangle CBA shares has this angle
- And the smaller triangle CDE has this angle
- So they definitely share that angle
- And now let's look at the ratios of the sides,
- We know that the ratio of CD
- We know that the ratio of CD to CB,
- the ratio of CD to CB is equal one over two
- This is half this entire side is equal to one over two
- And that's the same thing as the ratio of CE to CA
- CE is exactly half of CA,
- Cause E is the midpoint
- It's equal to CE over CA
- So we have an angle and corresponding angles that are congruent
- And then the ratio of two corresponding sides
- of either the side of that angle
- Or the same CD or CB as one half
- CE or CA as one half
- And the angle in between is congruent
- so by SAS so by SAS similarity,
- SAS similarity, similarity
- We know we know that triangle CDE, CDE
- is similar to triangle to triangle CBA, CBA
- And just from that you can get some interesting results,
- Because we know that the ratio of these
- sides of the shorter triangle or the smaller triangle
- To the longer triangle is also going to be one half
- Cause the other two sides have a ratio of one half
- What we are gonna do with similar triangles,
- so this is gonna be one half of that
- And we know one half of AB
- is just going be is just going to be the length of the FA
- So we know that this length right over here
- Is going to be the same as FA or FB
- We can get that straight from similar triangles
- Because these are similar,
- we know that DE, DE over BA over BA
- has got to be equal to these ratios,
- the other corresponding sides which is equal to one half,
- and so that's how we get that right over there
- Now let's think about let's think about that this triangle up here
- Let's thinks about this triangle up here
- Triangle we can call it BDF, triangle BDF
- So first when we compare triangle BDF to the larger triangle,
- they both share this angle right over here angle ABC,
- they both have that angle in common
- And we're gonna have exact same argument
- We can just you can just look at this diagram
- You know that the ratio of BA
- The ratio let me do it this way
- The ratio of BF to BA, the ratio BF to BA
- Is equal to one half, which is also the ratio of BD to BC BD to BC
- The ratio of this to that is the same ratio
- of this to that which is one half
- Cause BD is half of this whole length
- BF is half of this whole length
- And so you have corresponding sides have the same ratio
- On the two triangles and they share and angle in between
- So once again by SAS similarity,
- by SAS similarity SAS similarity
- We know that triangle, we know that triangle right this way
- DBF, triangle DBF is similar to triangle CBA,
- Is similar to triangle CBA
- And once again we use this exact same kind of argument,
- that we do with this triangle
- Well if it's similar with the ratio to all the corresponding sides
- Have to be the same and that ratio is one half
- So the ratio of this side to this side
- So the ratio of FD the ratio of FD to AC has to be one half
- or FD has to be one half of AC
- Which when one half of AC is just the length of AE
- So that is just going to be that length right over there
- I think you see where this is going
- And also, because it's similar
- to all of the corresponding angles have to be the same
- And we know that this is that, that the larger triangle
- has a yellow angle right over there
- So we have that yellow angle right over here
- And this triangle right over was also similar to the larger triangle
- So it will have that same angle measure up here,
- We are to show that in this in this first part
- So now let's go to this third triangle,
- I think you see the pattern
- I'm sure you might be able to just, pause this video,
- And just prove it for yourself
- But we see that the ratio of AF over AB
- Is going the same as the ratio AE over AC
- AE over AC which is equal to one half
- So we have two corresponding sides with the ratio
- is one half from the smaller to the larger triangle
- And that they share a common angle
- they share this angle in between the two sides
- So by SAS similarity so this is getting repetitive now
- We know that triangle we know that triangle EFA triangle EFA
- is similar to triangle CBA to triangle CBA
- And so the ratio of the corresponding sides it will be one half
- So the ratio of FB to BC needs to be one half
- or FE needs to be half of that
- Which is the length of BD
- So this is going to be that length right over there
- And you can also say that since we show all that this triangle
- This triangle and this triangle we haven't talked about this middle one yet
- They're all similar to the larger triangle,
- so they're all going to be similar to each other,
- so they're e all going to have the same corresponding angles
- So if the larger triangle had this yellow angle,
- Then all of the triangle will have this yellow angle right over there
- And the larger triangle will have this blue angle right over here
- Then the corresponding vertex all of the triangles
- were going has that blue angle
- All of the ones that we've shown are similar
- We haven't thought about this middle triangle just yet
- And of course, if this is similar to the whole,
- it'll also have this angle at this vertex over here
- because this corresponds to that vertex
- based on the similarity
- So that's interesting,
- Now let's look at the let's look at
- Let's compare the triangles to each other
- We've now shown that all of these triangles
- Have the exact same three sides,
- Has this blue side or I should have this one marked side
- This two marked sides of this three marked side
- One marked two marked three marked,
- And that even applies to this middle triangle right over here
- So by side, side, side,
- So by side, side, side congruency, congruency
- We now know and we wanna be careful
- when we get our corresponding sides right
- We now know that triangle, triangle CDE
- is congruent to triangle D to DBF, DBF
- When we look at the corresponding sides,
- I'm looking at the colors
- I want from the yellow to magenta to blue,
- The yellow to magenta to blue
- which is going to be congruent to triangle to triangle EFA, EFA
- Which is going to be congruent to this triangle and here,
- We wanna make sure that we're getting
- we're getting the right corresponding sides here
- So make sure when we do that,
- We just have to think about the angles
- So we know, we know this is interesting,
- That the ,the because the, the interior angles
- of a triangle add up to 180 degrees
- We know that this magenta angle plus this blue angle
- plus this yellow angle equal a hundred-eighty
- Here we have the blue angle and the magenta angle,
- And clearly they all add up to 180
- You must have the blue angle the blue angle
- The blue angle must be right over here
- Same argument blue angle as I,
- Yellow angle and blue angle
- we must have the magenta angle right over here
- They add up to 180,
- So this must be the magenta angle
- And then finally magenta and blue,
- This must be the yellow angle right over there
- And so when we wrote the congruency here,
- we started at CDE
- Yellow magenta blue,
- So here we're gonna go yellow magenta blue
- So this is going to be congruent to triangle FED to triangle FED
- And so that's pretty cool we just show that all three that this triangle
- This triangle this triangle and that triangle are congruent
- And also we can look at the corresponding
- and that they all have ratios relative to the larger triangle,
- They're all similar to the larger triangle the triangle ABC
- And that the ratio of between the sides is one to two,
- And also because we've look at corresponding angles
- We see for example, we see for example
- that this angle is the same as that angle
- So if you viewed DC if you viewed BC as transversal
- All of the side becomes pretty clear that FD is going to be parallel to AC
- Because the corresponding angles are congruent
- So this is going to be parallel to that right over there
- And you can use that as the same exact argument
- to say well then this side
- because once again, corresponding angles,
- corresponding angles here and here
- You can say that this is going to be parallel to that right over there
- And then finally,
- With the same argument over here you have
- You have I wanna make sure I get the right corresponding angles
- You have this this line and this line
- and this angle corresponds to that angle
- They are the same
- So this DE must be parallel to BA to BA
- So that's another neat property of this medial triangle
- I thought that all of these things just jumped out
- When you try to do something fairly simple with the triangle
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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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