Triangle similarity
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Similar Triangle Basics
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Similarity Postulates
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Similar triangles 1
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Similar triangles 2
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Similar Triangle Example Problems
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Similarity Example Problems
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Solving similar triangles 1
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Similarity example where same side plays different roles
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Solving similar triangles 2
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Challenging Similarity Problem
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Finding Area Using Similarity and Congruence
Similar Triangle Example Problems Multiple examples looking for similarity of triangles
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- What I wanna do in this video is
- See if we can identify similar triangles here
- And prove to ourselves that they really are similar
- Using some of the postulates that we've set up
- So over here I have triangle B, D, C it's inside of triangle A, E, C
- They both share this angle right over there
- So that gives us one angle
- We need two to angle -angle which gives us similarity
- And we know that these two lines are parallel
- And we know if two lines are parallel we have transversal
- That corresponding angles are gonna be congruent
- So that angle is gonna correspond to that angle right over there
- And we're done
- We have one angle and triangle A, E, C that is congruent
- to another angle in B, D, C
- and then we have this angle that is obviously congruent to itself,
- That's in both triangles
- So both triangles have a pair of corresponding angles
- that are congruent, So they must be similar
- So we can write triangle A, C, E
- A, C, E is gonna be similar triangle
- We wanna get the letters in the right order
- So where the blue angle is here, the blue angle there is vertex B
- We're gonna go to the white angles C
- and then we go to the unlabeled angle right over there B, C, D
- B, C, D
- So we did that first one
- Now let's do this one right over here
- This is kinda similar
- but at least it looks just superficial looking at it
- YZ is definitely not parallel to ST
- So we won't be able to do this corresponding angle argument
- Alright, especially cuz I didn't even labelled this parallel
- And so you wanna, you don't wanna look at things
- just by the way they look
- You definitely want to say what am I given and what am I not given
- These weren't labelled parallel,
- we wouldn't be able to make this statement
- Even though if they look parallel
- One thing we do have is that we have this angle right here,
- This common to the inner triangle and to the outer triangle
- And then given us a bunch of sides
- So maybe we could use SAS for similarity
- Meaning if we can show the ratio of the sides
- on either sides of this angle,
- And they have the same ratio from the smaller triangle
- to the larger triangle
- Then we can show similarity
- So let's go, we have to go on either side of this angle
- right over here
- Let's look at shorter side on on either side of this angle
- So the shorter is 2, and let's look at the shorter side
- on other side of the angle for the larger triangle
- Well then the shorter side is on the right hand side
- And that's gonna have, that's gonna be xt
- So we wanna compare is the ratio between, right this way,
- We wanna see is xy/ xt, over xt is that equal to the ratio
- of the longer side Or if we're looking relative to this angle,
- the longer of the two
- I'm sorry the longest of the triangle,
- though it looks like that as well
- Is that equal to the ratio of xz,
- xz over the longer of the two sides,
- When you're looking at this angle right here
- on either side of that angle
- for the larger triangle over xs, over xs
- And it's a little confusing cuz we've kinda flip with side
- But I'm just thinking about the shorter side
- on either side of this angle and between,
- and then the longer side of either side of this angle
- So these are the shorter side for the smaller triangle
- and the larger triangle
- These are the longer sides for the smaller triangle
- and the larger triangle
- And we see xy this is 2
- Xt is 3 +1 is 4, xz is 3, xz is 3 and xs is 6
- So you have 2/4 which is ? which is the same thing as 3/6
- So the ration between the shorter sides on either side of the angle
- And the longer side on either sides of the angle for both triangles
- The ratio is the same, so by SAS we know,
- so by SAS we know that the 2 triangles are congruent
- But we have to be careful on how we state the triangles
- We wanna make sure we get the corresponding sides
- So we could say that triangle, and I'm running out of space here
- Let me write it write above here
- We can write the triangle xyz, xyz is similar, is similar to triangle
- So we started up at x which is the vertex of the angle
- And we went to the shorter side first
- So now we want to start at x and go to shorter side of large triangle
- So x, so you go to xts, xts
- Xyz is similar to xts
- Now let's look at this over here,
- so in our larger triangle we have a right angle here
- But we really know nothing, we really know nothing about
- What's going on with any of these smaller triangles
- in terms of their actual angles
- You know this looks like a right angle
- We cannot assume it
- And it shares, if we look at this smaller triangle right over here
- It shares one side with the larger triangle,
- Well that's not enough to do anything
- And then this triangle over here also shares another side,
- But that also doesn't do anything
- So we really can't make any statement here
- About any kind of similarities
- So there's no similarity going on here
- If they, if they gave us, if they gave us,
- And well there are some, there are some shared angles
- This guy they both share that angle
- the larger triangles, small triangle
- So there could be a statement of similarity we can make
- If we knew that this definitely was a right angle
- Then we can make some interesting statement about similarity
- But right now we can't really do, we can't do anything as is
- Let's try this one out or this pair right over here
- So this the first ones that we've actually separated out the triangles
- So they've given us the 3 sides of both triangles,
- So let's just figure out the ratios
- between corresponding sides are a constant
- So let's start with the short side
- So the short side here is 3, shorter side is here is 9,
- So we have, wanna see whether this the ratio of 3 to 9 is equal to
- The next longest side over here is 3, is equal to 3 over
- the next longest side over here which is 27, which is 27
- And then if that's gonna be equal to,
- if that's gonna be equal to the ratio of the longest side
- So the longest side here is 6, 6
- and then the longest side over here is 18, 18
- So this is going to give us, let's see this is 3
- This is 8, let me do this in a neutral color
- So 3, we could this becomes 1/3
- This becomes which seems like a different number
- But we wanna be careful here
- And this right over here this becomes,
- this is a if you divide the numerator and denominator by 6
- this becomes a 1 and this becomes 3
- So you get 1 , 1/3 is to be equal to 1, needs to be equal of /9
- which needs to be equal to 1/3
- At first they don't look equal
- But we can actually rationalize this denominator right over here
- We show that 1/ 3 if you multiply it by a /,
- This actually gives you a numerator of / 9
- is 3 times 3 is 9
- So these actually are all the same
- This is actually saying this is 1/ 3 root 3
- which is the same thing as
- Which is this right over here which is the same thing as 1 /
- So actually these are similar triangles
- So we actually say it and I'll make sure I get the order right
- So let's start with E which is between
- the blue and the magenta side
- So that's between the blue and the magenta side that is H,
- right over here
- So triangle E, I'll do it like this,
- triangle E and then I'll go along the blue side F
- Then I go over along, along the blue side over here side
- Let me do it this way
- Actually let me just write it this way
- E triangle E, F, G we know is similar to triangle
- So E is between the blue and magenta side,
- blue and magenta side that is H
- And then we go along the blue side to F
- Go along the blue side to I
- And then we're go along the orange side to G
- Then you go along the orange to J
- So triangle E,F,J is, E,F,G is similar to triangle H,I,J
- by side, side, side similarity
- They're not congruent sides
- They all have just the same ratio or the same scaling factor
- Now let's do this last one right over here
- So we have, let's see, we have an angle
- that's congruent to another angle right over here
- And we have 2 sides,
- and so it might be attempting to use side angle side
- Because we have side angle side here
- And even the ratio's looks kind of tempting
- Because 4 times 2 is 8, 5 times 2 is 10,
- But it's tricky here because they aren't the same
- corresponding sides
- In order to use side angle side,
- the 2 sides that have the same corresponding ratios
- That could be on either side of the angle
- So in this case they are aren't on either of the angle
- In this case the 4's on one side of the angle
- but the 5 is not, the 5 is not
- So because if this 5 was over here,
- if it was over here then we could make it an argument for similarity
- But this 5 not being on the other side of the angle
- is not sandwiching the angle with the 4
- We can't use side angle side
- and frankly there's nothing that we can do over here
- So we can't make some strong statement
- about similarity for this last one
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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