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
Finding Area Using Similarity and Congruence Example of using similarity and congruence to find the area of a triangle
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- So this problem here we're told that the triangle ACE is isosceles
- so that's this big triangle right here
- Its isosceles which means it has two equal sides and we also know
- from isosceles triangles that the base angles must be equal
- So this two base angles are going to be equal
- and this side right over here
- is going to be equal in length to this side over here
- We can say AC is going to be equal to CE so we get all of that
- from this first statement right over there
- Then they gave us some more clues or some more information
- They say CG is equal to 24 so this is CG
- right over here it has length 24
- They tell us BH is equal to DF BH is equal to DF so those two things
- are going to be congruent they're gonna be the same length
- Then they tell us that GF is equal to 12
- so this is GF right over here
- so GF is equal to 12 that distance right over there
- And then they finally tell us that FE is equal to 6 so this is FE
- And then finally they ask us what is the area of CBHFD so CBHFD
- so they're asking us for the area they're asking us for the area
- of this part right over here
- Let me that part and that part right over there that is CBHFD
- So let's think about how we can do this
- We can figure out the area of the larger triangle and then from
- that we could subtract the areas of this little pieces at the end
- Then we'll be able to figure out this middle area this area
- that I've shaded then we don't have
- the information yet to solve that
- We don't know the height or the altitude of this triangle is
- We don't know it's base if we knew its base
- We would say hey! one half base times height we'd get the area
- of this triangle and then we have to figure out subtract this areas
- And we don't have full information there either
- We don't know this height once we know that height
- then we could figure out this height but we also don't quite yet know
- what this length right over here is
- So let's just take it piece by piece
- So the first thing we might want to do and you might guess
- cause we've been talking a lot about similarities
- making some type of argument about similarity here
- because there's a bunch of similar triangles
- For example, triangle CGE shares this angle with triangle DFE
- They both share this orange angle right here
- and they both have this right angle right over here
- So, they have two angles in common they're going to be similar
- by angle-angle you can actually show there's should be a third angle
- in common because this two are parallel lines
- So, we can write the triangle CGE is similar to triangle DFE
- And we know that by angle-angle
- We have one course set of corresponding angles congruent
- and then this angle is in both triangle so it is a set
- of corresponding congruent angles right over there
- And so then once that we know that they're similar,
- we can set up the ratio between sides
- 'Cause we have some information about some of the sides
- so we know that the ratio we know that the ratio between DF
- and this side right over here which is a corresponding side
- The ratio between DF and CG so the ratio between DF and CG
- which is 24 is going to be the same thing as the ratio between FE,
- FE which is 6 and GE which is not 12 it's 12 plus 6 it is 18
- And then let's see 6 over 18 this is just 1 over 3,
- you get 3 DF is equal to 24
- I just cross multiplied or you could multiply both sides by 24
- Multiply both sides by 3 you would get this
- Actually you could just multiply both sides 24
- and then you get 24 times one-third but we'll just do it this way
- divide both sides by 3 you get DF, DF is equal to 8
- So, we found out that DF is equal to 8 that length right over there
- And that's useful for us
- Because we know that this length right over here is also equal to 8
- And now what could we do?
- Well we can make another seems
- like we can make another similarity argument
- 'cause we have this angle right over here
- It is congruent to that angle right over there
- And we also have this angle which is going to be 90 degrees
- We have a 90-degree angle there and actually said that by itself
- is actually enough to say that we have two similar triangles
- We don't even have to show that they have a congruent side here
- and actually were gonna show this are two congruent triangles
- that we're dealing with right over here
- So we have two angles and actually we could just go straight to that
- 'cause when we talk about congruency
- if you have an angle that's congruent to another angle,
- another angle that's congruent to another angle
- and then a side that's congruent to another side you're dealing
- with 2 congruent triangles
- So, we can write triangle right over here I'll write it in pink
- Triangle AHB is congruent is congruent to triangle
- You wanna get the corresponding vertices right
- We can read the triangle EFD to triangle EFD and we know that
- because we angle-angle-side postulate for congruency
- And if that two triangles are congruent that makes things convenient
- that means if this side is 8, that side is 8 we already knew that
- That's how we established our congruency but that means
- if this side is 6 has length 6
- and the corresponding set on this triangle
- is also going to have length 6
- So we can write this length right over here is also going to be 6
- Now I can imagine, you can imagine where all of this is gonna go
- but we want to prove to ourselves
- we want know for sure what the areas
- We don't want to say "Hey maybe this is the same thing as that"
- Let's just actually prove it to ourselves
- So how do we figure out we've almost figure out the entire base
- of this triangle but we still haven't figure out the length of HG
- Well now we can use a similarity argument again
- because we can see that
- triangle ABH is actually similar to triangle ACG
- They both have this angle here and then they both have a right angle
- that one ABH has right angle there ACG
- has a right angle right over there
- So you have two angles two corresponding
- angles are equal to each other
- You're now dealing with similar triangles
- So we know that triangle ABH ill just write it this as AHB
- since I already wrote it this way
- AHB is similar to triangle AGC
- You want to make sure you got the vertices in the right order
- A is orange angle G is the right angle and C is the unlabeled angle
- This is similar triangle AGC
- Now what that does for is that we could use the ratios to figure out
- what HG is equal to so what can we say over here
- Well we can say that 8 over 24
- BH over its corresponding side of the larger triangle
- So we say 8 over 24 is equal to 6 is equal to 6 over not HG
- but over AG 6 over AG and I think you can see where this is going
- You have one-third is equal to 6 over AG
- or we can cross multiply here and we can get AG is equal to 18
- So this entire length right over is 18 if AG is 18 and AH is,
- then HG is 12
- And this is what you might have guessed if you were just trying
- to guess the answer right over here
- But now we have proven to ourselves that this base is has length of
- Well we have 18 here
- and we have another 18 here so has a length of 36
- so the entire base here is 36, so that is 36
- And so now we can find out the area of this larger
- of this entire isosceles triangle
- So, the area of ACE is going to be equal to one-half times the base
- which is 36 times 24
- And so this going to be the same thing as one-half times 36 is 18
- 18 times 24 I'll just do that over here at the top so 18 times 24
- 8 times 4 is 32, one times 4 is 4, plus 3 is 7,
- let me put a zero here
- because we're not dealing math with 2 but 20
- If 2 times 8 is 16, 2 times one is 2 plus 1 so it's 316
- and then you have 2, 7 plus 6 is 13, one plus 3 is 4
- So the area is ACE is equal to 432 but were not done yet
- This area that we that we care about
- is the area of the entire triangle minus this area
- and minus this area right over here
- So what is the area what is the area of this each
- of this little wedges right over here
- So it's going to be one-half times 8 times 6, so one-half times 8,
- is 4 times 6
- So this is going to be 24 right over there and this is going
- to be another 24 right over there
- So this is going to be equal to 432 minus 24 minus 24, or minus 48
- Which is equal to and we could try this to do this on our head
- If we subtract 32 we're gonna get to 400
- and then were gonna have to subtract another 16
- So if you subtract 10 from 400 its 390 so that you get to 300 you get
- to 384 whatever they units for this were
- If these were in meters and this would be meters squared
- If this centimeters this will be in centimeters squared
- And did I do that right?
- Let me go the other way if I add 8 if I add 40 to this
- 24 plus another 8 gets me to 432
- Yup and were done
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