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 Basics Introduction to what it means for triangles to be similar
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- When we compare triangle ABC to triangle XYZ,
- it's pretty clear they aren't congruent,
- that they have a very different lengths of their sides
- But, there does to be seem something interesting
- about the relationship between this two triangles
- One, all of their corresponding angles are the same
- So, the angle right here, angle BAC is congruent to XYZ
- Angle BCA is congruent to YZX,
- And angle ABC is congruent to XYZ
- So all of their angles, the corresponding angles are the same
- And we also see, we also see that the sides
- are just scaled up versions of each other
- So the goal from the length of XZ to AC,
- we can, we can multiply by 3, we multiplied by 3 there,
- to go from, to go from XY, the length of XY
- to the length of AB which is of course the corresponding side
- We are multiplying by 3, we have to multiply by 3
- And then, to go from the length of YZ to the length of BC,
- we also, we also, multiplied
- We also multiplied by 3
- So essentially, triangle ABC is just a scaled up version
- of triangle XYZ
- If they were the same scale,
- they would be the exact same triangles but,
- one is just a bigger, a blown up version of the other one or,
- this is miniaturize version of that one over there
- If you just multiply all of the sides by 3, you get to this triangle
- And so, we can't call them congruent but they're,
- there does seem to be a bit of a special relationship
- So we call this special relationship "similarity"
- So we can write that triangle, triangle ABC
- is similar, similar to triangle,
- and we wanna make sure we get the corresponding sides right,
- ABC is gonna be similar to XYZ
- To XYZ
- And so, based on what we just saw,
- there's actually kind of 3 ideas here,
- and they're all equivalent ways of thinking about similarities
- One way to think about it is that,
- one is a scaled up version of the other
- So scaled, scaled up or down of the other
- Down versions
- When we talk about congruency, they have to be exactly the same
- You could rotate, you could shift it, you could flip it,
- but when you do all those things,
- they would have to be essentially identical
- With similarity, you can rotate it, you can shift it, you can flip it,
- and you can also scale it up and down
- in order for something to be similar
- So for example, if you say, if something is congruent,
- if, if, let me say triangle, let's say triangle CDE,
- if we know that triangle CDE is congruent to triangle FGH,
- then we definitely know that they're similar
- They're scaled up by factor of one
- Then, we know for a fact that CDE
- Is also similar to triangle FGH,
- but we can't say the other way around
- If, if trianlge ABC is similar to XYZ,
- we can't say that it's necessarily congruent,
- and we see for this particular example,
- they definitely are not congruent
- So this is one way to think about similarity
- The other way to think about similarity is that,
- all the corresponding angles will be equal
- So, if something is similar, then all of the corresponding angles
- are going to be congruent
- Corresponding, corresponding angle
- Always having trouble spelling this
- It is two R's, one S
- Corresponding, corresponding angles, corresponding angles
- are congruent
- Are congruent
- So, if we say that triangle ABC is,
- triangle ABC is similar to triangle XYZ,
- that is equivalent to saying that angle, angle ABC,
- angle ABC is congruent,
- or we can say their measures are equal to angle XYZ,
- to angle XYZ,
- that angle, that angle BAC,
- BAC is going to be congruent to angle YXZ, to angle YXZ
- And then finally, angle ACB, ACB is going to be congruent to
- angle, to angle XZY
- XZY
- Angle XZY
- So if you have two triangles, all of their angles are the same,
- then you could say that they are similar
- Or if you find two triangles,
- and you are told they are similar triangles
- then you know that all of their corresponding angles are the same
- And the last, I guess the way to think about it is that,
- all the sides are just scaled up versions of each other
- So the sides, so sides, scaled by the same factor,
- scaled by same, same factor
- And the example we did here, the scaling factor was 3,
- it doesn't have to be 3
- It just to be the same scaling factor for every side
- If this side, if we scaled this side up by 3,
- and we only scaled this side up by 2 then,
- we would not be dealing with a similar triangle
- But, if we scaled all of these sides up by 7,
- then that's still similar
- As long as you have all of them scaled up by,
- or scaled down by the exact same factor
- So one way to think about it is, and I wanna, wanna keep having,
- well, I wanna still visualize those triangles
- Let me, let me re-draw em' right over here, a little bit simpler
- So I'm now talkig in, now in general terms,
- not even for that specific case
- So, if we say that this is A B and C,
- and this right over here is, X, XYZ
- I just re-drew em' so I can refer to them when we write down here
- If we're saying that this two things right over here are similar,
- that means that corresponding sides
- are scaled up versions of each other
- So we could say that the length of AB,
- we could say that the length of AB,
- AB, is equal to some scaling factor,
- and this thing could be less than one,
- some scaling factor times the length of XY, the corresponding sides
- And I know that AB corresponds to XY
- because of the order in which I wrote this similarity statement
- So some scaling factor times XY,
- we know that BC, the length of BC
- we know that the length of BC needs to be that same scaling factor
- The same scaling factor times the length of YZ,
- times the length of YZ, so that same scaling factor
- And then, we know the length of AC, the length of AC,
- is going to be equal to that same scaling factor times XZ
- So that's XZ and this could be a scaling factor
- So if AB is larger than, if ABC is larger than XYZ,
- than this K's will be larger than one,
- if they're the exact same size,
- if they are essentially congruent triangles,
- than this K's will be one
- And if XYZ is bigger than ABC,
- than this scaling factors will be less than one
- But another way to write the same,
- all I'm saying is corresponding sides
- are scaled up versions of each other
- This first statement right here if you divide both sides by XY,
- you get AB over XY, is equal to, our scaling factor
- And then the second statement right over here,
- if you divide both sides by YZ, you get B, let me do that same colour
- You get BC divided by YZ is equal to that scaling factor,
- is equal to that scaling factor
- Whatever, the example we just showed that scaling factor was,
- scaling factor was 3 but, now we're saying
- in a more general term, similarities,
- as long as you have same scaling factor
- And then finally, if you divide both sides here
- by the length between X,XZ , or segment XZ's length,
- you get AC over XZ is equal to K as well
- Is equal to K as well
- Or another way to think about it is,
- the ratio between corresponding sides,
- notice this is the ratio between AB and XY
- AB and XY
- The ratio between BC and YZ, BC and YZ
- The ratio between AC and XZ, AC and XZ
- That the ratio between corresponding sides
- all gives us the same constant
- Or you can rewrite this as, AB over XY is equal to BC over YZ,
- is equal to, is equal to AC over XZ,
- which would be equal to some scaling factor,
- which is equal to K
- So if you have similar triangles,
- let me draw an arrow right over here,
- similar triangles means that they're scaled up versions and,
- you can also flip and rotate, do all the stuff with congruency,
- and you can scale them up or down, which means,
- all of the corresponding angles are congruent,
- which also means that the ratio between corresponding sides
- is going to be the same concept for all the corresponding sides,
- or the ratio between the corresponding sides is constant
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