Similar triangles Introduction to similar triangles
- I will now introduce you to the concept of similar triangles.
- Let me write that down.
- 6 00:00:14,15 --> 00:00:16,35 So in everyday life what does similar mean?
- 8 00:00:26,89 --> 00:00:29,47 Well, if two things are similar they're kind of the same but
- they're not the same thing or they're not identical, right?
- That's the same thing for triangles.
- So similar triangles are two triangles that have
- all the same angles.
- 14 00:00:50,46 --> 00:00:57,35 For example, let me draw two similar triangles.
- I'll try to make them look kind of the same because they're
- supposed to look kind of the same, but just maybe
- be different sizes.
- So that's one, and I'll draw another one that's right here.
- I'm going to draw it a little smaller to show you that
- they're not necessarily the same size, they just are
- same shape essentially.
- One way I like to think about similar triangles are they're
- just triangles that could be kind of scaled up or down in
- size or flipped around or rotated, but they all have
- the same angles so they're essentially the same shape.
- For example, these two triangles, if I were tell you
- that this angle -- and this is how they do it in class.
- 29 00:01:39,99 --> 00:01:44,27 If I were to tell you this angle is equal to this angle
- and I told you that this angle here is equal to this angle.
- 32 00:01:52,52 --> 00:01:54,01 Well, a couple of things.
- You already know that this angle's going to be equal to
- this angle, and why is that?
- Well because if two angles are the same, then the third
- has to be the same, right?
- Because all three angles add up to 180.
- For example, if this is x, this is y, this one has to be
- 180 minus x minus y, right?
- That's probably too small for you to see.
- But that's the same thing here.
- If this is x and this is y, then this angle right
- here is going to be 180 minus x minus y, right?
- So if we know that two angles are the same in two triangles,
- so we know that the third one's also going to be to same.
- So we could also say this angle is identical to this angle.
- And if all the angles are the same, then we know that we are
- dealing with similar triangles.
- What useful thing can we now do once we know that
- a triangle is similar?
- Well, we can use that information to kind of figure
- out some of the sides.
- So, even though they don't have the same sides, the ratio
- of corresponding side lengths is the same.
- I know I've just confused you.
- Let me give you an example.
- For example, let's say that this side is -- this side is 5.
- Let's say that this side is, I don't know, I'm just going
- to make up some number, 6.
- And let's say that this side is 7, right?
- And let's say we know that, I don't know, let's say we know
- that this side here is 2.
- 64 00:03:37,99 --> 00:03:40,18 So we know the ratio of corresponding
- sides is the same.
- So, if we look at these two triangles, they have completely
- different sizes but they have corresponding sides.
- For example, this side corresponds to this side.
- How do we know that?
- Well, in this case, they just happen to have
- the same orientation.
- But we know that because these sides are opposite
- the same angle, right?
- This is opposite angle y, and then this side is
- opposite angle y again.
- This whole triangle might be too small for you to see, but
- hopefully you're getting what I'm saying.
- So these are corresponding sides.
- Similarly, this side, this blue side, and this blue side
- are corresponding sides.
- Not because they're kind of on the top left because we could
- have rotated this and flipped it and whatever else.
- It's because it's opposite the same angle.
- 86 00:04:32,81 --> 00:04:33,895 That's the way I always think about triangles.
- It's a good way to think about it, especially when you
- start doing trigonometry.
- So what does that us?
- Well, the ratio between corresponding sides
- is always the same.
- So let's say we want to figure out how long this side of
- the small triangle is.
- Well there's a bunch of ways we could do it.
- We could say that the ratio of this side to this side, so x to
- 7 is going to be equal to the ratio of this side to this side
- -- is equal to the ratio of 2 to 5.
- And then we could solve it.
- And the only reason why we can do this -- you can't do this
- with just random triangles, you can only do this with
- similar triangles.
- So we could then solve for x, multiply both sides but 7 and
- you get x is equal to 14 over 5.
- So it's a little bit less than 3.
- So 14 over 5, so 2.8 or something like that,
- that equals x.
- And we could do the same thing to figure out this yellow side.
- So if you know two triangles are similar, you know all the
- sides of one of the triangles, you know one of the sides of
- the other triangle, you can figure out all the sides.
- I think I just confused you with that comment.
- So, this side, so let's call this y.
- you're doing one triangle's going to be the denominator
- here, then that same triangle has to be the
- denominator on the--.
- If one triangle is the numerator on the left hand side
- of the equal sign, right, so the smaller one's
- the numerator.
- Then it's also going to be the numerator on the right hand
- side of the equal sign.
- I just want to make sure you're consistent that way.
- If you flip it then you're going to mess everything up.
- And then we can just solve for, so y is equal to 12 over 5.
- 127 00:06:30,736 --> 00:06:33,92 So, let's use this information about similar triangles
- just to do some problems.
- 130 00:06:44,75 --> 00:06:47,68 So let's use some of the geometry we've already learned.
- I have two parallel lines, then I have a line like that, then
- I have a line like this.
- What did I say, I said that the lines are parallel, so this
- line is parallel to this line.
- And I want to know if this side is length 5, what is -- well,
- let's say this length is length 5, let's say that this length
- is -- let me draw another color.
- This length is, I don't know, 8.
- 140 00:07:45,37 --> 00:07:48,33 I want to know what this side is.
- Actually no, let me give you one more side just to make sure
- you know all of one triangle.
- Let's say that this side is 6, and what I want to do is I want
- to figure out what this side is right here, this purple side.
- So how do we do this?
- So before we start using any of that ratio stuff, we have to
- prove to ourselves and prove in general, that these are
- similar triangles.
- So how can we do that?
- Let's see if we can figure out which angles are
- equal to other angles.
- So we have this angle here.
- Is this angle equal to any of these three angles
- in this triangle?
- Well, yeah sure.
- It's opposite this angle right here, so this is going to be
- equal to this angle right here, right?
- So we know that its opposite side is it's corresponding
- side, so we know that it corresponds to -- we don't know
- its length, but we know it corresponds to this
- 8 length, right?
- I forgot to give you some information.
- I forgot to tell you that this side is -- let me
- give it a neutral color.
- Let's say that this side is 4.
- Let's go back to the problem.
- So we just figured out these two angles are the same, and
- that this is that angle's corresponding side.
- Can we figure out any other angles are the same?
- Let's say we know what this angle is.
- 172 00:09:12,2 --> 00:09:15,1 I'm going to do kind of a double angle measure here.
- So what angle in this triangle -- does any angle here
- equal that angle?
- We know that these are parallel lines, so we can use alternate
- interior angles to figure out which of these angles
- equals that one.
- But I just saw the time and I realize I'm
- running out of time.
- So I will continue this in the next video.
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