Chapter 8 is similarity and 8.1 is similar polygons. So similar polygons are going to have the same shape, but they're going to be different sizes. So, you know, like a large pentagon and a small one that have the same angles.
So all the corresponding angles are going to be congruent. But if all the sides are congruent, the corresponding sides, then you'll end up with two figures that are the same size. So that's not going to work. So the corresponding sides are not congruent, but they are in proportion.
And I've got an example right here of two different triangles, obviously. So this says write a similarity statement, so that's kind of giving it away that these are similar triangles. But let's think about this. I can see the angles.
Hey, all of the corresponding angles are congruent. I've got that already. Now, the corresponding sides are not congruent, but they are in proportion.
Okay? So when I look at the smallest side on the small angle, and I compare that to the smallest side on the bigger triangle. Okay, those are my smallest. Then let's see four is my medium side here and eight is my medium side just underlying those I'll put a box around the largest sides in both triangles and if you compare those Those are all going to be in proportion. Okay?
So in other words, I can say that three is two six Those match up right as four is two eight right and so, you know, this reads out 3 is to 6 is 4 is to 8. That's one way to read this statement. And that's as 5 is to 10. And these are in proportion because 3 is half as big as 6, 4 is half as big as 8, and 5 is half as big as 10. In other words, those will all reduce to the same. All those fractions are equivalent, right?
So then we can say the sides are in proportion, okay? So now I can say that the The triangles themselves are similar. That's the symbol for similar. It just doesn't have, it's like the congruent symbol without the equal sign. Okay.
So writing a similarity statement, well, I forgot to name these, but I'll go back and do that on the note-taking guide. But let's say this one was ABC and this one is DEF. Okay, so if I'm writing a similarity statement, it's very similar to a congruent statement.
I just want to say what two triangles are congruent. Well, my smaller triangle is triangle ABC. And then I'm going to say that's similar to the larger triangle.
But you got to be careful again. A is going to match up with E, right? Those are both the right angles. So since I put A in the first slot here, I want to put E in the first slot.
Because then my similarity statement is not only going to tell me what triangles are similar, what angles are congruent. Okay. So then let's see, B comes next and B matches up with F, right?
Those are the ones with one dash in there and then C and D. Okay. So you can name the first triangle however you like, but then get the corresponding angles in the same positions in the second triangle.
Okay. Now this said, it says, um, find the side ratio. Well, the side ratio is just the ratio that these two sides are in, and that is one half, right?
You could say it's 3 sixths or 4 eighths or 5 tenths, but when we write side ratios, we reduce them, okay? They're usually written as fractions. You could write it, sometimes you might see it written like this, and that would be fine.
And actually, you could flip this around and say it's 2 to 1 if you're comparing those sides to those ones. Usually, I just go left to right, but you... could make an argument two to one would work.
So that's fine as well. Okay. And then I also want to point out the side ratio is the same as the is the same thing as the similarity ratio. Okay, so those mean exactly the same thing.
You could also call this the perimeter ratio, because the perimeters of those two triangles are going to be in a one to two relationship, right? This triangle is twice as large as that one, so that means the perimeter around the small one is going to be half the perimeter of the big one. So the perimeters are going to be in that same ratio.
And this is sometimes also called the scale factor. The scale factor tells you how much... bigger one of the similar figures is compared to the other one. Okay?
All right. So let's look at this example. So this one tells us right off the bat, gives us the similarity statement, tells us these are similar triangles, and I'm going to solve for x.
Okay? So, you know, I could say, oh, angle A is congruent to angle D, and that's nice, and I could go through and do that with the other triangles, the other angles. But it's not going to help me here because I'm interested in the sides.
Okay? So I want to see what part does the X match up with. Now some people might just look at this and say, oh, X and 15 match up.
And then these two sides. I think most people would get those two sides matching up. But the question is, does X match up with the 15 or the 17? So we don't really want to go based on looks.
Yes, these two sides are going to match up. But what I can do is I can think, okay, this side is BC. That's the second and third letters in that triangle. So then I'm looking at the second and third letters over here, EC.
So that means that X is going to match up with 17. Okay, that's important. And you could do the same thing. The 10 and the 14 are going to match up.
But now what I can do is write a proportion. These sides are going to be in proportion. So a proportion is just fractions that are equal to each other. Okay, so now I can say X is to 17. as 10 is to 14. Okay, and I could have done it different ways. There's lots of different ways you could have set up this proportion.
I could have said 14 is to 10 as 17 is to X, something like that. But however you write this proportion, you should wind up with the X and the 14 in opposite corners and also the 10 and 17. So, you know, you could switch the 10 and the 17, you could switch the X and the 14, but you should still end up with those. being diagonally across from each other.
And that kind of lends itself to how I'm going to solve this. There are a couple things I can do. I can multiply both sides by 17 to isolate the x.
But I like to use the cross products. When I have two fractions that are equivalent, then that means their cross products must be equivalent. So that means x times 14, or 14x, is going to equal 17 times 10, which is 170. No, it's not. Yes, it is. What am I thinking?
Okay. Oh, I see what I was thinking. Okay. So, um, I got myself confused for a second because I was looking at my notes. And, um, another thing you could have done with this fraction is to reduce that to five sevenths to begin with.
And that, that would just That's fine if you did that, because it will still work out to the same answer. Okay, so now I'm going to divide by 14. And this is not going to come out to an integer here. I could reduce 170 over 14, and that would reduce to 85 sevenths if you wanted to write it as a reduced fraction. Or you could do a decimal approximation, and this is going to come out to roughly 12.1. 4. Okay.
So if you do leave it as a fraction, it should be reduced. Okay. All right. Let's head on to the second page.
All right. So here I've got two different squares, right? Those are my geometric definitions of squares.
The side ratio, if I'm thinking about the sides, well, The sides are all congruent. This really should have, I'll fix this on the note-taking guide again. It should look like that, right?
Because 2 is not congruent with 3. All right. So when I'm looking at the sides, it doesn't matter which side I choose. So I'll just say that this side corresponds to this one. So my side ratio, all I have to do, 2 to 3, right? And it's already reduced.
So there we go. Okay. Now let's think about the area ratio. So I would get the area of a square by...
doing the base times the height. So the area here would be 4. 3 times 3 is 9, right? So the area there is 9. So my area ratio is 4 to 9. Okay, great. Now let's look for a pattern here.
So if you think about this, you might have noticed a pattern if you're comparing 2 thirds to 4 ninths. This is going to be equal to 2 squared over 3 squared, right? And hey, 2 thirds is the side ratio, and it actually always works like that.
So if you take the side ratio and you square it, that's going to be equal to the area ratio of any two similar figures. They have to be similar figures for this to work. All squares are going to be similar.
That would fit that definition of similar figures, right? So let's say you knew the area ratio and you wanted the side ratio. Well, you're going to do the opposite of squaring, right, which would be taking the square root.
So I can say the square root of the area ratio would be equivalent to the side ratio. So let's try out these examples. Let's say I've got two similar figures. I guess I should have put that in the directions.
But if the side ratio is 2 to 2 sevenths, well, then the area ratio is just going to be 2 squared over 7 squared, which is going to give me 4 49ths. And that's already reduced there. All right, so this one, all right, here I've got 49 over 100 for the area ratio. So I'll just take the square root of that, okay? Now, you could write it like this if you like, but just remember you can split that like so.
And this is going to be the side ratio, right? I shouldn't have read under that. under the area ratio. But I'm just taking the area ratio and taking the square root of it. Hey, the square root of 49 is 7, and the square root of 100 is 10. So I've got 7 tenths.
And that is my side ratio there. Okay. All right. So let's put that to use on this next example.
So I've got two figures here that I don't really have a good way to find the area of these two figures. At least not with the given info that I have here. I don't have enough info.
I'm told that this one is 20 square feet. I don't really have a way to find this one if I just had this, right? But what I can do, I'm told that these two figures are similar, so I can figure out what the area ratio is. Now, I can say the area ratio is 20 to x, but I don't know what x is.
So what I'm going to do is a little workaround here. I'm going to find the side ratio. first, which is 3 to 4. I meant to say this a minute ago.
So I want to show you how people miss this problem. A lot of people look at this and say, oh, 3 is to 4 as 20 is to x. 3 is to 4 as 20 is to x. And then they'll cross multiply. But this is wrong.
So don't do that. OK, that's the mistake I see all the time. Because then I'd be saying that these sides are in the same proportion as these areas. And the side ratio and the area ratio are not the same.
So what I want to do first is find the side ratio, because I do have two sides to work with. I would reduce it if I had to, but it's already reduced. And from that, I can find the area ratio.
So the area ratio is going to be 3 squared over 4 squared. That gives me 9 16ths. Okay? So now I've found the area ratio, and now I can write my proportion. Because my areas are not going to be in a 3 to 4 ratio.
They're going to be in a 9 to 16 ratio, because that is the area ratio. Okay? So 9 is to 16, as 20 is to x.
Okay? And make sure you get, it's pretty evident here, but make sure. 9 is smaller than 16, and I can see from the picture that 20 is going to be smaller than x. So I have the smaller one in the top position, so I want it set up the same way in the second one. And you can use the two sides to gauge which is your smaller and larger figure.
And then I'm cross-multiplying. 9x is going to equal... 320 divided by 9 off screen there.
Okay and then let's see X would equal 320 over 9 if you want to leave it as a fraction or as a actually I said should say this comes out to 35.5 repeating. And I do know the units here, right? The other area was in square feet, so this should be in square feet as well.
So there is the area of the larger figure. Okay, so you know there are ways sometimes that you could find the area of a quadrilateral like this But we definitely don't have any way to find the area of socks, right? But we can still if if we know that these are similar and that should be in here again I'll add that to the note-taking guide, but if I know that those socks are similar So in other words, they're the same shape just different sizes.
This one got shrunk in the laundry, I guess. Okay So now what I can do is I can say, oh, side ratio is going to be 4 fifths. From that, I can find the area ratio. So you've got to do this step first.
If you just say 4 is to 5 is x is to 30, you'll get the wrong answer. So I'm going to square the side ratio and give me 16 to 25. Okay, now I can write my proportion. Let me just separate those. So since I'm looking at two areas, I want to say 16 is to 25 as x is to 30. Just cross-multiply my way to success.
25x is going to equal 480. And then if I divide by 25, so 480 over 25. If you reduce that, it's going to give you 96 fifths. comes out to 19.2. And I do want to put my units in there. So there's the fraction and the decimal version. I don't really care between those two personally, but if it's fraction, it should be reduced.
Okay. And that is it for 8.1. See you next time.