Introducing triangle similarity

0:00- [Instructor] What we're gonna focus on this video is the idea of geometric figures being similar, or similarity. And in particular, we're gonna focus on similar triangles for the sake of this video. So, let me actually draw a triangle, and actually, while I draw it, I'm actually also going to draw its mirror image using this tool here, so let me get the little line tool going.

0:26All right, so I'm just drawing an arbitrary triangle here, so I'll go from maybe from here to there, to there, and then back over here. And so, with this little tool, it drew both that original triangle, and then it drew its mirror image here. And the whole reason why I'm doing a mirror image and I'm about to move things around and rotate them and scale them is if you can go from one triangle to another with the combinations of taking mirror images, scaling, rotating, shifting, et cetera, those two triangles are going to be similar.

1:02So, these are already the mirror image. Let me actually make a copy of one of these. Maybe I'll take the copy of the original one that I did, so Ctrl + C, Ctrl + V, so I'm now pasting it. So, these are still similar. I've just shifted it. Now, I can rotate this one.

1:20It is still similar. I can also scale this up. It is still similar. Now, you might say, "Well, what could I do to this triangle that would now not make it similar with either of these other two?" Well, if I were to distort it in some way, if I were to do something like that, where I'm distorting it more in one direction than another, I'm scaling it more in one direction than another, then that would no longer be similar.

1:49And so, one way to think about it is, it looks kinda the same. It's just oriented different. It could be a different size. Now, how do we say that a little bit more carefully using the language of math? Well, one thing that you might be tempted to say, and you would be right if you did, is it looks like the angles are the same in the two.

2:10So, for example, whatever angle this angle is right over there, if I were to eyeball it, I don't know, it looks like about 35 degrees. I would have to get a protractor out to measure it properly. But by doing all of these mirror images or shifting it or rotating it or scaling it, I have preserved this angle measure in all three of these situations, and that's not just true of that angle.

2:34It's true of this angle. This angle right over here corresponds to this angle right over here. I'll do two arcs to show that that's a different angle there. And over here, I'll do two arcs over there. And then, last but not least, and actually, once you know that two angles of a triangle are the same, you know the third one's going to be the same 'cause they have to add up to 180, but I can draw that.

2:58And maybe I'll do this in another color, actually, just so I don't keep... I'll do this in red, so. So, that angle corresponds to this angle. I'll do three arcs there to show that these are the ones that correspond to each other. So, if you have two triangles where, for sure, if three of the angles have the same measure, you can actually go the other way around.

3:20You can assume that they are similar. You can assume that you can go from one to the other with a combination of scaling, or I guess I should say dilations with rotations, with shifting. And if you have two angles that have the same measure, you know that the third one is going to be also congruent.

3:41So, actually, two angles are sufficient. If I didn't even know, if you didn't see that exercise that I did in the beginning of how I was able to get from one of these triangles to the other, if all I showed you is this triangle over here and this triangle over here, and all I told you is that these two angles have the same measure as these two angles, or another way to saying it, this one has the same measure as that one, and this one has the same measure as that one, that's sufficient to say these must be similar triangles.

4:14Now, another way that you can approach it, and once you take geometry, you'll learn many other ways of proving and justifying that two triangles are similar, but another way is to compare the ratio between sides. So, if all three sides of a triangle have the same ratio, then they're similar.

4:32For example, these first two that I drew, the ratio is 1:1. Whatever length this is, this is going to have the same length as this side right over here. This length is going to have the same length as this one over here. And then last but not least, maybe I'll go back to the purple, this side, this side has the same length as that side over there.

4:57And once you learn more about geometry, et cetera, you'll learn that these are more than just similar triangles. I'll introduce that word in the future. But it's not just 1:1 ratios. For example, I'll put some numbers on it. Let's say that this length right over here is...

5:14I'm trying to eyeball it, let's see. Compared to this one, it's about that long, so let's say this has a length of 2. And then this one over here, which corresponds to that side, this side right over there that corresponds to that one. And notice, it's right in between the same two corresponding angles.

5:32Let's say that that has a length of 3. So, in order to show that these are similar triangles, all of the other sides are going to have to have the same ratio of 2:3. For example, if this side is now, let's see, if this one is 2, this one looks like, and I'm just eyeballing it.

5:52Maybe this one here is... Well, it doesn't look quite exactly 3, but maybe it's 3. Well, then that means that the corresponding side over here, the corresponding side over here would have to be... Well, yellow and blue make green. (chuckles) The corresponding side over here, right over here, just as this one is one and a half times this one, this one would have to be one and a half times this one, so this would have to be 4.5.

6:22And by the exact same logic, if we look at this side right over here, which looks like the longest side, maybe this one is, let's see if 2 is there, maybe this one is 4. And I'm just eyeballing it. I don't know if this is actually what we are looking at here. But if this is 4, then this one, you could pause and think about it.

6:41It would have to be 6. So, notice, the ratio between corresponding sides is the same. This is a 2:3 ratio. This is a 2:3 ratio. When we go from the smaller triangle to the larger one, we're essentially multiplying each of the sides by 1.5. You can also compare the ratios within the triangles.

7:01For example, the ratio of this longest side to the shorter side is 6:3. That's a ratio of 2:1. 2:1. 4 divided by 2 is 2. 6 divided by 3 is 2. Another way to think about it, 3 divided by 2 is 1.5. 4.5 divided by 3 is 1.5. So, if you see these scenarios where you have at least two angles that have the same measure in a triangle, you can say they're similar.

7:29Or you don't need both of these points of evidence, although it's all gonna work out if you have sufficient evidence. Also, if the ratio between the three corresponding sides is the same, then you're also dealing with a similar triangle.