Side lengths in similar triangles

0:00- [Narrator] We're told the table shows the side lengths for three triangles, and then they ask us which pair of triangles is similar. Pause this video and have a go at that. Okay, so there's many different ways of approaching it. One way is to look at the ratio between the corresponding sides in the different triangles.

0:18So for example, we could look at the ratio of the short sides of the corresponding triangles. So if I go from RST, to the short side of RST to the short side of UVW, it looks like I'm multiplying by three. If I go from the medium side of RST to UVW, it looks like I'm multiplying by three.

0:40So, so far I have the same ratio. I have a three-to-one ratio, three-to-one ratio. And it looks like the long side, when I go from seven to 21, that's also multiplying by three. So the fact that all three of these, I'm multiplying by the same factor, or I have the same ratio between the corresponding sides, that tells me immediately that RST, RST, and UVW, these two triangles are similar.

1:10Now let's look at the other ones just to see or if any of the other combinations to see if they aren't similar, and maybe we'll also approach it a different way. So if we compare RST to XYZ, here we're multiplying by two to go from three to six, but here we're not multiplying by two.

1:27We are multiplying by two to go from seven to 14, but here we aren't to go to five by nine. All three of these ratios have to be the same. So we know that RST and XYZ are not similar. And then last but not least, we can look at these last two. If we go from UVW to XYZ, if I go from nine to six, I'm essentially multiplying by two-thirds.

1:52So times two-thirds or times six over nine is another way of thinking about it. If I were to multiply by two-thirds over here, I wouldn't get to nine. I would get to 10. So these two are also not going to be similar. So that makes us feel good that our only similar pair is RST and UVW.

2:15Now another way that we could have tackled this is by looking at the ratios between the sides. So what do I mean by that? So actually let me look at the ratio. Let me write it here in a different color here. I'm going to extend this table. I'm going to extend the table here.

2:33And I'm going to calculate three ratios. I'm going to calculate, maybe I'll use S, M, and L for the lengths of those sides. So I'll look at the ratio of the short to long. I'll look at the ratio of, and we could, that's the, actually, let's go it the other way.

2:49Let's do the longer sides top. So we have the long to the short. We have the long to the medium, and then we have the medium to the short side. So the long side to the medium side, sorry, the long side to the short side here is seven over three. Long side to the medium side here is seven over five.

3:14And the medium side to the short side is five over three. Five over three. Now let's do it for UVW. Long side to short side, 21 over nine, which is the same thing as seven to three. Seven to three. Long to medium. 21 over 15 is the same thing as seven to five.

3:38Seven to five. And then medium to short, 15 over nine is the same thing as five to three. You might already see where this is going, but I'm just going to keep going just for completeness. Now let's do it for XYZ. So XYZ, long side to short side is 14 over six, which is the same thing as eight over three.

4:00Eight over three. Long to medium, 14 over nine. And I can't simplify that one anymore. And then medium to short is nine over six, which is three over two. Now, we can see that RST and UVW have the same ratio between the corresponding sides. Long to short is seven over three.

4:25Long to medium, seven over five. Medium to short is five over three. So that's another way that we could conclude that RST and UVW are similar.