Checking triangle similarity with side length ratios

0:00- [Instructor] We are asked which triangles are similar to triangle ABC. So this is triangle, ABC here. Pause this video and try to figure this out on your own before we do it together. All right, now let's do this together. Now, in order for two triangles to be similar, the ratio between corresponding sides needs to be the same.

0:22So one way to think about it, let's look at all of the longest sides on each of them. So we have this side CB, I'll put in red, and let's see here, the longest is ED, right over there. The longest one here is GH. GH. And the longest one here is JL. And now let's look at the second longest sides.

0:45I'll do that in a different color, maybe purple. So the second longest side right here is BA, second longest side right over here is FD, second longest side here is GI, and second longest side here is KL. So what we're gonna do is look at the ratio of the longest side to the second longest side for every one of these and see if they match triangle ABC, and if none of them do, or if some of them don't, then we can immediately rule those out.

1:13So let's think about the ratio of the longest side to the second longest side. In triangle ABC that is 18 over 15, over 15, which is equal to, we have a common divisor of three so it's the same thing as 6 over 5, 6 over 5, which is the same thing as 1 1/5, which is equal to 1.2.

1:39So now let's figure out the ratios of the longest and second longest side for each of these. So in this one right over here, it's 16.2 over 14.4. I think I might take a calculator out for that one. This one over here it is 27 over 22.5. I'll use a calculator for that one as well.

2:00And this one over here, actually all of these have decimals in it so I'll use calculator for that, we have 10.8 to 9. So let me get a calculator out. So 16.2 divided by 14.4 is equal to, that's not 1.2, so I can immediately rule that one out. Let's look at the next one.

2:25So we have 27 divided by 22.5 is equal to 1.2, so that one's looking good so far. And I'll also look at this last one. 10.8 divided by 9, and this is equal to 1.2 as well. So these are both looking good but just to make sure, let's look at the ratio of the shortest, or the longest side to the shortest side.

2:53So the shortest sides I'll put in blue in every one of these. We didn't have to look at the second one anymore. We'll look at all of these. Now, the ratio in triangle ABC is 18 over 8. Over 8. Which is equal to, let's see, they're both divisible by two. That's 9 over 4, which is 2 1/4, which is equal to 2.25.

3:22And we can ignore this one now, so let's look at, use a calculator to figure out these ratios here. We want to look at the ratio of 27 over 12. 27 over 12. And we wanna look at the ratio of 10.8 over 4.8. 10.8 over 4.8. And 27 over 12, this is divisible by, both are divisible by three so this is the same thing as 9 over 4.

3:50Actually didn't need the calculator for it. This ratio holds up. So we just compared the ratio of both the longest side to the shorter side, and the longer side to the second longest side, and it is holding up. So this one is definitely similar to triangle ABC.

4:06Let's use a calculator to check this last one out. So we have 10.8 divided by 4.8 is equal to 2.25, so this one looks good as well. It is the same ratio of the longer side to the shorter side, so these two are similar to triangle ABC.