Solving similar triangles: same side plays different roles

In this problem, we're asked to figure out the length of BC. We have a bunch of triangles here, and some lengths of sides, and a couple of right angles. And so maybe we can establish similarity between some of the triangles. There's actually three different triangles that I can see here. This triangle, this triangle, and this larger triangle. If we can establish some similarity here, maybe we can use ratios between sides somehow to figure out what BC is. So when you look at it, you have a right angle right over here. So in triangle BDC, you have one right angle. In triangle ABC, you have another right angle. If we can show that they have another corresponding set of angles are congruent to each other, then we can show that they're similar. And actually, both of those triangles, both BDC and ABC, both share this angle right over here. So if they share that angle, then they definitely share two angles. So they both share that angle right over there. Let me do that in a different color just to make it different than those right angles. They both share that angle there. And so we know that two triangles that have at least two congruent angles, they're going to be similar triangles. So we know that triangle ABC-- We went from the unlabeled angle, to the yellow right angle, to the orange angle. So let me write it this way. ABC. And we want to do this very carefully here because the same points, or the same vertices, might not play the same role in both triangles. So we want to make sure we're getting the similarity right. White vertex to the 90 degree angle vertex to the orange vertex. That is going to be similar to triangle-- so which is the one that is neither a right angle-- so we're looking at the smaller triangle right over here. Which is the one that is neither a right angle or the orange angle? Well it's going to be vertex B. Vertex B had the right angle when you think about the larger triangle. But we haven't thought about just that little angle right over there. So we start at vertex B, then we're going to go to the right angle. The right angle is vertex D. And then we go to vertex C, which is in orange. So we have shown that they are similar. And now that we know that they are similar, we can attempt to take ratios between the sides. And so let's think about it. We know what the length of AC is. AC is going to be equal to 8. 6 plus 2. So we know that AC-- what's the corresponding side on this triangle right over here? So you could literally look at the letters. A and C is going to correspond to BC. The first and the third, first and the third. AC is going to correspond to BC. And so this is interesting because we're already involving BC. And so what is it going to correspond to? And then if we look at BC on the larger triangle, BC is going to correspond to what on the smaller triangle? It's going to correspond to DC. And it's good because we know what AC, is and we know it DC is. And so we can solve for BC. So I want to take one more step to show you what we just did here, because BC is playing two different roles. On this first statement right over here, we're thinking of BC. BC on our smaller triangle corresponds to AC on our larger triangle. And then in the second statement, BC on our larger triangle corresponds to DC on our smaller triangle. So in both of these cases. So these are larger triangles and then this is from the smaller triangle right over here. Corresponding sides. And this is a cool problem because BC plays two different roles in both triangles. But now we have enough information to solve for BC. We know that AC is equal to 8. 6 plus 2 is 8. And we know the DC is equal to 2. That's given. And now we can cross multiply. 8 times 2 is 16 is equal to BC times BC-- is equal to BC squared. And so BC is going to be equal to the principal root of 16, which is 4. BC is equal to 4. And we're done. And the hardest part about this problem is just realizing that BC plays two different roles and just keeping your head straight on those two different roles. And just to make it clear, let me actually draw these two triangles separately. So if I drew ABC separately, it would look like this. So this is my triangle, ABC. And then this is a right angle. This is our orange angle. We know the length of this side right over here is 8. And we know that the length of this side, which we figured out through this problem is 4. Then if we wanted to draw BDC, we would draw it like this. So BDC looks like this. So this is BDC. That's a little bit easier to visualize because we've already-- This is our right angle. This is our orange angle. And this is 4, and this right over here is 2. And I did it this way to show you that you have to flip this triangle over and rotate it just to have a similar orientation. And then it might make it look a little bit clearer. So if you found this part confusing, I encourage you to try to flip and rotate BDC in such a way that it seems to look a lot like ABC. And then this ratio should hopefully make a lot more sense.