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Mean Proportional Theorems

In mathematics, the mean proportional theorems are a pair of theorems that relate the arithmetic and geometric means of two (or more) numbers. They were first discovered by Nicomachus in the 2nd century AD.

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Video Transcript

Hey, guys, Mr. Kleinberg here and we're gonna talk about mean proportional, otherwise known as geometric means. Um, in most texts and most conversations with mathematicians that I know of, we refer. What I'm gonna talk about actually is geometric means, Um but it probably makes sense. Toe also much them as proportional or proportional, because we're gonna talk about similar triangles and proportional basically side values if you have a right triangle. So I have a right triangle right here. Triangle, ABC and from the right angle, you drop the altitude. What you done is you've created a set of similar triangles. You have the triangle B D. C. You have the triangle a D B. And obviously the bigger triangle a C B, all of which are similar to each other because they all have the right angle in them. So keep in mind they have this right angle this right angle and this strangle a D B. Is this right angle and then, if you think about it, the bigger triangle A, C B and the medium sized triangle A. D. B have this angle in common. So the bigger triangle, which will call ABC, is going to be similar to the medium triangle, which we call a D. B. Because they have these angles in common, they have a right angle in common. And if you recall from previous video, they have angle angle similarity. By that very same argument, the smaller triangle which will call D. C B whatever you like to call it, has the right England common. But the bigger right triangle and the tiny little right triangle have angle see in common, which again, by angle angle similarity makes them similar, which means all of their sides are proportional. And so we get some very special things going on. Namely, the biggest one is this is this is the biggest of the so called three proportional. The theory is that you give him here. But if you look at the medium triangle, I'm gonna do this in a different color. So if you look at the medium triangle and then the smaller triangle, you'll notice that this is the longer side. This is the shorter side. But this is the longer side of the smaller triangle, and this is the shorter side. So if you do, long is too short. As long is too short. That should be a true statement That proportions should be true because the triangles are similar So essentially a d is to be d as BD is to d c We've just done long is too short Eso eso long is too short as long is too short And so what happens is this side BD is very special because if you notice um what we get out of here is that we get if we do a cross multiplication. We ended up with a d times d c equals B d squared. So when you have a right triangle and you drop this altitude, this segment becomes a what we call a geometric mean Ah proportional constant between this side and this side of the of the pot news of the bigger triangle that's separated by the altitude. It's a very important and re occurring. The're amigos all the way back to the times of the Greek. When we're talking about construct herbal numbers and what numbers exist and what numbers don't exist or what numbers do we know exist? And what numbers do we not know yet exist? That's that altitude in a right triangles absolutely important in this. The're, um based upon similar triangles is absolutely key. But what we also get in a situation apartment in a situation where we have a right triangle, draw that again. A right triangle with the altitude drawn down is not only do we have this side is to this side, as the altitude is to this side what I just wrote up here. But now you have all kinds of things. Like the hypotenuse of the smaller triangle to its smallest side is gonna be equal to the high partners of the bigger triangle to its smallest side. So of the three triangles that air formed, which are the biggest triangle than the medium triangle and the smaller triangle, all of them have their own high pod news that at any given time, behave as different sides. For example, this is the high pot news of this of this medium triangle. But it's the longer leg of the bigger triangle that makes any sense. This is the high pot news of the smaller triangle, but it's the smaller leg of the bigger triangle. This is the longer leg of the smaller triangle with the smaller leg of the medium triangle. So every one of these segments is playing multiple characters, depending on how what the context ISS. So because they're all similar to each other, you have all these proportional relationships, and because you have these triangles inside of triangles, one side might act as the long side of the long leg of one triangle. But then this smaller leg of the other or the high partners have one triangle with a larger leg of the other. And so because of that, we have all these different proportions based upon this very special situation. When you have a right triangle, the altitude drawn from the right angle to the opposite side, obviously perpendicular to the opposite side, and you create the bigger triangle, which is Triangle one. The medium triangle will call triangle to, and then the smaller triangle, which will call Triangle three. Now the special segment. This is one of between. It's the geometric mean between this side and this side, which will do some more examples of in a second. But you essentially, you have three similar triangles, and by similar triangles you proportionality between respective sides. So as long as you can rely on the proportionality of similar triangles, and you can understand that we're dealing with three different triangle shapes, the bigger the medium and the smaller. And you can differentiate which side is acting as the small vs large versus the iPod news versus the leg. Depending on the situation, then you can set up all kinds of proportions and solve all kinds of problems. So my next video is going to basically be about that. I'll see you, then take it easy.