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Mean Proportional Examples

In mathematics, the mean proportional is a geometric mean, and is the number which is the arithmetic mean of the reciprocals of the members of a set of numbers.

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Heather Z.

Oregon State University

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Boston College

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Utica College

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Video Transcript

All right. So we're here to talk about how to use geometric means or the proportional relationships involved specifically with a right triangle on the altitude drawn. There are lots of similar triangles, three to be exact, um, the larger the medium and the smaller depending on how the right triangle is drawn. But the fact that you have three similar triangles if you looked at the previous video because you essentially have ankle angle similarity between any two of the triangles, you have tons of proportional situations going on. So how can you use those proportions from similar triangles to solve for a given side? So let's dissect a couple these questions. So if you have thats right triangle here. Hi pot news 100. You can see the smaller leg of the smaller triangles 36 you can see that altitude X, which is the geometric mean between the two triangles. If you recall, this segment had a very special relationship between the medium and the smaller triangle. In fact, uh, if you recall from the previous video, this'll leg is to the altitude as the altitude is to this leg. So 100 minus 36 is 64. So we get this value here. I'm gonna raise this year. So this has a length of 64 you know, we know is 36 is two X as ex is to 64. Why? Because small leg is too large. Leg as small leg is too large leg based upon similar triangles and proportional relationships. So you're going to get 36 times 64? That's 2304 equals X squared. If you take the square root of that value, you end up with conveniently 48. And that's the value of this link. So, depending on you know what's labeled what triangles we have and so forth. So tell you this problem again. You have a smaller triangle, medium triangle. And of course, the larger trend. I want to ask yourself, What are we looking for? What do we have? We're looking for either the high pot noose of the smaller triangle or the smaller side or leg of the bigger triangle. Then ask yourself, What do we have? Well, we have hi pot news of the larger triangle. So think about this high pot news of the larger triangle which is 25 over X, which is the high pot. News of the smaller triangle equals well, Think about this. This is the smaller leg of the smaller triangle. But the crazy thing is, this is the smaller leg of the bigger triangle. So end up with X over nine and again, you can see that the value we're looking for ends up being a geometric mean or a mean proportional between the two triangles. And so you get 25 times nine, which ends up being to 25 equals X squared. And when you square that you ended with 15 equals X squared. So what I recommend is not just jumping in and using a rule, but really looking into what triangles do we have what's labeled what we're looking for and just trying to match up sides. So let's take this next example again. We're looking for this geometric mean. This is very similar to the first example 25 minus nine gets us 16 and as you recall, nine is Toe X as X is to 16. Why small is too long as small is still long. So it nine times 16 that's 1 44 equals X squared and then therefore X equals 12. Let's do a couple more examples example for All right. So you can see we have the hypotenuse apartment. We have the hypotenuse of the larger of the largest triangle. We're looking for what looks to be the high pot news of the medium triangle. So what I'll do there is I'll say, I already know that I could say 81 is two X because I'm saying hi pot news of the large is to the high partners of the medium. Okay, then we're gonna look for what do we What do we know what we have in common? Well, we do know that the one of the legs of the medium is 45 and that looks to be the larger of the two legs. But we also know the larger of the two legs of the bigger triangles. X. So once again, I can say access to 45. And once again, I could say, Oh, look at that. And if you're ever having one of those situations, guys, where you have a right triangle, altitude dropped. Pardon me. Altitude dropped. You'll know if you set your proportion, right? Because the unknown is going to always be in this relationship. So 81 times 45 that gets us 3645 equals X squared. And if we take the square root of that answer, we end up with roughly 64 is your answer. So every one of these problems is essentially just setting up a proportional situation between similar sides. Because we have similar triangles because we have angle, angle, similarity, like question eight Here, for example, this is the proportional mean between this side and this side, which is the main the're, um with mean proportional. So you could say X is 2 48 as 48 is to 64 that's gonna get us 48 square, which is 2304 equals 64 X. When we go ahead and cross, multiply, and then when we solve, we get a nice value of 36 equals X. So all of these problems air again, such that you know, you have three triangles, you have the larger triangle, you have the medium triangle and you have the smaller triangle. All of them are right triangles, all of them are similar by angle angle similarity, which means all of their corresponding sides are similar to each other. But it might just be that one's hi pot noose of the medium is the high partners of the you know the sides might vary in terms of how you define them, based upon what triangle decide is belong to you. So hopefully this helps, um, check my other videos on similar triangles, angle, angle, similarity and mean proportional fear. Um, essentially the geometric mean given a right triangle and the altitude and the three similar triangles that are created. I'll talk to you later. Thanks.