Medians Centroid and Examples

In mathematics, the median is a measure of the "middle" value of a data set. The median is a commonly used measure of the properties of a set in statistics and probability theory. The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest, and finding the middle one. For example, the median of the list [3, 4, 6, 7, 8, 9] is 6. If there is an even number of elements, then there is no single middle value. In this case, one may take the mean of the two middle values. For example, the median of the list [3, 4, 4, 6, 7, 8, 9] is (4+6)/2 = 4.5.

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Video Transcript

Hi. So we're back to talk about Mawr Characteristics of triangles and we're gonna focus on special segments related two triangles and the first one we're gonna talk about this immediate. The median draw a triangle here. A medium by definition is a is a segment who is one end point is on the Vertex of the Triangle and Houthi other end point is located at the midpoint of the opposite side. My take marks indicate that we have now bisected that opposite side. And by connecting these two points, I have what we call a median. And so I could do this three times because there are three Verte season three opposite sides, respectively. And so there's another medium and our third one would be from this end point to the midpoint of the opposite side. It looks something like that. Okay, so every triangle, regardless of it being obtuse, acute or right, is going to have three medians, couple special things. One, The medians will always meet at the same point. They'll always intersect each other at the same point. That point is known as the century brother. A little nicer, known as the central right. It is known as the balancing point of the triangle. So if you had, like, a was a a wooden triangle you were gonna have as a coffee table. If you were to construct the medians and then find the Centro oId, you would technically only need one leg for the table. Stand on. Obviously, that would not be practical, but it would balance if it was constructed properly. So the Seine tried is the point of concurrency. That's how that's what we call it, the point of concurrency, where all three medians intersect. Another special thing that happens, um, based upon how meetings intersect with each other, is they cut each other in a special ratio. This length here is half of this length here, so that's always gonna be a 1 to 2 ratio. So this is going to be one to to and again, the one to two ratio. Another way of looking at it is doing different color. This part of the median is one third its entire length, and therefore this part here is two thirds the entire length. So if I were to label this as a B c de e f, and we'll call the Centro oId G. And here's what I know. If a D was equal to 12 than a G would be two thirds of 12 or in other words, eight and G D would be one third of 12, which would be four. And then notice that for Plus eight gets you 12 and also noticed that four is half of eight. Because earlier I said there was a 1 to 2 ratio in terms of this part to this part, which is wanted to. So that's a special relationship that exists when medians intersect and cross each other. So a couple of examples I could give you would be. Here's a triangle. First of all, I could say this is a median, and this portion here is X Plus two. In this portion, here is three X Plus eight, and I could ask you to find X Well, since this is a median, both sides are equal because this is the midpoint, and so X plus two would be forced to equal three x plus eight. And in this case you would get negative. Six equals two X or, in other words, negative three equals X. Now this is a nonsensical problem. By the way, I'm just using this to demonstrate. But if you were to plug negative three back in, you get negative one so we would get negative. One goes negative one, but obviously we can't have a negative side length. That's not the point of this problem. I'm glad. I'm glad it came up, because if your answer is negative at a doesn't mean it's not a right answer in this case, it does Algebraic Lee workout, but contextually or in reality, that would not be the situation. This could not be a triangle with such algebraic relationships. Just wanna prove a point that I could say Hey, find X Because that point is the midpoint. I consent those two seconds of each other and solve. Another thing I could do is I could have, let's say, a triangle. I could have to medians intersecting, and I could say, Hear me out on this one. I could say that a B the entire median is, let's say, 10 X plus 12, and I could say that C B is X plus three and I could say solve for X. Remember, this is going to be one third of the entire thing. So what I can say is, alright, three of these better equal the entire thing. So in other words, three of the X plus threes, or to the words three CBS equals the entire A B. All right. So, again, this is not meant to necessarily make practical sense. But more so provide an example of what to dio. We would have three X plus nine equals 10 X plus 12. And this is pure coincidence. But we get negative. Three equals x again. Pardon me and take that back Negative three equal seven x and then we'd have negative 3/7 equals X and other. If you wanted to figure out, you know, the side lengths or the length of each segment, respectively, you would take this and you would plug it back in, respectively, two x here or X here. And in this case, negatively seven is a totally viable answer. So I could have asked you any sort of thing I could. I could have said, you know, find me the length of C B. I could have said finding the length of a C. I could have said finding the length of a B, and it's a matter of taking finding a relationship. So if you had the one third part, you would say, Well, three of them are required for the whole. Or you might say, two thirds of the poll equals a C, depending what you're given. But two major things here a median is the line segment who's one point is on the verge of Vertex of a triangle, and the other is the midpoint of the opposite side. All of the medians intersect at the same point point of concurrency, which we call the central I'd and that will cut the median in a ratio of 1 to 2 in terms of the bottom part to the upper part. Or if you look at each part, respectively, to the entire median, you're gonna have a 1 to 3 ratio or 2 to 3 ratio, depending on which section you're talking about. So here we have explored what a median is, what the central it is, and then a few examples of some problems you might see regarding the medians and century