 # In mathematics, a partition of a set "S" is a collection of non-empty subsets of "S" such that every element is contained in exactly one of the subsets. The partition (or disjoint-set data structure) problem is the problem of finding a partition of a given set "S" into a minimum number of non-empty subsets. The problem was first formulated in 1936 by Frank Rubin and is also known as the "Rubin partition problem".

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Hey, guys, we're gonna do some practice with partitioning segments. So let's suppose I have points A which set to Foreign Point B, which is 8. 10. And I want to divide a B into a ratio of 5 to 1. So we're looking. How are the segment? Looks, Let's say this is a Let's say this is B. If we're gonna find a ratio from 5 to 1, that means that point we're looking for will call C. We want this to be a five units with respect to this distance to be over you. One unit, so 51 That means we're dealing with a total of six units. We also want our what? We would usually have a 1 to 1 ratio with mid points. We're gonna pull that up. Okay, We want the point. We pulled towards B. Okay, which is 8 10 versus pulled towards A, which is 24 on Soto. Find the point that we're looking for. That's gonna partition this into a segment of five. Yeah, toe One ratio is I'm gonna first take. Okay, Hear me out five times eight plus one times two because I wanna I wanna wait. It So one of those six pieces. I want five of them to be towards the eight x value. And I'm a divide by six because there are a total of six chunks here. Okay, After calculating all this, we end up with 40 plus two. It's 42 divided by six. We did with seven. Likewise, I'm gonna do five times 10 for the white coordinates. Plus one times four again divided my six. That's gonna get us 54/6, which is nine. So the point that's gonna partition this particular line segment into a 5 to 1 ratio of 79 It should be very believable, because if I want this to be five minutes long compared to this, be one year long, I want the point to be closer to point B, closer to the 10.0.0.8. Come in 10. And that's why you wait the point that you're wanting to be closer to or farther away. Then okay. With the opposite. Basically ratio. You would think so. Let's try another example. Let's say I have the points. A which is it? Negative five four and the point B, which is at seven negative for again this is not drawn to scale. Bunny means Here's a Here's B this time I want the ratio to be 123 which means we're gonna be we're gonna find a point. See, it's gonna closer to a because I wanted to be one and three, which means obviously have a total of four parts altogether. So this time, when I go to find my ratio, I'm gonna do one time seven because I wanna be farther away from seven plus three times negative five that all over for Okay, that equals seven minus 15/4, which is negative. Eight or four, which is negative, too. Then for the Y values would do the same thing one times negative for plus three times four all over for which is going to result in a value to because you're gonna have negative for plus 12 gets you a positive +88 divided by for us to the point that will partition this segment into a one to three ratio will be the point negative to to. And it makes sense that negative 22 is closer to this point than it is this point in terms of your X and Y coordinates. Let's go ahead and do one more example. Okay, so let's say I want to do partition the statement that has endpoints a of negative nine negative nine. And B has the coordinates of five negative, too. And I want the ratio to be 3 to 4. So how do you want to write the segment? This is obviously not drawn to scale up intended every time. Here's a here's B. Since I wanted 3 to 4 ratio, we're not gonna be quite in the middle, so we're gonna be somewhere to the left of that. That's where I'll call my point C. This will have three units compared to this of length four units. So we wanna pull down towards a. So we're gonna write is we're gonna say for the X coordinate, we're gonna say, All right, three times five plus four times negative night, All that over seven because there's a total of seven unit pieces here this evaluated gets his negative three. Likewise, we're gonna have three times negative too. Plus four times nine again, all over seven. That evaluated gets us 30 70. I'm skipping. I've had this pre calculated 37th So the point that we're looking at is three comma 30 70. Now, after doing several examples, let me show you a shortcut of where and how to multiply. What do you see? How There's a three here on this site here in a foreign this side and noticed that we're we've been multiplying the three by the opposite point and the four by the opposite point. That's how you will set up all these weighted averages. So whatever the breakdown is, you're gonna take this component of the breakdown and multiply these values versus is taking this component of the other breakdown and multiplying it by the opposite point. That's how you can set up your particular multiplication in case you're curious about which goes to what? So hopefully this is helpful. I'll see you next time. Take it easy. Webster University
##### Top Geometry Educators ##### Lily A.

Johns Hopkins University  ##### Samuel H.

University of Nottingham