Hello Geometers! I am here to just show you a quick lesson on the Segment Addition Postulate, just to show you how it works and why we need it sometimes. The first thing I want you to know about the Segment Addition Postulate is what it actually says.
The first thing about it is that it relies on having two points and then another point that's between them. As a matter of fact, it's sometimes considered, or can be considered really, the definition of betweenness in geometry. that we can only say that R is between Q and S if it's on the same segment from Q to S somewhere in between them. Like if R was out here somewhere we wouldn't say we wouldn't call that between. So basically you have any two points somewhere on a segment or somewhere on a line and that makes a segment and then you have a point that is somewhere between them.
Now notice what I have done. It's important to me that when I draw a point that is supposed to be between two other points I deliberately did not draw it in the middle and here's why. If you draw it in the middle, sometimes you'll start thinking of it as the midpoint, but we're not being told that it's the midpoint. This isn't supposed to be the midpoint. It could be, but most of the time it won't be.
Most of the time it's just some random point in between two other points. So the segment addition postulate technically says if you have this situation where a point is between two other points, then the distance from here to here, which we say this way, plus the distance from here to here, which we say this way, is equal to the total distance. And remember when you write a segment name like that with nothing over the top, that means the distance between these two points or another way of saying that is the length of the segment QR.
Okay, so that's what the segment of addition postulates tells us. Alright, it's really not as, um, I guess formal or unintuitive as it seems. It really kinda makes sense.
Like if I said, okay, what if QR is 3 centimeters and RS is 8 centimeters? And then I asked you how long all of QS would be. Well, then you would probably sort of instinctively say, well, if this part's 3 and this part's 8, I'm going to put them together.
And the whole thing is 3 plus 8, which is 11. So that's really all the segment addition policy says. It's how we use that that makes it a little bit more than that. So ways we can use that. really we can use this to write an equation. And one way I like to summarize this is what this says is that one part plus the other part equals the whole thing.
Okay, we basically just have a part plus part equals whole situation. Part plus part equals the whole thing. All right, so let's look at that in a couple of different contexts, a couple of different ways we might see a problem. Alright, one way I might see a problem is I might see something that has, oh I don't know, a 10 here and an X here, and then maybe they'll tell me over here this.
Now remember, this means the length of segment AC is 17. So they told me this part had a length of 10, and this part has a length of X, and this part has 17. Anytime you have to solve, in order to solve, technically, and some of these you may be able to do in your head, but to solve something you need an equation. That's what we solve, is that we solve equations. So in order to be able to solve, you need to be able to write an equation. So my part plus part equals the whole thing is how I write my equation. So I'm going to go ahead and write that.
I wouldn't write it out every time necessarily, but I'm writing it for you this time. This is what's going on in my head. Part plus part equals whole. So AB plus BC equals all of AC.
So what I'm going to do is I'm going to plug in what I know. I know that AB is 10 plus BC is X, and AC has a length of 17. And then I have an equation that I can pretty easily solve in order to find out what X is. Okay, notice that one thing to remember, anytime you solve for a variable, make sure at the end of the problem that you know what you're actually being asked for. Are you being asked for X?
Or... Are you being asked for, for instance, maybe the length of the whole thing? Well, in this case, that's already given to us. But sometimes you're asked questions that actually mean you have to take that x and go back and plug it in.
So let's look at another example, a little bit less numerical. Let's say that I am told on this segment where n is between m and p that mn has a length of 2x plus 1. and I know that means length because the segment name isn't on the top, and that NP has a length of 5X, and that MP... has a length of 29. Okay, and I deliberately wrote them out to the side because they're not always going to be written on your segment for you, but I would probably think about putting them on there.
Like if this is what I'm writing on my paper, I might go ahead and move the mn down here and np, so this is 5x, and then I might even do this, like draw arrows to show that I'm talking about the whole thing and say this whole thing is 29. So part plus part equals the whole thing tells me the way I'm going to set this up. And again, if you want to, if it helps you to write out those words, then do and just substitute what you need to for each of those things. But part plus part equals the whole means 2x plus 1 plus 5x.
That's one part. There's the other part equals the whole. And there's my equation. So now I have the equation that I can solve to find x.
I'm going to combine my like terms, so this is 7x, and then I'm going to subtract 1 from both sides, and I get that x is equal to 4. So now what if I wasn't asked to solve for x? What if I was asked for maybe this? Maybe if it said this? Remember, if this means the length of NP, then this expression does not mean locate NP.
It means find the length of NP. So I'm being asked, what is the length of NP? Well, if X is 4, I'm going to take that 4 and go back and say, then 5 times X means 5 times 4. So the length of NP is 20. So always when you solve for X, go back and look and see if you need to plug in to get your final answer.
All right, last example we're going to do. And here it is. This one's a little bit more algebraic.
Okay, you notice that none of the lengths are given as just a number. So I might go ahead and write it on the segment. This is 3x plus 2. RS is 6x minus 1. And you don't have to write this one on there, but if you do, make sure you don't write it in the place where it looks like it's just this part or just this part.
You can do the arrow thing like I did in the last one, like that. Or you could use, like, a bracket. Not the prettiest bracket I've ever drawn, but there it is.
And remember now we're saying that our part, one part, plus the other part equals the whole. Another way of saying that is the whole is the sum of the parts. Okay, and that's actually a fairly common expression.
So this is something you could use, by the way. And I'm just taking part, part, plugging it in, and then the whole. is 12x minus 8. You could use this if you had three parts or four parts or five parts.
It's not just something if you only have two parts. Basically, it's just saying all your parts add up to the whole thing. So now I need to combine my like terms, and I get this. Now collect my variables on one side, collect my numbers or constants on the other side. And I get that x is 3. Okay?
Very often, again, you're wanting to go back and plug it in. So I'm going to go back and I'm going to put a 3 in each of these. And I'm going to find out that qr is 3 times 3 plus 2. I can write that out if I want to. 3 times 3 plus 2 is 11. rs is 6 times 3 minus 1. 6 times 3 minus 1 is...
18 minus 1 is 17. And QS is 12 times 3 minus 8. 12 times 3 is 36. 36 minus 8 is 28. Which, by the way, we can kind of check ourselves here by saying, does 11 plus 17 equal 28? And yes, it does. This part is 11, this part 17, and the whole thing's 28. Now, I deliberately did something when I set up this problem that you might be throwing you just a little bit.
Okay, this is on purpose. I deliberately set it up so that the bigger part was over here, because I want to remind you that it doesn't matter when you're drawing this, it really doesn't matter where R goes, as long as you don't put it in the middle and start thinking it's the midpoint. The reason it doesn't matter is because in geometry, we're not allowed to assume that something is bigger or smaller than something else just from the way it looks. We have to go with the information we have. So even though it looks like this side is smaller, The information we have tells me that this side is bigger, so it doesn't really matter what it looks like.
This is the bigger side. RS is 7 has a length of 17 and QR has a length of 11. So sometimes we kind of have to ignore what our eyes are telling us and make sure we're relying on the info. Hope that helped and I'll see you later.