College Algebra: Inclusive and Exclusive Events

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Okay, so we know that if I were to give you two points we could always find that midpoint and of course, that midpoint would be collinear with those other two points, just meaning that they'd all be on the same line. I'd have a point, another point, and that midpoint would be right in between along that invisible line.
But what if I just gave you three points and I just want to know are they collinear or not? How could I find out? Well, one way to do that is the following. If I think about them graphically for a second, so just suppose that I have three points, maybe one, two, three. If I compute the individual distances between those points, two at a time, so find this distance, find this distance, and find that distance. Well, if it turns out that no matter how you take those distances two at a time, when you add those two numbers up they always exceed the third one, the leftover one, that is, this length plus that length is bigger than that length, and this length plus that length is bigger than this length, and finally this length plus that length is bigger than this length, then we know they can't be collinear.
But consider the following. If the points actually were collinear and I compute this distance, and this distance, and then this big distance, what would I see? There would be a combination, this distance plus that distance would actually equal this distance. So in that case then I would know those three points are collinear. So if I want to find out if three points on the plane are collinear, all I have to do is find their pair wise distances from each other and see if there's two of them that can be added together to produce the third length. If there are two lengths that can be added together to produce the third length, I know they're collinear. Otherwise, I know I have a triangle. And, in fact, if I have a triangle I can ask myself, I wonder if it's a right triangle and I could find that out pretty easily, too, because I know the Pythagorean theorem if the right triangle precisely if one of the sides squared plus one of the other sides squared equals the third side squared. So you could actually see if the triangle is a right triangle or not by doing that little Pythagorean calculation.
So let's take a look at this in particular with some examples. So let me ask the following question, are these three points collinear? One of them is minus 1(5), one of them is 2 - 4, and one of them is 4 - 10. In fact, I'm going to plot them for you in my white box over there to my left and let's take a look and see. Now I'm not going to connect them with lines or anything, I'm just going to put down those three points so let's take a look and see if they look collinear or not. Well, what do you think? They look like they're on the same line, but let's actually see for sure and make sure that these things aren't just off by a pixel or something, see for sure, but it looks pretty good to me.
So what am I going to do? I want to compute the pair wise distances so let me take this point and this point and find the distance between those two points. So, in fact, let me give these names, maybe I should call this point A, I'll call this point B, and I'll call this point C. So let's find the distance between A and B. I'm just going to write this for shorthand for saying the distance between the points A and B. What does that equal? Well, that equals the square root of, now what do I do? I take remember the differences in the Xs, so I'll take 2 - -1 squared and add it to the differences of the Y's Minus 4, minus 5 squared, and what does that equal? Well, that equals the square root of, that's 3 squared, which is 9, plus, and this is going to be minus 9 squared is 81 and so this looks like this is the square root of 90 which I can simplify a little bit because that's just the square root of 9 x the square root of 10. The square root of 9 is 3, so I see 3 square root of 10. So, what that means, in fact, let me make a little note here and I'll keep bringing this back, that the distance between A and B, these two points, actually equals 3 square root of 10.
Okay, let's see if we can compute the distance between A and C. Let's try to do that right now. The distance between A and C would equal what? Well, it would be the square root of, I take X - X so I see 4 - -1 squared, plus the differences in the Y's minus 10 - 5 squared and what does that equal? Well, that equals the square root, well 4 - -1 = 4 + 1 = 5, 5 squared is 25, so I have 25 here and then I add minus 10 and minus 5 is minus 15, what's minus 15 squared? Well, let's see. Well, I could use the calculator, in fact, I'm just going to show you because we should do hi-tech stuff here, so you might actually know what 15 squared is, but let's just make sure you're right. Okay, 15 x 15 = 225. So if I add these things up, what do I get? I see the square root of 250, which you'll notice is the square root of 25 x the square root of 10. The square root of 25 is 5, so I see 5 x the square root of 10. When we come back, I can record that distance here. The distance between A and C is equal to 5 square root of 10.
This is actually a pretty involved problem because now I've got to compute one last distance. I've now got to compute the distance between B and C. So let's do that right now and see how that goes. Remember, all I'm after here is to see if two of them add up to the third one. Now let's compute the distance between B and C. That equals the square root of, well I take the changed distance and change an X, so it's 4 - 2 squared plus Y - 10 - -4 squared and what does that equal? That equals the square root of, well 4 - 2 = 2 squared is 4 and this is going to be - 10 plus 4, which is - 6 squared is 36 and so this equals the square root of 40, which you'll notice that the square root of 4 x the square root of 10 which equals 2 square root of 10. So the distance between B and C, that's the third distance, is 2 square root of 10.
Well, now the question is are there two of these numbers that add up to give the third one. And you see, yes there are. These two numbers, 2 square root of 10 plus 3 square root of 10 actually equals 5 square root of 10. So, in fact, I see that they do all line up and that picture that I see right there now is, in fact, accurate in the sense that they all live on the same line because I see that the distance between A and B plus the distance between B and C actually equals the distance between A and C because 3 square root of 10 plus 2 square root of 10 equals 5 square root of 10. So now I know for absolutely certain that, in fact, these three points are collinear. We'll try another example on the next lecture.
Further Topics in Algebra
Combinations and Probability
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