Dividing line segments
Find the point B on segment AC, such that the ratio of AB to BC is 3 to 1. And I encourage you to pause this video and try this on your own. So let's think about what they're asking. So if that's point C-- I'm just going to redraw this line segment just to conceptualize what they're asking for. And that's point A. They're asking us to find some point B that the distance between C and B, so that's this distance right over here. So if this distance is x, then the distance between B and A is going to be 3 times that. So this will be 3x. That the ratio of AB to BC is 3 to 1. So that would be the ratio-- let me write this down. It would be AB-- that looks like an HB-- it would be AB to BC is going to be equal to 3x to x, which is the same thing as 3 to 1, if we wanted to write it a slightly different way. So how can we think about it? You might be tempted to say, oh, well, you could use the distance formula to find the distance, which by itself isn't completely uncomplicated. And then this will be 1/4 of the way. Because if you think about it, this entire distance is going to be 4x. Let me draw that a little bit neater. This entire distance, if you have an x plus a 3x, is going to be 4x. So you'd say, well, this is 1 out of the 4 x's along the way. This is going to be 1/4 of the distance between the two points. Let me write that down. This is 1/4 of the way between C and B, going from C to A. B is going to be 1/4 of the way. So maybe you try to find the distance. And you say, well, what are all the points that are 1/4 of the way? But it has to be 1/4 of that distance away. But then it has to be on that line. But that makes it complicated, because this line is at an incline. It's not just horizontal. It's not just vertical. What we can do, however, is break this problem down into the vertical change between A and C, and the horizontal change between A and C. So for example, the horizontal change between A and C, A is at 9 right over here, and C is at negative 7. So this distance right over here is 9 minus negative 7, which is equal to 9 plus 7, which is equal to 16. And you see that here. 9 plus 7, this total distance is 16. That's the horizontal distance change going from A to C, or going from C to A. And the vertical change, and you could even just count that, that's going to be 4. C is at 1. A is at 5. Going from 1 to 5, you've changed vertically 4. So what we can say, going from C to B in each direction, in the vertical direction and the horizontal direction, we're going to go 1/4 of the way. So if we go 1/4 in the vertical direction, we're going to end up at y is equal to 2. So I'm just going, starting at C, 1/4 of the way. 1/4 of 4 is 1. So I've just moved up 1. So our y is going to be equal to 2. And if we go 1/4 in the horizontal direction, 1/4 of 16 is 4. So we go 1, 2, 3, 4. So we end up right over here. Our x is negative 3. So we end up at that point right over there. We end up at this point. This is the point negative 3 comma 2. And if you were really careful with your drawing, you could have actually just drawn-- well, actually you don't have to be that careful, since this is graph paper. You actually could have just said, hey, we're going to go 1/4 this way. Where does that intersect the line? Hey, it intersects the line right over there. Or you could have said, we're going to go 1/4 this way. Where does that intersect the line? And that would have let you figure it out either way. So this point right over here is B. It is 1/4 of the way between C and A. Or another way of thinking about the distance between C and B, which we haven't even figured out. We could do that using the distance formula or the Pythagorean theorem, which it really is. This distance, the distance CB, is 1/3 the distance BA. The ratio of AB to BC is 3 to 1.