Dividing line segments according to ratio

We're told point A is at negative one, four and point C is at four, negative six. Find the coordinates of point B on segment line segment AC such that the ratio of AB to AC is three to five. So, pause this video and see if you can figure that out.

All right, now let's work through this together. And to help us visualize, let's plot these points. So first let us plot point A, which is at negative one, four. So negative one, one, two, three, four. So that right over there is point A.

And then let's think about point C, which is at four, negative six. So one, two, three, four, comma negative six. Negative one, negative two, negative three, negative four, negative five, negative six, just like that.

And so the segment AC— I'll get my ruler tool out here— segment AC is going to look like that. The ratio between the distance of A to B and A to C is three to five, or another way to think about it is B is going to be three-fifths along the way from A to C.

Now, the way that I think about it is, in order to be three-fifths along the way from A to C, you have to be three-fifths along the way in the x-direction and three-fifths along the way in the y-direction.

So let's think about the x-direction first. We are going from x equals negative one to x equals four. To go from this point to that point, our change in x is one, two, three, four, five. And so if we wanna go three-fifths of that, we went a total of five—three-fifths of that is going just three.

So that is going to be B's x-coordinate. And then we can look on the y-coordinate side. To go from A to C, we are going from four to negative six. So we're going down by one, two, three, four, five, six, seven, eight, nine, ten. And so three-fifths of ten would be six.

So B's coordinate is going to be one, two, three, four, five, six down. So just like that, we were able to figure out the x and the y coordinates for point B, which would be right over here.

And you could look at this directly and say, "Look, B is going to have the coordinates—this looks like this is two, negative two," which we were able to do with the graph paper.

So another way you could think about it, even algebraically, is the coordinates of B. We could think about it as starting with the coordinates of A, so negative one, four, but we're going to move three-fifths along the way in each of these dimensions towards C.

So it's going to be plus three-fifths times how far we've gone in the x-direction. So in the x-direction, to go from A to C, we're going from negative one to four, and so that distance is four minus negative one, and this, of course, is going to be equal to five.

And then, on the y-dimension, this is going to be our A's y-coordinate plus three-fifths times the distance that we travel in the y-direction. And here, we're going from four to negative six, so we say negative six minus four; that is negative ten.

And so the coordinates of B are going to be negative one plus three-fifths times five is going to be plus three, and then four plus three-fifths times negative ten—well, three-fifths times negative ten is negative six, and so that gets us two, comma negative two.

And we are done, which is exactly what we got right over there.