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Dilating lines

Sal shows how we can use dilations to map a line into another, parallel, line. Created by Sal Khan.

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Video transcript

Define a dilation that maps line a onto line b by choosing a center and a scale factor. So let me get my scratch out to think about this a little bit. So let's, just for fun, let's imagine that we pick a point that sits on line a as our center of dilation. And for simplicity, let's pick the origin. So let's imagine that our center of dilation is at 0, 0. And let's just pick an arbitrary scale factor. Let's say our scale factor is 2. And let's think about what happens. This means that each of the points on point a are going to get twice as far from our center of dilation after the dilation than they were before the dilation. Twice as far. So for example, this point right over here, it's 3 away in the x direction before dilation. So it's going to be 6 away in the x direction after dilation. Likewise, it's 3 away in the y direction before dilation. So it's going to be twice as far in the y direction after dilation. So it's going to go from the point (3,3) to the point (6,6). Notice, it got twice as far away from the center of dilation. It essentially got pushed out along the line. Same thing for this point. This point is 3 less in the x direction than our center, and 3 less in the y direction than our center. So after a scaling, after dilation centered at the origin, with a scale factor of 2, it's going to be twice as far away. So it's going to be 6 less in the x direction, and 6 less in the y direction. Once again, just pushed out. So notice when we picked a center of dilation that sits on the line, it really doesn't matter what scale factor we pick, we'll just essentially be stretching and shrinking the points along the same line. And since this line just keeps going on and on and on forever in both directions, it's really just going to map onto itself. If this was a line segment, we would be mapping it to a different line segment that would have the same slope, but its length would change. But a line has an infinite length. So as you stretch and squeeze it, it still has an infinite length, and it's just mapping it onto itself. So this dilation right over here, a, is going to map onto itself. It's just going to map onto a. So that's not going to suit our purposes. But what can suit our purpose is if we pick another point that's not on a, and not on b. So let's say we pick this point right over here. And I like this point, and so let me pick that as my center. I'll tell you why I like it in a second. That is the point 3, 2. The reason why I like that, is we can see that this point on point a right now is 1 away, and if I want to map it to the corresponding point on point b, well that's going to be 3 away from our center of dilation. So I could have a scale factor of 3. If I have a scale factor of 3, it's going to go 3 times as far away. And that'll work for other points as well. For example, this point right over here. It's 1 to the left of our center of dilation. If we have a scale factor of 3, it's going to be 3 to the left. So it's going to map. If we think of it that way, you could think of it as going right over there. So I'm going to go with center at 3, 2, scale factor of 3. And you could have picked many, many other-- in fact an infinite number of other centers, and then you would have had to calculate the appropriate scale factor for each of those. But I like this one because it fit nicely on the grid, and I was able to figure out how much to scale it. So I picked my center, so I want to dilate, I want to pick my center at 3, 2. And I want to scale it by 3. So let's check our answer. We got it right.