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Dilations: center

Determining the center of dilation, given a figure and its image under a dilation.

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Video transcript

- [Instructor] We are told that triangle N' is the image of triangle N under a dilation. So this is N' in this red color, and then N is the original N is in this blue color. What is the center of dilation? And they give us some choices here, choice A, B, C, or D as the center of dilation. So pause this video and see if you can figure it out on your own. So there's a couple of ways to think about it. One way I like to just first think about, well what is the scale factor here? So in our original N, we have this side here, it has a length of two, and then once we dilated it by, and used that scale factor, the corresponding side has a length of four. So we went from two to four. So we can figure out our scale factor, scale factor is equal to two. Two times two is equal to four. Now what about our center of dilation? So one way to think about it is, pick two corresponding points. So let's say we were to pick this point and this point. So the image, the corresponding point on N', is going to be the scale factor as far away from our center of dilation as the original point. So in this example we know the scale factor is two, so this is going to be twice as far from our center of dilation as the corresponding point. Well you can immediately see, and it's going to be in the same direction, so actually if you just draw a line connecting these two, there's actually only one choice that sits on that line, and that is choice D right over here as being the center of dilation. And you can also verify that notice, this first point on the original triangle, its change in x is two and its change in y is three, two three, to go from from point D to point to that point. And then if you wanna go to point D to its image, well now you gotta go twice as far. Your change in x is four, and your change in y is six. You could use the Pythagorean Theorem to calculate this distance and then the longer distance, but what you see is, is that the corresponding point is now twice as far from your center of dilation. So there's a couple of ways to think about it. One, if you connect corresponding points, your center of dilation is going to be on a line that connects those two points. And that the image should be the scale factor as far away from the center of dilation, in this case it should be twice as far from the center of dilation as the point that it is the image of.