Example identifying the center of dilation

We are told the triangle N prime is the image of triangle N under a dilation. So this is N prime 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 is the center of dilation.

So pause this video and see if you can figure it out on your own. There are a couple of ways to think about it. One way I like to just first think about what is the scale factor here.

So in our original N, we have this side here; it has a length of two. Once we dilated it by and used that scale factor, that corresponding side has a length of four. So we went from two to four. We can figure out our scale factor: the scale factor is equal to two. Two times two is equal to four.

Now, what about our center of dilation? One way to think about it is to pick two corresponding points. Let's say we were to pick this point and this point. The image, the corresponding point on N prime, is going to be the scale factor as far away from our center of dilation as the original point.

In this example, we know the scale factor is 2, so this is going to be twice as far from our center of dilation as the corresponding point. Well, you can immediately see it’s going to be in the same direction. So actually, if you just draw a line connecting these two, there’s only one choice that sits on that line, and that is choice D right over here as being the center of dilation.

You can also verify that. Notice this first point on the original triangle: its change in x is 2, and its change in y is 3. To go from point D to point 2, that point, and then if you want to go from point D to its image, well now you’ve got to go twice as far. Your change in x is 4, and your change in y is 6.

You could use the Pythagorean theorem to calculate this distance and then the longer distance. But what you see is that the corresponding point is now twice as far from your center of dilation.

So there are 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.