Dilations: scale factor

- [Instructor] We are told that pentagon A'B'C'D'E', which is in red right over here, is the image of pentagon ABCDE under a dilation. So that's ABCDE. What is the scale factor of the dilation? So they don't even tell us the center of the dilation, but in order to figure out the scale factor you just have to realize when you do a dilation, the distance between corresponding points will change according to the scale factor. So for example we could look at the distance between point A and point B right over here. What is our change in y? Our change in, or even what is our distance? Our change in y is our distance because we don't have a change in x. Well this is one, two, three, four, five, six. So this length right over here is equal to six. Now what about the corresponding side from A' to B'? Well this length right over here is equal to two, and so you can see we went from having a length of six to a length of two, so you would have to multiply by 1/3. So our scale factor right over here is 1/3. Now you might be saying okay that was pretty straightforward because we had a very clear, you could just see the distance between A and B. How would you do it if you didn't have a vertical or a horizontal line? Well one way to think about it is, the changes in y and the changes in x would scale accordingly. So if you looked at the distance between point A and point E, our change in y is negative three right over here, and our change in x is positive three right over here. And you can see over here between A' and E', our change in y is negative one, which is 1/3 of negative three, and our change in x is one, which is 1/3 of three. So once again you see our scale factor being 1/3. Let's do another example. So we are told that pentagon A'B'C'D'E' is the image, and they don't, they haven't drawn that here, is the image of pentagon ABCDE under a dilation with a scale factor of 5/2. So they're giving us our scale factor. What is the length of segment A'E'? So as I was mentioning while I read it, they didn't actually draw this one out. So how do we figure out the length of a segment? Well I encourage you to pause the video and try to think about it. Well they give us the scale factor, and so what it tells us, the scale factor is 5/2. That means that the corresponding lengths will change by a factor of 5/2. So to figure out the length of segment A'E', this is going to be, you could think of it as the image of segment AE. And so you can see that the length of AE is equal to two. And so the length of A'E' is going to be equal to AE which is two times the scale factor, times 5/2, this is our scale factor right over here. And of course what's two times 5/2? Well it is going to be equal to five, five of these units right over here. So in this case we didn't even have to draw A'B'C'D'E'. In fact they haven't even given us enough information. I could draw the scale of that, but I actually don't know where to put it because they didn't even give us our center of dilation. But we know that corresponding sides, or the lengths between corresponding points, are going to be scaled by the scale factor. Now with that in mind, let's do another example. So we are told that triangle A'B'C', which they depicted right over here, is the image of triangle ABC, which they did not depict, under a dilation with a scale factor of two. What is the length of segment AB? Once again they haven't drawn AB here, how do we figure it out? Well it's gonna be a similar way as the last example, but here they've given us the image and they didn't give us the original. So how do we do it? Well the key, and pause the video again and try to do it on your own. Well the key realization here is that if you take the length of segment AB and you were to multiply by the scale factor, so you multiply it by two, then you're going to get the length of segment A'B'. The image's length is equal to the scale factor times the corresponding length on our original triangle. So what is the length of A'B'? Well this is straightforward to figure out. It is one, two, three, four, five, six, seven, eight. So this right over here is eight, so we have two times the length of segment AB is equal to eight. And then you get the length of segment AB, just divide both sides by two, is equal to four. And we're done.