High school geometry
Dilating doesn't just shrink or grow a figure. The scale factor also changes the distance from each point to the center of dilation. Created by Sal Khan.
- [Instructor] We are told triangle A-prime, B-prime, C-prime is the image of triangle ABC under a dilation whose center is P and scale factor is 3/4. Which figure correctly shows triangle A-prime, B-prime, C-prime using the solid line. So pause this video and see if you can figure this out on your own. All right, now before I even look at the choices, I like to just think about, what would that dilation actually look like? So our center of dilation is P. And it's a scale factor of 3/4. So one way to think about it is, however far any point was from P before, is now going to be 3/4 as far, but along the same line. So I'm just going to estimate it. So if C was there, 3/2 would be this far. So 3/4 would be right about there. So C-prime should be about there. If we have this line connecting B and P like this, let's see, half of that is there. 3/4 is going to be there. So B-prime should be there. And then on this line, halfway is roughly there. I'm just eyeballing it. So 3/4 is there. So A-prime, A-prime, should be there. And so A-prime, B-prime, C-prime should look something like this. Which we can see is exactly what we see for choice C. So choice C, it looks correct. So I'm gonna just circle that, or select it just like that. But let's just make sure we understand why these other two choices were not correct. So choice A, it looks like it is a dilation with a 3/4 scale factor. Each of the dimensions, each of the sides of these triangles, of this triangle, looks like it's about 3/4 of what it originally was. But it doesn't look like the center of dilation is P. Here the center of dilation looks like it is probably the midpoint of segment AC. Because now it looks like everything is 3/4 of the distance it was to that point. So they have this other center of dilation in choice A. The center of dilation is not P, and that's why we can rule that one out. And then for choice B right over here, it looks like they just got the scale factor wrong. Actually they got the center of dilation and the scale factor wrong. It still looks like they are using this as a center of dilation. But this scale factor looks like it's much closer to 1/4 or 1/3, not 3/4. So that's why we can rule that one out as well. We like our choice, C.