Dilating in 3D | Solid geometry | High school geometry | Khan Academy

Let's say I have some type of a surface. Let's say that this right over here is the top of your desk, and I were to draw a triangle on that surface. So maybe the triangle looks like this, something like this. It doesn't have to be a right triangle, and so I'm not implying that this is necessarily a right triangle, although it looks a little bit like one. Let's call it triangle ABC.

Now, what I'm going to do is something interesting. I'm going to take a fourth point P that's not on the surface of this desk. It's going to be right above point B, so let me just take that point, go straight up, and I'm going to get to point P right over here. Now, what I can do is construct a pyramid using point P as the peak of that pyramid.

Now, what we're going to start thinking about is what happens if I take cross sections of this pyramid. In this case, the length of segment PB is the height of this pyramid. Now, if we were to go halfway along that height, and if we were to take a cross section of this pyramid that is parallel to the surface of our original desk, what would that look like?

Well, it would look something, it would look something like this. Now you might be noticing something really interesting. If you were to translate that blue triangle straight down onto the surface of the table, it would look like this. When you see it that way, it looks like it is a dilation of our original triangle centered at point B, and in fact, it is a dilation centered at point B with a scale factor of 0.5.

You can see it right over here. This length right over here, what BC was dilated down to, is half the length of the original BC. This is half the length of the original AB, and then this is half the length of the original AC. But you could do it at other heights along this pyramid. What if we were to go 0.75 of the way between P and B?

So if you were to go right over here, so it's closer to our original triangle, closer to our surface, then the cross section would look like this. Now, if we were to translate that down onto our original surface, what would that look like? Well, it would look like this. It would look like a dilation of our original triangle centered at point B, but this time with a scale factor of 0.75.

And then what if you were to go only a quarter of the way between point P and point B? Well, then you would see something like this. A quarter of the way, if you take the cross section parallel to our original surface, it would look like this. If you were to translate that straight down onto our table, it would look something like this, and it looks like a dilation centered at point B with a scale factor of 0.25.

The reason why all of these dilations look like dilation centered at point B is because point P is directly above point B. But this is a way to conceptualize dilations or see the relationship between cross sections of a three-dimensional shape, in this case, like a pyramid, and how those cross sections relate to the base of the pyramid.

Now let me ask you an interesting question: what if I were to try to take a cross section right at point P? Well, then I would just get a point; I would not get an actual triangle. But you could view that as a dilation with a scale factor of zero. And what if I were to take a cross section at the base? Well, then that would be my original triangle, triangle ABC, and then you can view that as a dilation with a scale factor of one because you've gone all the way down to the base.

So hopefully, this connects some dots for you between cross sections of a three-dimensional shape that is parallel to the base and notions of dilation.