Properties perserved after rigid transformations

What we're going to do in this video is think about what properties of a shape are preserved or not preserved as they undergo a transformation. In particular, we're going to think about rotations and reflections. In this video, both of those are rigid transformations, which means that the length between corresponding points does not change.

So, for example, let's say we take this circle A. It's centered at point A, and we were to rotate it around point P. Point P is the center of rotation, and just say for the sake of argument we rotate it clockwise a certain angle. So let's say we end up right over here. So we're going to rotate that way, and let's say our center ends up right over here.

Our new circle—the image after the rotation—might look something like this. I'm hand drawing it, so you got to forgive that it's not that well hand drawn of a circle, but the circle might look something like this. So, the clear things that are preserved—or maybe it's not so clear—and we're going to hopefully make them clear right now.

Things that are preserved under a rigid transformation like this rotation right over here—this is clearly a rotation. Things that are preserved? Well, you have things like the radius of the circle. The radius length, I could say to be more particular, the radius here is 2. The radius here is also two right over there. You have things like the perimeter—well, if the radius is preserved, the perimeter of a circle, which we call a circumference, well, that's just a function of the radius.

We're talking about two times pi times the radius, so the perimeter, of course, is going to be preserved. In fact, that follows from the fact that the length of the radius is preserved, and of course, if the radius is preserved, then the area is also going to be preserved. The area is just pi times the radius squared, so if they have the same radius, they're going to have all of these in common. You can also feel that intuitively right.

So what is not preserved? Not preserved—and this is in general true of rigid transformations—is that they will preserve the distance between corresponding points. If we're transforming a shape, they'll preserve things like perimeter and area—in this case, like instead of perimeter, I could say circumference. So they'll preserve things like that. They'll preserve angles. We don't have clear angles in this picture, but they'll preserve things like angles.

But what they won't preserve is the coordinates of corresponding points. They might sometimes, but not always. So, for example, the coordinate of the center here is for sure going to change. We go from the coordinate negative three comma zero to here. We went to the coordinate negative one comma two. So the coordinates are not preserved. Coordinates of the center!

Let's do another example with a non-circular shape, and we'll do a different type of transformation. In this situation, let us do a reflection. So we have a quadrilateral here—quadrilateral ABCD—and we want to think about what is preserved or not preserved as we do a reflection across the line L. So let me write that down. We're going to have a reflection in this situation.

We could even think about this without even doing the reflection ourselves, but let's just do the reflection really fast. So we're reflecting across the line y is equal to x. What it essentially does to the coordinates is it swaps the x and y coordinates, but you don't have to know that for the sake of this video.

So B prime would be right over here. A prime would be right over there. D prime would be right over here, and since C is right on the line L, its image C prime won't change. So our new, when we reflect over the line L—and you don't have to know for the sake of this video exactly how I did it, I really just want you to see what the reflection looks like.

The real appreciation here is to think about, well, what happens with rigid transformations. So it's going to look something like this. The reflection looks something like this. So what's preserved? And in general, this is good to know for any rigid transformation.

What's preserved? Well, side lengths—that's actually one way that we even used to define what a rigid transformation is—a transformation that preserves the lengths between corresponding points. Angle measures—angle measures! So, for example, this angle here, the angle A, is going to be the same as the angle A prime over here. Side lengths—the distance between A and B is going to be the same as the distance between A prime and B prime.

Perimeter—if you have the same side lengths and the same angles, then perimeter and area are also going to be preserved, just like we saw with the rotation example. These are rigid transformations. These are the types of things that are preserved.

Well, what is not preserved? Not preserved—and this just goes back to the example we just looked at—well, coordinates are not preserved. So as we see the image of A, A prime has different coordinates than A. B prime has different coordinates than B. C prime in this case happens to have the same coordinates of C because C happened to sit on our line that we're reflecting over.

But D prime definitely does not have the same coordinates as D. So most of—or let me say—coordinates of A, B, C, or A, B, D? Coordinates of A, B, D not preserved after transformation or their images; they don't have the same coordinates after transformation.

The one coordinate that happened to be preserved here is C's coordinates because it was right on the line of reflection. And you could also look at other properties of how it might relate, how different segments might relate to lines that were not being transformed.

So, for example, right over here, before transformation, CD is parallel to the y-axis. You see that right over here? But after the transformation, C prime, D prime—so this could be C prime, D prime—is no longer parallel to the y-axis. In fact, now it is parallel to the x-axis.

So when you have the relations to things outside of the things that were transformed, that relationship might not—no, those relationships may no longer be true after the transformation.