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Rigid transformations: preserved properties

Rigid transformations, like rotations and reflections, change a shape's position but keep its size and shape. These transformations preserve side lengths, angle measures, perimeter, and area. But they might not keep the same coordinates or relationships to lines outside the figure.

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Video transcript

- [Instructor] 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 gonna think about rotations and reflections in this video. And both of those are rigid transformations which means that the length between corresponding points do 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 for the sake of argument we rotate it clockwise a certain angle. So let's say we end up right over so we're gonna rotate that way. And let's say our center ends up right over here. So our new circle, the image after the rotation might look something like this. And 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. And so, the clear things that are preserved or maybe it's not so clear, we're gonna hope we 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 two. The radius here is also 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 and then the area is also going to be preserved. The area is just pi times the radius squared. So they have the same radius. They're gonna have all of these in common. And you could also that feels 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. And this case, I can set a perimeter. I can say circumference. 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. 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 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 gonna have a reflection in this situation. And we can 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 XYZ equal to X. So 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 now its image, C prime, won't change. And 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 that fairly quickly. I really just want you to see what the reflection looks like. The real appreciation here is think about, well, what happens with rigid transformations. So, it's gonna look something like this. The reflection. 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 use 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 gonna 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, the 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 as C because C happens to sit on our, the 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 AB ABD. Coordinates of A B D not preserved. After transformation, or their images don't have the same coordinates. After transformation. The one coordinate that happened to be preserved here is Cs coordinates. Because it was right on the line of reflection. And you can also look at other properties so how it might relate how different segments might relate to lines that would not be that were not being transformed. So, for example, right over here before transformation, CD is parallel to the Y axis. You see this 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 relationship to things outside of the things we're transformed that relationship might not. Those relationships may no longer be true after the transformation.