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# Rigid transformations: preserved properties

CCSS.Math:

## Video transcript

what we're going to do in this video is think about what properties of a shape are preserved or not preserved as they under grow under 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 say 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 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 and we're gonna 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 to the radius here is also is also 2 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 of the length of the radius is preserved and of course if the radius is preserved and then the area is also going to be preserve the area is just pi times the radius squared so if they have the same radius they're gonna have all of these in common and you can also net 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 poins for transforming a shape they'll preserve things like perimeter and area in this case I can set a perimeter I could 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 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 could even think about this without even doing the reflection ourselves but let's let's just do the reflection really fast so we're reflecting across the line X Y is equal to X so what it essentially does to the coordinates 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 I did that fairly quickly I'd 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 gonna look something like this the reflection the reflection looks thing like this so what's preserved and in general this is good to know for any rigid transformation what's preserved will 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 and 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 to see because c s-- happened 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 our let me say coordinates of a B or a BD coordinates of a B D not preserved after transformation are there images they don't have the same coordinates after trance formation 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 on how it might relate how different segments might relate to lines that were not be trans that we're not being transformed so for example right over here before transformation C D 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 cast 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 know that those relationships may no longer be true after the transformation