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# Rotating shapes: center ≠ (0,0)

## Video transcript

triangle Sam s a.m. and this is this one Revere s a.m. is rotated 270 degrees about the point 4 comma negative 2 so this is 4 comma negative 2 right over here draw the image of this rotation using the interactive graph and we have this little interactive graph tool where we can draw points so if we want to put them in the in the trash we can put them there the direction of rotation by a positive angle is counter clockwise so this would be 270 degrees in the counterclockwise direction so to help us think about that I've copy and pasted this on my scratch pad and we can we can draw through it and the first thing that we might want to think about is if you rotate and I've talked about this in the previous video when we were rotating around the origin if you rotate something by last time we talked about a negative 270 degree rotation but if we're talking about a 270 degree rotation if you imagine a point right over here this would be 90 degrees 180 and then that is 270 degrees that is a 270 degree rotation we're going in a counterclockwise direction you see that that is equivalent that is equivalent to a 90 degrees to a 90 degrees clockwise rotation or a negative 90 degree rotation and 90 degree rotations are a little bit easier to think about so let's instead of thinking of this in terms of rotating 270 degrees in the positive direction in the counterclockwise direction let's think about let's think about this rotating this is 90 degrees in the clockwise direction they are going to be equivalent but how do we do that well like we've done in previous videos what I'm going to do is I'm going to take each point let's say point is in a color you can see Point M I'm having trouble switching go point M right over here and I'm going to construct a right triangle between point M and our center of rotation where the hypotenuse of the right triangle is going to be the line that connects these two so let me draw that so that's the hypotenuse and let me draw a right triangle like this so it's going to go it's going to be like it's going to look like that and then it's going to look like whoops that wasn't to press my command button before okay there you go right so I have this right triangle so if you imagine imagine rotating this right triangle this magenta one that I just constructed by negative 90 degrees by negative 90 degrees or 90 degrees in the clockwise direction well what's going to happen to each of these sides well this side right over here if I rotate it by negative 90 degrees it is going to end up it is going to end up it is going to end up pointing up and how long is it going to be well see this one is one two three four five six seven units long so it's still going to be seven years long one two three four five six seven it's going to put us right over here it is going to put us right over there and then this side let me make this clear this side that I'm highlighting in green which we see is four units long it forms a right angle it forms a right angle so if you were to rotate this entire triangle 90 degrees in the clockwise direction whether this is going to go for in this direction just like that just like that and then your new triangle the image of I guess what we just this this right triangle I've constructed if I rotate it by negative 90 degrees 90 degrees in the clockwise direction is going to look like this so after rotation so what did I just do I just took this I just took this triangle that I constructed and I rotated it negative 90 degrees to get to this triangle right over here and so the point M it's image after the 90 degree rotation is would be right over here and I could call that I'll call that M Prime what I'm essentially going to do I'm going to that for point M I'm going to do that point for point a and I'm going to do that for point s and I'm going to connect the three the M Prime the a prime of the s prime and then that will give me the image that will give me the vertices of my new triangle the image of triangle Sam after rotation so let's keep going let's do this let's do this now with point a with 0.8 and I can construct another right triangle and I could do it I could do it several I could do it several ways actually let me just do it this way let me let me draw it I could I could draw this the hypotenuse we know is going to go from here to from point A to our Center of origin and I could draw a right triangle that's up here like this or I could draw a right triangle that is down here like this maybe I'll focus on this one let me draw this right triangle so this right triangle right like this so if I draw this right triangle like this so this side let me do this in a color but I can see more clearly or that is let's see this side no you won't be able to see that all right I'm gonna stick with the purple so this side right over here that I am drawing in purple if you rotate it clockwise in the 90-degree direction or negative 90 degree rotation well it's going to instead of being straight horizontal is going to be straight vertical so it's going to be it's going to be it's going to overlap with what we had already drawn and how long is it well right now it goes from x equals negative 5 to x equals 4 so it's 9 long and you can count that 1 2 3 4 5 6 7 8 9 or you could say it's 2 longer than what we had already drawn so let's see if you if you start y equals negative 2 and you add 9 you're going to get 2 y equals 7 so this is going to go all the way like this and then and then let's see this side this side forms a right angle with it so if you so it's going to go like this it's going to be a right angle and how long is this it's 1 2 3 4 5 units long so now we're going to go 1 2 3 4 5 units in that direction and just like that we get to our a prime we get we get 2 we get to our a prime and we could draw the hypotenuse of of this triangle so let me see if I can draw it so press you can draw the straight line all right it's going to look like going to look like that so we know where the image of point M is after the rotation we know where the image of point a is after rotation and we see how these triangles you constructed we've just rotated them we've just rotated them 90 degrees in the clockwise direction now we just have to do this for point s and then we have the three points for our image so Point s once again we can construct a right triangle let's do that so hypotenuse will connect point s to our center of rotation and I could do this several ways let me think about the best way to do it well let me construct this right triangle so it looks like this and then it looks like this so this top side this top side right over here if I were to rotate it if I were to rotate it negative 90 degrees or 90 degrees in the clockwise direction where does that put us well once again it's going to put us straight up and down and this length is it goes from x equals negative 3 to x equals 4 it's so it's seven long so it gets us to this point right over here now this side let me be careful when I just say this let me do it in a different color this side right over here forms a right angle and so it's and if you see when you go from here you go down so if you're going to go left and you see this is three long so it's three it's three long this is a right angle right over here and then that gets us to our s prime that gets us to our s prime we can now draw the hypotenuse so notice this whole like kind of crazy complex that we have drawn of these different triangles I've just rotated the whole thing by negative 90 degrees that this triangle that we have that we were we had just made when you rotate it it maps to that one right over there but now we have in our image our s prime or a prime on our M prime and we can connect them so it will look like this so let's just connect them make the dots so that that and that and so you see you might you might have lost our original our original try our original triangle I'm just going to shade it in so you can see it clear although what we really care about is mapping the points mapping actually let me I don't want you to think that we're mapping that we're mapping everything in-between we're just mapping the points that define the triangle so our original triangle that we cared about was this thing that I'm going to draw really bold it was this thing and I'm gonna draw really really really bold and you rotate it by negative ninety degrees which is equivalent to positive 270 you could have rotated positive 270 you could have rotated it positive 270 to get to this other one right over here but it maps to this this is its image now we just have to go back on to the exercise now we have to go back on to the exercise and put it in so let's remember these points we have the point 8 comma 5 so we have the point 8 comma 5 in our image we have the point 9 comma 7 9 comma 7 and we have the point and we have the point 1 comma 5 so we have the point 1 comma 5 and now I just have to finish it by going to this point again and we're all done we've rotated it by 270 degrees which is equivalent to a negative 90 degree rotation we can check that we indeed got it right