Rotating algebraically about a point

0:00- [Instructor] We're told to graph the image of triangle A, B, C after a rotation by 90 degrees counter clockwise about the point 3, -5. And then later when we're done they're gonna want us to describe the rotation in coordinate notation. So, pause this video, see if you could have a go at this before we do this together.

0:21All right, now let's work on this together. So, in other videos, other examples, we've done pure rotations about the origin. What's different here is that we're rotating about the point 3, -5. So, one way to handle this, and in my mind this is the easiest way to handle this, is to start by shifting the entire access, do a translation so that 3, -5 goes to the origin, then we do our rotation the way that we have seen in other videos, and then we shift everything back.

0:53We essentially undo that translation. And the whole time we should be keeping track of the rotation in coordinate notation. So, what do I mean by that? So, let's just start first by shifting. So, if we're rotating around 3, -5, so 3, -5 is right over here. If we wanted to move that point, and everything along with it to the origin, I want to shift to the left by three, so whatever X coordinate I have I wanna subtract three, and whatever Y coordinate I have I want to add five to move that to the origin.

1:33So the first transformation I wanna do is this translation, so I'll say X, Y. Whatever X is I wanna shift it to X-3, and whatever Y is I wanna shift it to Y+5. So, now, the point 3, -5. it gets mapped onto the origin. And let's think about where everything else gets mapped to.

1:57Well, point C right over here, we're shifting three to the left. One, two, three and up five. One, two, three, four, five. So it will go right over there. Point B, three to the left, and up one, two, three, four, five, it goes there. And then point A, three to the left, and one, two, three, four, five up.

2:19So now our triangle is right over here. And, now, the next thing we wanna do is rotate 90 degrees counterclockwise around this point, which is now the origin, which we have seen before. And I always like to remind myself, some of you already might know what that transformation looks like, but I always remind myself if I'm at, say the point, let's just say the point 4, or 0,4.

2:49So, let's say I'm at the point 0,4 right over here, I'm rotating 90 degrees counterclockwise, it's going to rotate over here. I'm going counterclockwise by 90 degrees, so it becomes, it goes to the point -4,0. So that reminds me, and you could try that with other points, but we've seen this before, I'm not proving it to you, but I'm just making sure I'm doing, that transformation in my mind is the one that makes sense here.

3:15Whatever my X and Y is, when I do a rotation 90 degrees counterclockwise, I then take the negative of the Y, and that becomes my X. So the Y was 4, I now have a -4 as my X, and whatever was the X now becomes my Y. And we could try this with other points if you like.

3:35Now, we aren't just doing this rotation only, we're doing this rotation on top of this translation that we've already done. So, if we wanna do that rotation, let us now write the coordinate notation, so X, Y is now going to be, after we do the translation and rotation, we're going to do this transformation with these coordinates here.

4:00So what was my Y, I'm going to take the negative of that, so negative Y plus five. And what was my X now becomes my Y, X-3. And mow let us draw that. So, if this point, let me do this in a red color. If we have this point here, which was originally our C, we could call this C prime, or our first C prime, whatever you wanna call it.

4:33If we now do this transformation, so my coordinate right over here, let me write it. This is -4, 5, it's now going to be -5, -5, -4. So, it's negative five, negative four. It goes right there. This point right over here is the point 1, 11. After the transformation, it's going to become the point

5:04- 11, 1. - 11, 1. It goes right over there. This is the point 4, 8. After transformation, it is going to be the point -8, 4.

5:21- 8, 4. So now our triangle has been mapped to this. This is after that rotation. Now about the origin. But now we wanna shift things, now we wanna shift things back, so we have one more translation. We essentially wanna do that previous translation. So, we now wanna do the last one where we add three to the X, and we subtract five from the Y to be able to shift our origin back to -3, or I should say 3, -5.

5:52And so our last one... Actually, let me just draw it. We can shift everything to the right by three and down five. To the right by three, and one, two, three, four, five. To the right by three. One, two, three. Down five. One, two, three, four, five. To the right by three.

6:15Down one, two, three, four, five. And so that's where our triangle will end up on the drawing part. I copy and pasted this from Khan Academy. You have a tool there to actually move things around with your mouse. But this is where we end up. And then that last transformation we just did, we shifted everything back, we undid that.

6:36So our last transformation when we combine all of these transformations, I am now going to take whatever my X is, which was that, and now I'm going to shift it to the right by three. I'm undoing this right over here. And then I'm gonna take whatever my Y value is, x-3, and I'm going to shift it down by five, minus five.

7:03And then we could simplify this. We could say X, Y, it gets transformed to, let's see, this is negative Y. Let me just write it out. Negative Y minus 5+3, and then X-3-five. This is X-8. And then let's see this becomes, let me just write it over here. This is negative Y-5+3 is minus two.

7:30And then this is X-8. And we are done.