Rotating line segments

[00:00:00]We're told line segment JK is plotted on the coordinate plane below. Graph the image of line segment JK rotated rotated 90 degrees clockwise around the origin.

[00:00:14]Of course that's at 0 comma 0. Pause this video and think at least how you would approach this ideally do it before I do it with you. All right. Now let's do this together. So, we're going to be rotating about the origin, about this point right over here by 90°.

[00:00:30]So, it's sometimes a little bit hard to visualize it when there are these points out there. One way that I think about it is imagine, let me do this in another color. Imagine a line that just goes along the y-axis like this and it forms a 90° angle with this line. You're not always going to be able to do that, but at least for this one, I could imagine that. Now imagine if we were to take this and we were to rotate it. Now it's easy to imagine rotating this red line 90° clockwise around the origin. That would just make it look like this. Let's see. It goes from 0 to four. It has a length of four along the y- axis. Now it'll have a length of four along the x axis. So if you just rotated that, that's 90 degrees clockwise. We're going in the same direction that you would rotate a clock. And then if you did that, what would happen if you took this brown line JK with you? What would that look like? Let me do that. Maybe I'll do it in a similar color. So this you had you went two units to the left at a perpendicular direction and you had three units to the right. So now if you go here, those two units to the left are going to be two units up right there.

[00:01:43]And those three units to the right are going to be three units down. So it would look like this where this would be we could call that J prime and this could be K prime. There's other ways to do it but that would do the trick. We have now rotated JK 90° clockwise around the origin. Let's do another example.

[00:02:06]So here we have line segment ST. Graph the image of line segment ST rotated 90° counterclockwise with the center at end point t which is at 1 comma -2. So I often would phrase this counterclockwise around the end point or maybe you would say the center of rotation at the end point. I would actually prefer if they said center of rotation at endpoint t or if they said around the end point t. But either way, we are going to be rotating around this right over here. And we are going to be going 90° counterclockwise.

[00:02:43]So for my brain, I still like to imagine lines that go straight to the side, straight horizontally or straight vertically. And you could do that by constructing a little bit of a right triangle here. So if we did this, and I'll maybe I'll do this in different colors. So if I constructed a right triangle and now if I just imagined that this line st is a side of that right triangle the hypotenuse if you will and now at least in my brain it's much easier to visualize rotating it around t because if I did that this orange line is now let's see that has a length of four to the right. If I rotate it counterclockwise, so in the opposite direction that the hands of a clock turn, that that line or that side I should say would look like this. And similarly, if I were to if and that one's actually probably the easiest one to imagine, but if that line rotated like this, this other purple line also has a length of four and it's perpendicular. So it would look like this. It would look like this. One, two, three, four. It would look like this.

[00:03:54]And so the rot after the rotation, we would be dealing with a I guess we could call it st prime or something like this.

[00:04:02]It would look it would look like this.

[00:04:05]So t and t prime are in the same place.

[00:04:07]So this is also t prime at 1, -2. But now this over here is s prime. And of course all they wanted us to do is graph this line. And I did these other horizontal and vertical lines to just help me visualize because it's at least in my brain easier to do the rotations with horizontal and and and vertical lines. But if you just focus on ST, you can see you can feel pretty good about that as well. If we just look at this rotation over here, it's like, oh yeah, that looks like a 90° rotation. And this I could maybe even call this a right angle if you will, but I like doing it with the horizontal and vertical lines.