Rotating line segments

0:00- [Instructor] We're told line segment JK is plotted on the coordinate plane below. Graph the image of line segment JK rotated 90 degrees clockwise around the origin. Of course it, that's at zero, zero. Pause this video and think at least how you would approach this.

0:18Ideally do it before I do it with you. Alright, now let's do this together. So we're going to be rotating about the origin, about this point right over here by 90 degrees. So it's sometimes a little bit hard to visualize it when there are these points out there.

0:35One way that I think about it is imagine, lemme just see another color. Imagine a line that just goes along the Y axis like this and it forms a 90 degree angle with this line, you're not always going to be able to do that, but at least for this one, I can imagine that.

0:51Now imagine if we were to take this, and we were to rotate it. Now it's easy to imagine rotating this red line, 90 degrees clockwise around the origin, that would just make it look like this. Let's see, it goes from zero to four. It has a length of four along the Y axis.

1:07Now 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, line JK with you, what would that look like?

1:27Well, 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.

1:43And 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 degrees clockwise around the origin.

2:03Let's do another example. So here, we have line segment ST, graph the image of line segment ST, rotated 90 degrees counterclockwise with the center at endpoint T, which is at one, negative two. So I often would phrase this counterclockwise around the endpoint, or maybe you would say the center of rotation at the endpoint.

2:27I would actually prefer if they said center of rotation at endpoint T, or if they said around the endpoint T. But either way, we are going to be rotating around this right over here, and we are going to be going 90 degrees counterclockwise. So for my brain, I still like to imagine lines that go straight to the side, straight horizontally or straight vertically.

2:50And 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 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.

3:20If I rotate it counterclockwise, so in the opposite direction that the hands of a clock turn that line or that side, I should say, would look like this. And similarly, if I were to, 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.

3:47So it would look like this. It would look like this. One, two, three, four. It would look like this. And so after the rotation, we would be dealing with a, I guess we could call it ST Prime or something like this. It would look like this. So T and T Prime are in the same place.

4:07So this is also T Prime at one, negative two. But now this over here is S prime. And of course all they wanted us to do is graph this line. I did these other horizontal and vertical lines to just help me visualize 'cause it's at least in my brain, easier to do the rotations with horizontal and vertical lines.

4:27But if you just focus on ST, 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 degree 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 line.