Determining angle of rotation

We're told that triangle A'B'C' (so that's this red triangle over here) is the image of triangle ABC (so that's this blue triangle here) under rotation about the origin. So, we're rotating about the origin here. Determine the angle of rotation.

So, like always, pause this video, see if you can figure it out.

I'm just going to think about what how did each of these points have to be rotated to go from A to A' or B to B' or from C to C'. So, let's just start with A. This is where A starts. Remember, we're rotating about the origin—that's why I'm drawing this line from the origin to A.

Where does it get rotated to? Well, it gets rotated to right over here. So, the rotation is going in the counterclockwise direction, so it's going to have a positive angle.

So, we can rule out these two right over here. The key question is, is this 30 degrees or 60 degrees? There are a bunch of ways that you could think about it. One, 60 degrees would be two-thirds of a right angle, while 30 degrees would be one-third of a right angle. A right angle would look something like this, so this looks much more like two-thirds of a right angle.

So, I'll go with 60 degrees. Another way to think about it is that 60 degrees is one-third of 180 degrees, which this also looks like right over here. If you do that with any of the points, you would see a similar thing.

So, just looking at A to A' makes me feel good that this was a 60-degree rotation. Let's do another example.

So, we are told quadrilateral A'B'C'D' (in red here) is the image of quadrilateral ABCD (in blue here) under rotation about point Q. Determine the angle of rotation.

So, once again, pause this video and see if you can figure it out. Well, I'm going to tackle this the same way. I don't have a coordinate plane here, but it's the same notion. I can take some initial point and then look at its image and think about, well, how much did I have to rotate it?

I could do B to B', although this might be a little bit too close. So, I'm going from B to (let me do a new color here just because this color is too close to...) I'll use black.

So, we're going from B to B' right over here. We are going clockwise, so it's going to be a negative rotation. So, we can rule that and that out. And it looks like a right angle. This looks like a right angle, so I feel good about picking negative 90 degrees.

We could try another point and feel good that that also meets that—negative 90 degrees. Let's say D to D'. This is where D is initially, this is where D is, and this is where D' is.

Once again, we are moving clockwise, so it's a negative rotation. This looks like a right angle—definitely more like a right angle than a 60-degree angle.

So, this would be negative 90 degrees. Definitely feel good about that.