Composing transformations

0:00- [Instructor] We're told that Ray ST is the image of ray QR after the following sequence of transformations. I guess they didn't have to write ray twice at the beginning. A dilation by a scale factor of 3/2 about the point -7, 7 followed by 180 degree rotation about the origin.

0:19And so what we wanna do is figure out what ray ST looks like after we do these transformations. So pause this video and have a go at that before we do this together. All right, so let's do the first transformation. Maybe I'll do this in red. A dilation by a scale factor of about 3/2 about the point -7, 7.

0:38So let's see where -7, 7 is right over there. And the way that I think about the dilation, I'll think about the X and Y distance of each of these points from that point about which we are doing the dilation. And then I'll scale that distance down. So what do I mean?

0:54Well, to go from our point of dilation, I guess I could say, to this point Q, I'm going from seven to one. So I'm going down by six. And then I am going up in... Or I'm going to the right in the X direction. I'm going from -7 to -3. So I'm going up by four. So if I were to scale each of these by 3/2, I would instead go down four.

1:21'cause three, or sorry, 3/2. I was thinking 2/3. I make mistakes sometimes. 3/2 times -6 is -9. So we'll go down by nine. So that'll be one, two, three more, we'll go down by nine. And then 3/2 times four is six. And so then we'll go up by siX or to the right by six.

1:41So if we go from -7, if you increase by six, you get to -1. So we would go right over here. Let's call that point S. And then let's do the same thing right over here for point R that we did for point Q. Lemme do this in another color. So to go from our point of dilation to R, we're going down by two, so we're doing down by two.

2:05And then we are going to the right by one, two, three, four, five, six, seven, eight, nine. So plus nine. So now if we wanna multiply both of those by 3/2, instead of going down by two, you multiply that by 3/2. You're gonna go down by three, and then you're going to go to the right instead of by nine.

2:243/2 times nine is going to be... Let's see, did I do that right? 'Cause that seems like a shady number. Let's see. We increased one, two, three, four, five, six, seven, eight, nine, 10. Oh, there you go. This isn't a nine, this is plus 10. 'Cause I was like, this is gonna give us a fraction.

2:44We don't like that. So if we increased by 10 before, and we wanna go 3/2 of the 10, that's by 15. So we're gonna go all the way to the 10 and then five more. one, two, three, four, five. And so this is going to be our image. once we've mapped point R, it goes to point T.

3:03And so just once we do that, our ray looks like this. Our ray looks like this but we are not done. We now need to rotate 180 degrees about the origin, 180 degrees about the origin. So what's going to happen there? So I actually like to think about what happens to say the axes when you rotate 180 degrees about the origin?

3:35So for example, if I were to say this point right over here, this becomes, say this Y value, this is the .0, -4. This would get rotated all the way to 0, 4. And if I were to take this point right over here, this would get all the way rotated back to X equaling -4.

3:59And so what you could see is it essentially just takes the negative of both your X and Y coordinates. And so if we're at this point right over here, lemme just in another color, it's getting confusing. if we're at the point -1, -2, if we take the negative of both of those, we end up at the point positive one, positive two.

4:20So that's where our S goes to at the end. And actually, this was an intermediate area S, so this is our final S right over there. And then let's think about that point T. That point T, what it gets mapped to at 180 degrees is right now it's at the .8, 4. Well, it's gonna get mapped to -8, -4.

4:42- 8 and -4, which is right over there. And so this is actually the T that we care about when we think about Ray ST is the image after the following sequence of transformations. And so the ray starts at S and it goes through T and it keeps going. So it's going to look like this.

4:59It's going to look like this. Now I took a screenshot of this problem on Khan Academy, but if you're on Khan Academy, you can actually use the tools and move this around and it'll be a little bit neater. But there we did it. We created Ray ST based on performing the sequence of transformations.