Rotations: description to algebraic rule

0:00- [Instructor] We're told that Julia rotated triangle ABC counterclockwise about the origin by 180 degrees to create triangle A Prime B Prime C Prime. Write a rule that describes this transformation. So why don't you pause this video and see if you can do that on your own before we do this together.

0:18All right, now let's do this together and to help us visualize this, let's graph some triangle ABC. I'm just going to make up the points because they didn't tell us what the points are, but then we can see what happens to those points when we rotate it counterclockwise about the origin by 180 degrees.

0:36So let's say that this is point A right over here at two, two. So that's A right over here, I'm gonna write these down. So A is at point two two. Let's say that I have B, that is at point three four. And actually let me make it, let me even make A different because I want them, it might be confusing if both the X and the Y coordinate are the same.

1:02So let me put A right over here. So A is at the point three comma two. Three comma two. Let's put a B at the point. Oh, I don't know. Let's put it right over here, one comma four. So it's at one comma four. And then let's put a C at the point right over here. Six comma five, C is at six comma five.

1:33And we could clearly draw a triangle out of these if we want. I could draw it just so we can visualize it as a triangle. And now let's imagine what rotating counterclockwise about the origin by 180 degrees would look like. So this is the origin, not a surprise.

1:49And we are going to rotate 180 degrees. So we are going to rotate. If we rotate that much, that would be 90 degrees and so we're gonna go all the way around like that. Or actually, let me just go point by point. That might make it a little bit easier. So if I start with point A, and if I were to go 180 degrees about the origin, I would be, I would be right over here.

2:19That is A prime. Now where is, and you can see that that's 180 degrees. We haven't changed our distance from the origin. Now what are the coordinates of A prime? A prime is at the coordinates negative three, comma negative two, negative three comma negative two.

2:39Some of you might already be seeing a pattern here. Now, let's think about where B prime is going to be. So if this is where B is, if this is where B is, if you rotate at 180 degrees counterclockwise about the origin, it's going to be right over here. Just going really essentially the same distance on the other side of the origin.

3:04So that's going to be B prime right over there, which is at the point negative one comma negative four, negative one comma negative four. And then last but not least, let's think about where C prime is going to be. So if you were to take this point and go all the way 180 degrees about the origin, we would see that right over, let's see, it would get right over here.

3:34If we were to just extend that line. So this is where C prime is. I know this is getting a little bit messy now, but C prime is at negative six, negative five, negative six, negative five. And we clearly see a pattern. Whatever our X's and Y's are in our pre-image for A, B, and C, in our image at the new point that we've rotated into, it's just the negative of each of those.

4:00So the rule I would say is for any X and Y, our rotation is going to result in and point in the image that's gonna be negative x, comma negative Y and we are done.