Rotations: description to algebraic rule | Transformational geometry | Grade 8 (TX) | Khan Academy

We're told that Julia rotated triangle ABC counterclockwise about the origin by 180° to create triangle A'B'C'. 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?

All 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 we can see what happens to those points when we rotate it counterclockwise about the origin by 180°.

So let's say that this is point A right over here at (2,2). So that's A right over there. I'm going to write these down, so A is at (2,2). Let's say that I have B that is at point (3,4). Actually, let me make it a little different because I want them; it might be confusing if both the x and the y-coordinate are the same. So let me put A right over here, so A is at the point (3,2).

Let's put B at the point, oh, I don't know, let's put it right over here at (1,4). So it's at (1,4), and then let's put C at the point right over here at (6,5). C is at (6,5). 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.

Now, let's imagine what rotating counterclockwise about the origin by 180° would look like. So this is the origin, not a surprise, and we are going to rotate 180°. So, we are going to rotate. If we rotate that much, that would be 90°, and so we're going to 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 right over here. That is A'. Now where is— and you can see that that's 180°. We haven't changed our distance from the origin.

Now, what are the coordinates of A'? A' is at the coordinates (-3,-2). Some of you might already be seeing a pattern here. Now let's think about where B' is going to be. So if this is where B is, if you rotate at 180 degrees counterclockwise about the origin, it's going to be right over here.

It's going really essentially the same distance on the other side of the origin, so that's going to be B' right over there, which is at the point (-1,-4). And then last but not least, let's think about where C' is going to be. So if you were to take this point and go all the way 180° about the origin, we would see that right over here.

If we were to just extend that line, so this is where C' is. I know this is getting a little bit messy now, but C' is at (-6,-5), 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.

So the rule I would say is for any (x, y), our rotation is going to result in a point in the image that's going to be (-x, -y), and we are done.