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

We're told that Eduardo rotated triangle ABC by 90 degrees clockwise about the origin to create triangle A'B'C'. So what Eduardo did is took this triangle right over here, rotated it 90° clockwise. So it's rotating at 90 degrees clockwise about the origin.

So the rotation was like that, which does look right. Write a rule that describes this transformation. So pause this video and have a go at that before we do this together.

All right, so just to remind ourselves what a rule for transformation looks like, it starts with each x or y coordinate in what we could call the pre-image before we have transformed. Then it says, well, what will that coordinate become in the image? And so there’s going to be some new point that’s going to have some x's or y's here and going to have some x's or y's there.

Now, before we do that, let's just think about what's happening to the different points in the pre-image when they get rotated onto that image. To do that, I will set up a little table here. So this is going to be the pre-image points—pre-image. So before we do the transformation, and then I'll try doing the similar color, this will be the points on the image in red.

So let's write these points down. So point A right over here, what are its coordinates before the rotation? A is at the coordinate (4, 5). So (4, 5). What is point B in the pre-image? It is (4, 3). (4, 3). Point C in the pre-image is (7, 5). (7, 5).

Now what happens to these once they get rotated? So the corresponding point on the image to A—so I'll call that A'—that is A' right over there, that is at the point (5, -4). (5, -4). B' is at (3, -4). And then C' is at (5, 7). (5, 7).

So can we see a pattern here? Well, we have a 4 for x here, and then we have a -4 for x there. So it looks like what was the x in A became the negative of that, became the y in A'. So let me actually just write that down. I’ll write that in purple to show how tentative I am.

So whatever is the x here, maybe the negative of that becomes the new y, and it looks like whatever was the old y becomes the new x—at least that’s what I’m seeing for point A. So maybe we take the old x in the pre-image, take the negative, and then make that the image’s y; and then what was the y, we make that into the new x.

Let’s see if that holds up. So for point B, if we took this 4, took the negative of it, and made that the new y, that’s exactly what happened there. And then if we take the old y and just put it in for x, that is exactly what happened there.

So this is holding up, and it also holds up with point C right over there. So there you have it. We have got our rule. This rotation—you can view as if you’re starting with an (x, y) point. You can get the new coordinate of where that point is, where its image is going to be after a rotation, by putting the old y in place for the x and then putting the negative of the old x in place for the y.