Mapping shapes example

So I'm here on the Khan Academy exercise for mapping shapes, and I'm asked to map the movable quadrilateral onto quadrilateral ABCD using rigid transformations.

Here in blue, I have the movable quadrilateral, and I want to map it onto this quadrilateral in gray. We have a series of tools here: rigid transformations of translation, rotation, and reflection on the Khan Academy tool. Of course, we can undo it.

The technique I'm going to use to do this is: I'm going to first use translation to make one of the corresponding points overlap with the point that corresponds to it. So, for example, it looks like this corresponds to point C right over here. I'm going to translate.

Once I click that, I can translate this around. So I'm going to translate right over here to point C. Now let's see: to make these two overlap, I really can't do any more translation. I've made one point overlap. Do I rotate or do I reflect? Well, if I eyeball it right over here, it looks like I am doing a rotation. Let me try to make use of rotation to make this segment right over here overlap with segment CD.

So let me do a rotation now. Let's see... Yep, this is looking good. There you go; we did the rotation, and we are done. Now, let's do another example.

So here, what do we need to do? All right, I'm going to do the same technique. This seems to correspond to point C, so I'm going to translate first. I translate first, and then there's something interesting going on right over here because I've actually been able to overlap points C and A by shifting it, by translating it, I should say.

It's not clear that if I were to rotate it, then I would lose the fact that the point that corresponds to A is now sitting on top of A, and the point that corresponds to C is now sitting on top of C. It feels like a reflection, and it looks like a line that would actually contain the points A and C. If we reflect over that line, then we'll be in good shape.

So let me see: reflection. Let me move the line... see how it—oops—that's not what I wanted to do. Let me move my line. So, that is, I think, a good line of reflection. Then let me actually try to reflect. There you go; I was able to reflect over that line.

My clue that I had to reflect over the line that contained A and C is that the points A and C and their images after the transformation were all sitting on top of each other. So that was a good clue: that on a reflection, if they're both sitting on the line of reflection, they wouldn't move, so to speak. And there you have it: so this was a translation and then a reflection.