Performing sequences of transformations

Starting with the figure shown, perform the following transformations-- reflection over the line y equals x minus 2, translation by 3 in the x direction. So we're going to shift to the right by 3 and then shift down by 9. Is a transformed figure congruent to the original figure? And actually, we could answer this question before we even do the transformation, because this transformation only involves reflections and translations. As long as we're only reflecting, translating, and rotating, we're going to get to a congruent figure. The thing that will make it not necessarily congruent is when we dilate it. So dilating it is going to actually scale it up or down. The figure will still be similar, but it won't necessarily be congruent anymore. So I could already say yes, the figures are congruent. But let's actually perform the transformation that they want us to. So a reflection over the line y equals x minus 2. And so let's click on translate, and this isn't popping up-- this isn't allowing us just to do it by hand, or do it with our mouse. They want us to think about how much-- actually we don't want to translate first, we want to reflect first. So once again, the reflection line tool doesn't show up. Instead, for this exercise, they want us to specify two points on the line that we want to reflect around-- because two points define a line. So what are two points that are on the line y equals x minus 2? Well when x is equal to 0, y is negative 2. And then another point on it that might jump out at you-- when x is 2, y is 0. So we just need any two different points on that line. And then notice, y equals x minus 2 is a line that looks like this. I'm just going to kind of trace it. I don't have my drawing tool out right now, so it's a line that looks something like that. Actually, let me do it, just because I think it'll be interesting to look at. Let me copy and paste this right over here. And then let me put it onto my actual scratch pad. So that's from a previous problem. So let me clear all of this business out. So let me paste it right over here. And notice, we just reflected around the line y equals x minus 2. y equals x minus 2. Let's see, it has its y-intercept right over there, it has a slope of 1. So this is the line that we just reflected around. And it looks pretty clear that that is what happened. So just like that, I know my line isn't as straight as it could be, but you get the general idea. That is the line y is equal to x minus 2. Notice each of these points-- you drop a perpendicular from that point, whatever distance that is, go the same distance on the other side, and we have reflected over. Drop a perpendicular, same distance on the other side. And so that is our reflection about the line y equals x minus 2. But we are not done yet. We now have to perform the translation by 3, negative 9. So we're going to translate. So when we do it by 3, it's going to shift this to the right by 3. And then negative 9, which is going to shift it down by 9. And we see that it did. And we already said, these are definitely congruent. We haven't scaled this up or down, we've just translated it, rotated it-- actually, we didn't even rotate it. We've just translated it, and reflected-- or reflected it, and translated it. But let's check our answer, just to feel good about ourselves. And we got it right.