Determining mappings (incl. dilations)

Perform transformations on the movable quadrilateral until it matches quadrilateral NEUT, quadrilateral NEWT right over here. Are the two figures congruent? So we have our little tools here. But I'm actually going to do it on my scratch pad first to think about how we might want to transform it. So the first thing-- it looks like this point right over here would correspond to point E once we shift it over and then maybe dilate it down and then reflect it over. So let's try to do that. So the first thing, if we were to translate, so how much are we going to translate by? So we're going to move to the right by 1. So in the x direction, we're going to increase by 1. And in the y direction, we're going to go up by-- we're going to go from negative 4 to 1. So we're going to translate by 1, 5. So that's going to put our figure like this. So that point is going to be right over here. Everything is going to go in the x direction by 1, and then 1, 2, 3, 4, 5 in the y direction. So this line is going to look like that. So 1, and then 1, 2, 3, 4, 5 is going to go right over there. So that's going to go right over there. I'll try my best to draw it. And then 1 in the x direction, and then 1, 2, 3, 4, 5 in the y direction. So it's going to look something like this. Or it's going to look like this right after that transformation. And now, what we could do-- let's see, we could either reflect first, or we could try to scale it down first. So let's try to scale it down. Let's try to dilate it. So let's dilate it. And let's make our center. I don't want to change this point right over here. I like it sitting there because that seems to be the point that it corresponds to. Let's put the center there. So let's put the center at the point 5 comma 1. So the center is going to be right over there. And we essentially want this line, which has length 6, to be scaled down to have a length 3. Or essentially, we want this point right over here, which is 6 to the left of this point-- we want it to be 3 to the left of this point. So we want it to get half as far. So we want to scale by 1/2. So then that's going to get the figure-- this point is going to be over here. This point, which is in the y direction, 2 away, is now going to only be 1 away. Let me make it clear. In the x direction, it sits at negative 3 relative to 5. So it's 8 away. So now it's only going to be 4 away after a scale of 1/2. In the y direction, it's 2 away. So now, it's only going to be 1 away. So that point is going to go right over there. And then this point-- or actually, let me focus on this point right over here. We're not looking up there just yet. This point right over here, in the x direction, is 4 away and, in the y direction, is 4 away. So at a scale of 1/2 in the x direction, it's now going to be 2 away. And in the y direction, it's going to be 2 away. So it's going to look like this after the scaling. And now it looks pretty clear. We just have to do a reflection around this line. We just have to do a reflection around the line y equals 1. Reflect around the line y is equal to 1. So let's do all of that on this actual tool now. So first, I did the translation. I'm going to translate in the x direction by 1 and the y direction by 5. That got me right over there. And then I wanted to dilate it down. So let me scroll down a little bit. So dilate it about the point 5 comma 1. And then by a scale of 1/2. So that scaled me down. And now I'm going to reflect over y equals 1. So I'm going to reflect over the line. Well, the line y equals 1, it goes from the point 0-- I'm just picking two points on it-- so it goes from the point 0, 1 to 1, 1. And there we have it. We've gotten to that point. Now, we have to answer their question. Are the figures congruent? Well, no. The first figure was much bigger than this one. And the way that we got it to be able to squeeze into this one is that we dilated it down. These figures would have been congruent if all of my transformations were just translations, reflections, and rotations. But I had to dilate it. I had to scale the figure. Because I had to scale the figure, these figures are not congruent. No, the figures are not congruent.