Mapping shapes

We're told that triangles. Let's see, we have triangle PQR and triangle ABC are congruent. The side length of each square on the grid is one unit, so each of these is one unit. Which of the following sequences of transformations maps triangle PQR onto triangle ABC?

So we have four different sequences of transformations. And so why don't you pause this video and figure out which of these actually does map triangle PQR, so this is PQR, onto ABC? It could be more than one of these, so pause this video and have a go at that.

All right, now let's do this together. So let's first think about sequence A, and I will do sequence A in this purple color. So remember, we're starting with triangle PQR. So first it says a rotation 90 degrees about the point R. So let's do that and then we'll do the rest of this sequence.

If we rotate this 90 degrees, one way to think about it is that a line like that is then going to be like that. So we're going to go like that, and R is going to stay where it is since you're rotating about it; but P is now going to be right over here. One way to think about it is to go from R to P; we went down one and three to the right. Now when you do the rotation, you're going to go to the right one and then up three. So P is going to be there, and you could see that that's the rotation. So that side will look like this.

So that is P, and then Q is going to go right over here. It's going to, once again, also do a 90-degree rotation about R, and so after you do the 90-degree rotation, triangle PQR is going to look like this. So that is Q. So we've done that first part, then a translation six units to the left and seven units up.

So each of these points are going to go six units to the left and seven up. So if we take point P, six to the left, one, two, three, four, five, six, seven units up, one, two, three, four, five, six, seven, it'll put it right over there. So that is point P. If we take point R, we take six units to the left, one, two, three, four, five, six, seven up, one, two, three, four, five, six, seven, it gets us right over there.

And then point Q, if we go six units to the left, one, two, three, four, five, six, seven up is one, two, three, four, five, six, seven, puts us right over there. So this looks like it worked. Sequence A is good; it maps PQR onto ABC. This last one isn't an R; this is a Q right over here, so that worked. Sequence A, now let's work on sequence B. I'll do some different color, a translation eight units to the left and three up.

So let's do that first. So if we take point Q, eight to the left and three up, one, two, three, four, five, six, seven, eight, three up, one, two, three. So this will be my red Q for now. Now if I do this point R, one, two, three, four, five, six, seven, eight, let me make sure I did that right: one, two, three, four, five, six, seven, eight, three up, one, two, three.

So my new R is going to be there, and then last but not least, point P, eight to the left, one, two, three, four, five, six, seven, eight, three up, one, two, three, goes right there. So just that translation will get us to this point; it'll get us to that point. So we're clearly not done mapping yet, but there's more transformation to be done. So it looks something like that. It says then a reflection over the horizontal line through point A.

So point A is right over here; the horizontal line is right like that. So if I were to reflect point A, wouldn't change. Point R right now is three below that horizontal line; point R will then be three above that horizontal line. So point R will then go right over there. Just from that, I can see that this sequence of transformations is not going to work; it's putting R in the wrong place, so I'm going to rule out sequence B.

Sequence C, let me do that with another color; I don't know, I will do it with this orange color. A reflection over the vertical line through point Q. Oh sorry, a reflection over the vertical line through point Q. So let me do that; the vertical line through point Q looks like this. I'm just going to draw that vertical line. So if you reflect it, Q is going to stay in place. R is one to the right of that, so now it's going to be one to the left once you do the reflection, and point P is four to the right, so now it's going to be four to the left: one, two, three, four.

So P is going to be there after the reflection, and so it's going to look something like this after that first transformation. Unless it's getting a little bit messy, but this is what you probably have to go through as well. So I'll go through it with you. All right, so we did that first part, the reflection, then a translation four to the left and seven units up.

So four to the left and seven up. So let me try that. So four to the left, one, two, three, four, seven up: one, two, three, four, five, six, seven. So it's putting Q right over here; I'm already suspicious of it because sequence A worked where we put P right over there. So I'm already suspicious of this, but let's keep trying. So four to the left and seven up: one, two, three, four, seven up: one, two, three, four, five, six, seven.

So R is going to the same place that sequence A put it, and then point P: one, two, three, four, one, two, three, four, five, six, seven. Actually, it worked; and it works because this is actually an isosceles triangle. So this one actually worked out; we were able to map PQR onto ABC with sequence C. So I like this one as well.

And then last but not least, let's try sequence D. I'll do that in black so that we can see it. So first we do a translation eight units to the left and three up, eight to the left and three up. So we start here, one, two, three, four, five, six, seven, eight, three up: one, two, three. So I'll put my black Q right over there.

So eight to the left: one, two, three, four, five, six, seven, eight, three up: one, two, three. I'll put my black R right over there. That's actually exactly what we did in sequence B the first time, so P is going to show up right over there. So after that translation, that first translation in sequence D, it gets us right over there.

Then it says a rotation negative 270 degrees about point A. So this is point A right over here, and negative 270 degrees, it's negative, so it's going to go clockwise. And let's see, 180 degrees; let's say if we were to take this line right over here, if we were to go 180 degrees, it would go to this line like that.

And then if you go to another 90 degrees, it actually does look like it would map onto that. So this is actually looking pretty good. If you were to take this line right over here, well then if you go negative 270 degrees, we'll map onto this right over here and then that point R will kind of go along for the ride, is one way to think about it, and so it'll go right over there as well. So I'm actually liking sequence D as well.

So all of these work except for sequence B.