Identifying transformation described with other algebra and geometry concepts

We're told that a certain mapping in the x-y plane has the following two properties: each point on the line ( y = 3x - 2 ) maps to itself. Any point ( P ) not on the line maps to a new point ( P' ) in such a way that the perpendicular bisector of the segment ( PP' ) is the line ( y = 3x - 2 ).

Which of the following statements is true? So is this describing a reflection, a rotation, or a translation?

So pause this video and see if you can work through it on your own.

All right, so let me just try to visualize this. So I'll just do a very quick sketch. If that's my y-axis, and this right over here is my x-axis, ( 3x - 2 ) might look something like this. The line ( 3x - 2 ) would look something like that. What we're saying is what they're telling us is any point on this, after the transformation, maps to itself. Now that by itself is a pretty good clue that we're likely dealing with a reflection because remember, with a reflection you reflect over a line. But if a point sits on the line, well, it's just going to continue to sit on the line.

But let's just make sure that the second point is consistent with it being a reflection. So any point ( P ) not on the line—let's see, point ( P ) right over here—it maps to a new point ( P' ) in such a way that the perpendicular bisector of ( PP' ) is the line ( y = 3x - 2 ).

So I need to connect this. The line ( 3x - 2 ) would be the perpendicular bisector of the segment between ( P ) and, well, let's see, I’d have to draw a perpendicular line. I would have to have the same length on both sides of the line ( y = 3x - 2 ). So ( P' ) would have to be right over there.

So once again, this is consistent with being a reflection. ( P' ) is equidistant on the other side of the line as ( P ). So I definitely feel good that this is going to be a reflection right over here.

Let’s do another example. So here we are told—and I'll switch my colors up—a certain mapping of the plane has the following two properties: point ( O ) maps to itself. Every point ( V ) on a circle ( C ) centered at ( O ) maps to a new point ( W ) on circle ( C ) so that the counterclockwise angle from segment ( OV ) to ( OW ) measures ( 137 ) degrees.

So is this a reflection, rotation, or translation? Pause this video and try to figure it out on your own.

All right, so let’s see, we’re talking about a circle centered at ( O ). So let’s say this is circle ( C ) centered at point ( O ). I’m going to try to draw a decent-looking circle here. You get the idea—this is not the best hand-drawn circle ever.

All right, so every point, let’s just pick a point ( V ) here. So let’s say that that is the point ( V ) on the circle centered at ( O ), and it maps to a new point ( W ) on circle ( C ) so that the counterclockwise angle from ( OV ) to ( OW ) measures ( 137 ) degrees.

Okay, so we want to know the angle— the angle from ( OV ) to ( OW ) going counterclockwise is ( 137 ) degrees. So this right over here is ( 137 ) degrees, and so this would be the segment ( OW ). ( W ) would go right over there.

What this looks like is, well, if we're talking about angles and we're rotating something, this point corresponds to this point. Essentially, the point has been rotated by ( 137 ) degrees around point ( O ). So this right over here is clearly a rotation.

This is a rotation. Sometimes reading this language at first is a little bit daunting; it was a little bit daunting to me when I first read it. But when you actually just break it down and you try to visualize what's going on, you'll say, “Okay, well look, they’re just taking point ( V ) and they’re rotating it by ( 137 ) degrees around point ( O ),” and so this would be a rotation.

Let’s do one more example. So here we are told, they’re talking about, again, a certain mapping in the x-y plane: each circle ( O ) with radius ( r ) and centered at ( (x,y) ) is mapped to a circle ( O' ) with radius ( r ) and centered at ( (x + 11, y - 7) ).

So once again, pause this video. What is this: reflection, rotation, or translation?

All right, so you might be tempted, if they’re talking about circles like we did in the last example, to think maybe they’re talking about a rotation. But look, what they’re really saying is that if I have a circle—let's say I have a circle right over here centered right over here—this is ( (x,y) ).

It’s mapped to a new circle ( O' ) with the same radius, so if this is the radius, it’s mapped to a new circle with the same radius, but now it is centered at ( (x + 11) ). So our new x-coordinate is going to be 11 larger, ( x + 11 ), and our y-coordinate is going to be 7 less.

But we have the exact same radius. So what just happened to this circle? Well, we kept the radius the same and we just shifted our center to the right by 11, plus 11, and we shifted it down by 7. So this is clearly a translation.

So we would select that right over there, and we're done.