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Properties & definitions of transformations

Videotranskripsjon

- [Instructor] We're told that a certain mapping in the xy-plane has the following two properties. Each point on the line y is equal to three x minus two 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 is equal to three x minus two. 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, and I'll just do a very quick, so if that's my y-axis, and that this right over here is my x-axis. Three x minus two might look something like this. The line three x minus two would look something like that. And so what we're saying is, or 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 gonna 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, so 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 equals three x minus two. So I need to connect, so the line three x minus two, y is equal to three x minus two, would be the perpendicular bisector of the segment between P and what? Well let's see I'd have to draw a perpendicular line. It would have to have the same length on both sides of the line y equals three x minus two. 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 in the plane has the following two properties. Point O maps to itself. Every point V on a circle C centered at O, all right, 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 circle centered at O. So let's see, let me just say, so I have this point O. It maps to itself on its transformation. Now every point V on circle C centered at O. So let's see, let's say this is circle C centered at point O, so I'm gonna 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 a circle centered at O maps to a new point W on the circle C. So maybe it maps to a new point W on, actually let me keep reading, W on circle C so that the counterclockwise angle from OV to OW measures 137 degrees. Okay so we wanna know 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. And so what this looks like is well if we're talking about angles and we are rotating something, this point corresponds to this point, it's 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 (laughing) read it. But when you actually just break it down and you actually try to visualize what's going on, you'll say well okay 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, so they're talking about, again a certain mapping in the xy-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 plus 11 and then y minus seven. 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, maybe they're talking about a rotation. But look, what they're really saying is, is that if I have a circle, let's say I have a circle right over here, centered right over here. After it, so this is x comma y, centered at x comma 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, now it is centered at x plus 11, so our new x-coordinate is gonna be 11 larger, x plus 11, and our y-coordinate is gonna be seven less. But we have the exact same radius. We have the exact same radius. So our circle would still, so we have the exact same radius right over here. So what just happened to the circle? Well we kept the radius the same, and we just shifted, we just shifted our center to the right by 11, plus 11, and we shifted it down by seven. We shifted it down by seven. So this is clearly a translation. So we would select that right over there. And we're done.