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Үндсэн товъёог

Цаг: 0:00Нийт үргэлжлэх хугацаа:6:25

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 Prime in such a way that the perpendicular bisector of the segment PP prime is the line y is equal to three X minus two which of the following statements is true so is this describing a reflection or rotation or a translation so pause this video and see if you can work through it on your own alright so let me just try to visualize this so I'll just do a very quick so 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 3x minus two would look something like that and so what we're saying is where 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 well 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 Prime in such a way that the perpendicular bisector of PP prime is the line y equals 3x minus 2 so I need to connect so why the the line 3x minus 2y is equal to 3x minus 2 would be the perpendicular bisector of the segment between P and what 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 equals 3x minus 2 so P prime would have to be right over there so once again this is consistent with being a reflection P prime 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 Oh maps to itself every point V on a circle C centered at O maps to a new point W on a circle on circle C so that the counterclockwise angle from zero segment Oviedo W measures 137 degrees so is this a reflection rotation or translation pause this video and try to figure it out on your own alright so let's see we're talking about circle centered at O so let's see if let me just say so I have this point O it maps to itself on its transformation now every point V on circle C centred at O so let's see let's say this is circle C centred at point O so I'm going to try to draw a decent-looking circle here you get the 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 see so maybe it maps to a new point W on X let me keep reading W on circle C so that the counterclockwise angle from ovie to o W measures 137 degrees okay so we want to know the angle the angle from o v 2o w going counterclockwise is 137 degrees so this right over here is 137 degrees and so this would be the segment o w 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 theya is 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 actually 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 so they're talking about an again a certain mapping in the XY plane each circle o with radius R and centered at X wise mapped to a circle o prime with radius R and centered at X plus 11 and then Y minus 7 so once again pause this video what is this reflection rotation or translation all right so it 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 prime with the same radius so if this is the radius its 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 going to be 11 larger X plus 11 and our y coordinate is going to be 7 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 this 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 7 this is we shifted it down by 7 so this is clearly a translation so we would select that right over there and we're done