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Current time:0:00Total duration:8:12

Video transcript

we're told the triangle P I n is rotated negative 270 degrees about the origin so this is the triangle P I n we're going to rotate it negative 270 degrees about the origin draw the image of this rotation using the interactive graph the direction of rotation by a positive angle is counterclockwise so positives counterclockwise this is the standard convention and well this is negative so a negative degree would be clockwise and we want to use this tool here and this tool I can I can put points in or I could delete points I can draw a point by clicking on it so what we want to do is think about well look if we rotate the points of this triangle around the origin by negative 270 degrees where is that going to put these points and to help us think about that I've copied and pasted this on our scratch pad so actually let me let me go over here so I can actually draw on it so let's just first think about what a negative 270 degree rotation actually is so if I were to start if I were to let me draw some coordinate axes here so that's the x-axis and that's a y-axis if you were to start right over here and you were to rotate around the origin by negative ear by negative 270 degrees what would that be well let's see this would be rotating negative 90 this would be rotating negative another negative 90 which would together be negative 180 and then this would be another negative 90 which would give you in total negative 270 degrees that's negative 270 degrees now notice that would get that point here which we could have also gotten there by just rotating it by positive 90 degrees we could have just said that this is equivalent to a positive 90 degree rotation so if they want us to rotate the points here around the origin by negative 270 degrees that's equivalent to just rotating all of the points and I'll just focus on the vertices because those are the easiest ones to to think about to visualize I can just vote rotate each of those around the origin by positive 90 degrees but how do we do that and to do that what I am going to do to do that what going to do is I'm going to draw a series of right triangles so let's first focus on actually let's first focus on point I right over here and if I were to let me draw a right triangle and I could draw it several ways but let me draw it like this so it's a right triangle where the line between the origin and I is its hypotenuse so let me see if I can it's I could probably draw I could use a line tool for that me so that's the hypotenuse of the line now if I'm going to rotate I 90 degrees about about the origin that's equivalent to rotating this right triangle 90 degrees so what's going to happen there well this side right over here if I rotate if I rotate this 90 degrees where is that going to go well instead of going 7 along the x axis it's going to go 7 along the y axis so it's going to it's going to if you rotate it positive 90 degrees that side is going to look like this so that's rotating it 90 degrees just like that now what about this side over here what about this side let me do this in a different color what about what about this side right over here well this side over here notice we've gone down from the origin we've gone down 7 but if you were to rotate it up notice these this forms a right angle between this magenta side of this blue side so you're going to form a right angle again and so from this point you're going to go straight as a right angle and instead of going down 7 you're going to go to the right 7 so you're going to go to the right 7 just like this and so you're the point I or the corresponding point in the image after the rotation is going to be right over here so that green line let me draw the hypotenuse now it's going to look it's going to look like whoops I wanted to do that in a different color I'm going to do the green of trouble changing colors all right so there you go it's going to look it's going to look like that so your new point I if you if you rotate this triangle this right triangle 90 degrees your new point I maybe I shouldn't say I'll call it i prime which is what is the image of this point after I've done the 90 degree rotation is going to be right over here and now we can do that for each of the points we can do that for n here let's draw a right triangle let's draw a right triangle and I could do it a bunch of different ways I could draw it I could draw a right triangle like this let me draw it like this so that's one side of my right triangle so let me draw the hypotenuse first so I have the PI potenuse connects the origin to my point just like that and then I could I could either draw it up here or I could draw it down here like this I could draw it like this so if you rotated this 90 degrees if you rotate this 90 degrees this side which is 7 units long and we're going seven below the origin it's not going to be seven to the right right if I rotate it by 90 degrees it's going to be right over here it's going to be here and this side which has length this let me switch colors this side which has length two it forms a right angle so we're going to form a right angle and have length two right over here and so you're the image of point n is going to be let me get the right color it is going to be just like that it is going to be the image of point n is going to be right here I'll call that n prime so we know where the new where the image of I is the image of n is so now we just have to think about where does the image of piece it and once again we can do our little right triangle idea so let's draw a right triangle so just like that I can draw that side and I can do this side right over here so if I were to rotate if I were to focus on this let me do this in a color I haven't so if I were to focus on already use that color if I were to focus on this right over here and if I were to rotate it by 90 degrees instead of going to to the right it's going to go to straight up if I rotate this by 90 degrees it's going to be just like this now this side on let me let me pick another color this side right over here forms a right angle and it has a length of three so we're going to form a right angle here and have a length of three and just like that we know where the image of P is going to be it is going to be it is going to be right over here so this is P Prime and I know it's kind of confusing but now we just we don't have to connect the P Prime the I Prime and the M Prime to figure out what the image of my triangle is after rotation so let me do that so if I connect these two I get that if I connect these two I get that and if I connect these two I connect those two I have that and there you have it I have the image and now I just have to input it on the actual problem so let's see point negative 3 comma 2 let me get it out so negative 3 comma 2 is there I have 7 comma 2 so let me put that in there 7 comma 2 and then I have 7 comma 7 is there so let me get this out so 7 comma 7 and then 2 I'll draw here again to connect the lines and I'm done I've rotated it through an angle of 90 degrees or negative 270 degrees which is what they originally they originally asked me for and I can check to make sure I got the right answer