Determining rotations

we're told the triangle a prime B prime C prime so that's this red triangle over here is the image of triangle ABC so that's this blue triangle here under rotation about the origin so we're rotating about the origin here determine the angle of rotation so like always pause this video see if you can figure it out so I'm just gonna think about what how did each of these points have to be rotated to go from a to a prime or be it a B prime or from C to C prime so let's just start with a so this is where a starts remember we're rotating about the origin that's why I'm drawing this line from the origin to a and where does it get rotated to well it gets rotated to right over here so the rotation is going in the counterclockwise direction so it's going to have a positive angle so we can rule out these two right over here and the key question is is is this 30 degrees or 60 degrees and there's a bunch of ways that you could think about it 160 degrees would be two-thirds of a right angle while 30 degrees would be one-third of a right angle a right angle would look something like this so this looks much more like two-thirds of a right angle so I'll go with 60 degrees another way to think about it is that 60 degrees is one-third of 180 degrees which this also looks like right over here and if you do that with any of the points you would see a similar thing so just looking at aid a prime makes me feel good that this was a 60 degree rotation let's do another example so we are told quadrilateral a prime B prime C prime D prime in red here is the image of quadrilateral ABCD in blue here under rotation about point Q determine the angle of rotation so once again pause this video and see if you can figure it out well I'm going to tackle this the same way I don't have a coordinate plane here but it's the same notion I can take some initial point and then look at its image and think about well how much did I have to rotate it I could do B to B prime although this might be a little bit too close so we're going from B to let me do it a new color here just cuz it's this color is too close to I'll use black so we're going from B to B prime right over here we are going clockwise so it's going to be a negative rotation so we can rule that and that out and it looks like a right angle this looks like a right angle so I feel good about picking negative 90 degrees we could try another point and feel good that that also meets that negative 90 degrees let's say D to D prime so this is where D is initially this is where D is and this is where D prime is and once again we are moving clockwise so it's a negative rotation and this looks like a right angle definitely more like a right angle than a 60 degree angle and so this would be negative 90 degrees definitely feel good about that