Notes 7.3 Rotations

[00:00:00]hey guys today we're going to talk about note 7.3 which is on rotations so starting us off here a rotation or turn is a transformation about a fixed point called the center of rotation each point on the original figure and its image are the same distance from the center the rotation can be clockwise which i have drawn there where you go around like you would a clock or counterclockwise which would be um anti-clock you'd be going backwards on a clock um then i have two other things i want to show you guys here on the left what i have here this is called a 90 degree clockwise rotation because we're turning 90 degrees like one quarter turn one quarter of a circle um around our center and then i also have a 90 degree clockwise rotation would be the same as a 270 degree counterclockwise rotation but you see in either in either method we end up at the same point here so i can go 90 degrees clockwise so i can go 270 degrees counterclockwise now it's important to remember that this property is kind of in play because we know a circle is 360 degrees so to get around the full circle we'd be traveling 360 degrees 90 degrees in one direction plus 270 in the other direction makes the total of 360 there all right so let's go over some rules for you guys to do rotations on your own so down here we have different types of rotations that you're going to see we have the 90 degrees counterclockwise a 90 degree clockwise and a 180 degree rotation i don't have the counter clockwise or clockwise with that because it doesn't matter you're going to end up in the same spot so it's 180 degrees so first of all 90 degrees counterclockwise as you can see we're following a path that would be like anti-clock would be going backwards in time and the pattern for it is that you have your x y become a negative y x so one way of putting that into words is that you could change the position of x and y and let me put a comma there i guess and change the sign of y so you're going to switch them around and then we're going to change the sign of y if y is negative make it positive if it was negative sorry if it's positive make it negative all right so that's what you're going to be doing there all right our second rule is for a 90 degree clockwise rotation so what you see here is we have our x y our point whatever we have some xy becomes y negative x and again we see a rotation from this point up here at p and we rotate it nine degrees clockwise then we get p prime and again you can see that the x and the y have traded places and then i changed the x value to be negative and so again putting that to words you should you could say again change the position of x and y and change the sign of x okay our final one is the 180 degree rotation and this one's really the most convenient because you don't have to worry about which direction you're going but 180 degree rotation all you need to do is just change the x and y signs you don't have to change their location or anything you just change their signs and so here you can see my point p here at 5 2 if i change the x to be negative and the y to be negative it ends up 180 degrees on the other side right over here so we're just changing the signs change sign of x and y all right so make sure you get these rules down uh they're going to be really helpful for you guys to do some problems but let's go show you how to do a problem all right so our first example down here we have triangle pqr has vertices 1 1 4 7 and 6 2.

[00:03:52]graph triangle pqr and its image after a 90 degree counterclockwise rotation about the origin list the coordinates of the image okay so i've gone ahead and i've already graphed triangle pqr for you at the coordinates that they've listed one one four seven and six two and what i'm gonna do is i'm actually going to use my formula to to find the new points and then i'll do the points afterwards here so my formula for a 90 degree counterclockwise rotation if you need to go back go back to go back to the other page of notes and look what you're going to do you're going to take your xy and a 90 degree counterclockwise rotation turns this into a negative y comma x so i'm going to take those points that they gave me one like 1 1 and i'm going to switch them and i'm going to make the y value negative so this 1 1 for p will become p prime and as i switch them well one and one don't really change very well but we're gonna have a negative one one because i make my negative in my front there q a similar pattern here q prime if i change them around i'd have a seven and a four but we need to make sure we make that y value of negative so we get negative 7 positive 4.

[00:04:59]and then r again we're going to start by changing the round 2 and 6 and i'm going to make sure i get that first one to be negative which is our previous y value okay so all i need to do now is graph these points on my coordinate plane and then i'll have my rotation finished so let's do that p prime is at negative 1 1 so left one up one it's going to be p prime right there q prime is at negative 7 4 negative 7 and 4 let me get to q prime and then r prime is at negative 2 a positive 6 put r prime on there and if i connect these points which i'm going to get my line tool out for you guys if i connect these points then i'm going to get the triangle triangle's image here being rotated it's been 9 degrees counterclockwise rotation and that's how you do it there is one more example for you guys to check out sorry i'm drawing extra lines here there's one more example for you guys check out down here at the bottom but you'll find that one in your quick check so if you guys have any questions don't be afraid to reach out