 So chapter 9 is all about transformations and symmetry some of the neatest stuff in all of geometry In your notes, I'll put this in your table of contents introduction to chapter 9 So to start we have four basic types of transformations. There are translations rotations reflections and dilations a Translation is also called a slide transformation. You're probably familiar with this one The red object is the pre-image the blue is the image in other words The red is where you start the blue is where you end up The arrow is called the translation vector a Vector is an important mathematical idea And it shows both the direction and the distance of the transformation plus the word vector is really fun to say vector Next rotations so a rotation is a turn or a spin Here we've got the pre-image in the bottom right corner rotated to Whoa whoa Rotated to that upper left corner So again rotations They rotate the pre-image a fixed number of degrees about some center point So when you define a rotation you need that center point, which is that red dot in the bottom left the amount of rotation in other words the the degree measure and Then the direction at a clockwise or counterclockwise Next reflections flip transformations or mirror transformations Here we have a reflection in the the lake. There's that nice little mountain and it creates a nice line of symmetry Also, here's another reflection a b d e the blue is the pre-image Reflected into the green the the image Ah, let's see you probably remember reflections from the geojabra project here we had a reflection of a point in a line and we'll spend more time talking about coordinates x y grids and reflections in a bit Next transformations so a dilation is a stretch or a shrink transformation a Stretch or shrink requires a center of dilation kind of a bouncing point and a scale factor So here I have cd is shown In order to stretch with that scale factor r equals 2 we're just going to double up cd to get cd prime It's possible to also have a dilation in the the other direction right have a scale factor less than 1 That would be a shrink dilation Here the rectangle in blue is the pre-image and the green is the image Then we have congruence transformations right the translations rotations reflections those are all congruence Transformations because they prefer preserve distances they preserve lengths in other words the pre-image and image are always congruent to each other Whereas dilations are similarity transformations Right the pre-image and image don't necessarily have to be congruent But they are indeed similar and then you can compose transformations if you string together more to or more Transformations, then you create what's called a composite transformation Here we start at the triangle on the top we reflect over the horizontal line and then translate To the right so that so the end product is kind of a glide transformation next symmetry Three basic types of symmetry First line symmetry also known as mirror or reflexional symmetry Sometimes called bilateral symmetry here. This butterfly shape is indeed symmetric because if you were to fold it in half The left side and the right side look the same This pentagon also has line symmetry But it has more than just that up-down line of symmetry. It has five lines of symmetry Here's a picture of the or those called the The blue angels two airplanes flying one on top of the other kind of nice picture of reflection symmetry Now another type of symmetry is called rotational symmetry Rotational symmetry exists when the pre-image and the image when you rotate the pre-image the Two shapes coincide and so this picture here is an example of rotational symmetry That star shape has rotational symmetry because if you rotate 72 degrees The image and the pre-image look the same Same thing happens again when you rotate 144 degrees and so on and so we say this has This particular shape has five-fold rotational symmetry The magnitude of rotational symmetry is 72 degrees 72 degrees means if I rotate 72 degrees then the image will look the same as the pre-image You'll probably want to write that down Magnitude of rotational symmetry is 360 degrees divided by the order of rotation and then finally point symmetry Point symmetry a point of symmetry is the midpoint of all segments joining pre-image and image points A way to check for point symmetry is rotate the object 180 degrees if it looks the same then it has point symmetry So here's an example this kind of a Six-sided star has point symmetry because if we put a point on Put a point on the edge and we take and reflect that point L Move it around a little bit first So there's a tool in Geogebra called reflect. Where is it? There it is reflect object about a point So that is the tool we want. We're gonna reflect L through P didn't work reflect L Through point P and we can see L prime the image of pre-image L L prime is stuck to Stuck to the outside of the object as well And so that indicates that this object has point symmetry here we can Show you a segment connecting the two You can even return on the trace feature and show that the image traces out the pre-image So you get the idea This Pentagon does not have point symmetry. So if we do the same thing with this Pentagon, I'll reflect point H Move around a bit reflect point H About point P Come on reflect point H About point P we see that the image is not coinciding with the pre-image In fact, if we turn the trace feature on you'll see that the image is just the sort of the flip The rotation of 180 degrees the rotated image of the pre-image So Pentagon does not have point symmetry finally Take a look at this word decode All of these letters have horizontal symmetry, right? They've got reflectional symmetry The word tomato if you line it up down the middle has a vertical line of symmetry What I'd like you to do is I'd like you to find at least Sorry find words at least five letters long that have horizontal and vertical lines of symmetry bring those to class tomorrow And maybe the longest words will win valuable prizes