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Transformations and Isometries - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A transformation changes the size, shape, or position of a figure and creates a new figure. A geometry transformation is either rigid or non-rigid; another word for a rigid transformation is "isometry". An isometry, such as a rotation, translation, or reflection, does not change the size or shape of the figure. A dilation is not an isometry since it either shrinks or enlarges a figure.

Transformations in Geometry basically what they are is changing an original size, shape or position of a figure to create a new image so you're going to start with something and you're going to change it in some way and end up with a new image. Now there's 4 types of transformations.
The first type is the dilation. And dilation is taking a figure and either enlarging it or making it small and reducing it but you're going to keep the dimensions those ratios the same so you're going to create similar figures.
The second type, rotations, so you're going to start with a figure and then picking a point and an amount you're going to rotate that figure so you're not going to change its size or its shape.
The next type is the translation or sliding it, so basically you're going to have a figure and you're going to slide it in some direction.
And the last type of transformation is reflection or flipping it so with a reflection you're going to need a line that you're reflecting, so notice here I reflected this triangle over this dotted line.
Certain transformations are more specifically described and we call those isometries. An isometry is a transformation where the original shape and new image are congruent. Another way of saying this is to call it a rigid transformation not "regeed" but "rigid" transformation, so only 3 transformations are isometries, rotations I'm going to write an "I" are isometries translations are isometries and reflections. The reason why dilations are not isometries is because you're changing the size of the shape, so these 2 are never going to be congruent when you have a dilation unless your scale factor is equal to 1.
How do we describe translations? Well we're going to use an arrow to show the original image going to our new image so if you just made one transformation you would write this as triangle abc maps onto triangle a prime, b prime, c prime and I've written that down below, so those little apostrophes actually mean prime. Let's say you did another transformation then that will become triangle a double prime, b double prime, c double prime, so every time you go through a transformation you're going to have one more prime on each of your vertices, so keeping this in mind you can perform any type of transformation.

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