Geometry Ch 9 - Transformations Flashcards
A transformation of a geometric figure is…
…a change in its position, shape, or size. When you assemble a jigsaw puzzle, you often move the puzzle pieces by flipping them, sliding them, or turning them. Each move is a type of transformation. The photos below illustrate some basic transformations that you will study.
Preimage
The original figure in a geometric transformation.
Image
The resulting figure of a geometric transformation.
Isometry
A transformation in which the preimage and image are congruent.
Translation (slide)
An isometry that maps all points of a figure the same distance in the same direction.
Composition of transformations
a combination of two or more transformations. Each transformation is performed on the image of the preceding transformation. The composition of any two translations is also a translation.
Reflection (flip)
An isometry in which a figure and its image have opposite orientations. thus a reflected image in a mirror appears “backwards.”
Rotation
A transformation that turns a figure about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.
A figure has symmetry if…
… there is an isometry that maps the figure onto itself.
Reflection / Line / Mirror / Mirror-Image Symmetry
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. … An object or figure which is indistinguishable from its transformed image is called mirror symmetric.
Rotational Symmetry
A figure that is its own image for some rotation of 180º or less.
Point Symmetry
A figure that has 180º rotational symmetry.
Dilation
A transformation whose preimage and image are similar. Thus, it is a similarity transformation. It is not, in general, an isometry. Every dilation has a center and a scale factor n, n > 0. the scale factor describes the size change from the original figure to the image. The dilation is an enlargement if the scale factor is greater than 1. The dilation is a reduction if the scale factor is between 0 and 1.
A translation or rotation is…
… a composition of two reflections.
A translation is…
… a composition of reflections across two parallel lines.
A rotation is…
… a composition of reflections across two intersecting lines.
Fundamental Theorem of Isometries
In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections.
Glide Reflection
The composition of a glide (translation) and a reflection across a line parallel to the direction of translation.
Isometry Classification Theorem
There are only four isometrics. They are the following:
Reflection
Translation
Rotation
Glide Reflection
Tessellation (Tiling)
A repeating pattern of figures that completely covers a plane, without gaps or overlaps. You can create them with translations, rotations, and reflections.
Objects that tessellate
Every triangle tessellates.
Every quadrilateral tessellates.
Semiregular Tessellation is made from…
… two or more regular polygons.
