When transformations are combined, the resulting transformation is a composition of transformations. Having a thorough understanding of the individual transformations is key to dissecting compositions of transformations. Using visual aids also helps a lot on performing composition of transformations. When learning about transformations, picture each step along the way.
Transformation: An operation that moves, flips, or changes a figure to create a new figure.
Rigid: A transformation that preserves size and shape.
Isometry: Another word for rigid transformation, a transformation that does not change the shape or size of a figure.
Composition of Transformations: To perform more than one rigid transformation on a figure
Theorem: A composition of two (or more) isometries is an isometry.
Each isometry is a rigid transformation, so after performing several isometries, the figure does not change the shape or size of a figure. We can think of the series of isometries as a single isometry.
A glide reflection consists of a reflection and a translation in a direction parallel to the line of reflection.
It doesn’t matter whether the reflection or the translation is done first. The composition can also be written as a single rule.
A composition of reflections over parallel lines is equivalent to a translation.
Reflections over Parallel Lines Theorem: If there are two reflections over parallel lines that are h units apart, it is the same as a single translation of 2h units
Reflection over the Axes Theorem: If there are two reflections over each axis, then the final image is a rotation of 180° of the original
Two reflections across two intersecting lines is equivalent to a rotation.
Reflection over Intersecting Lines Theorem: If there are two reflections over lines that intersect at xº, then the resulting image is a rotation of 2xº where the point of intersection is the center of rotation.
Theorem: Any translation or rotation is equivalent to a composition of two reflections.