Transformations in math

Translations, reflections, rotations

A transformation is a general term for four specific ways to manipulate the shape of a point, a line, or shape.

The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation.

A compositions of transformations means that two or more transformations will be performed on one object. For instance, we could perform a reflection and then a translation on the same point.

There are several interesting theorems involving the composition of reflections.

Important Terms & Definitions

  • Preimage: the figure prior to the transformation.
  • Image: The figure after the transformation.
  • Isometry: a transformation that preserves congruence. In other words, a transformation in which the image and preimage have the same side lengths and angle measurements. The following transformations maintain their mathematical congruence.
    • Translations (a translation is considered a 'direct isometry' because it not only maintains congruence, but it also, unlike reflections and rotations, preserves its orientation.
    • Rotations
    • Reflections
    • Dilations
  • On the other hand,a dilation is not an isometry because its image is not congruent with its pre-image.
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