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Transformation in Geometry

A series of free High School Geometry Video Lessons from Brightstorm.

Geometry Transformations and Isometries
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.

Geometry Translation
A geometry translation is an isometric transformation, meaning that the original figure and the image are congruent. Translating a figure can be thought of as "sliding" the original. If the image moved left and down, the rule will be (x - __, y - __) where the blanks are the distances moved along each axis; for translations left and up: (x - __, y + __), for right and down (x + __, y - __), for right and up (x + __, y + __).

Geometry Reflection
A reflection is an isometry, which means the original and image are congruent, that can be described as a "flip". To perform a geometry reflection, a line of reflection is needed; the resulting orientation of the two figures are opposite. Corresponding parts of the figures are the same distance from the line of reflection. Ordered pair rules reflect over the x-axis: (x, -y), y-axis: (-x, y), line y=x: (y, x).

Geometry Rotation
A rotation is an isometric transformation: the original figure and the image are congruent. The orientation of the image also stays the same, unlike reflections. To perform a geometry rotation, we first need to know the point of rotation, the angle of rotation, and a direction (either clockwise or counterclockwise). A rotation is also the same as a composition of reflections over intersecting lines.

Composition of Transformations
A composition of transformations is a combination of two or more transformations, each performed on the previous image. A composition of reflections over parallel lines has the same effect as a translation (twice the distance between the parallel lines). A composition of reflections over intersecting lines is the same as a rotation (twice the measure of the angle formed by the lines).

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