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.
How to determine the types of transformations, the definition of isometry, and how to say and write a transformation's new image.
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 + __).
This tutorial reviews how to perform a translation on the coordinate plane using a triangle.
This tutorial reviews how to translate a given polygon on the coordinate plane.
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).
This video shows reflection over the x-axis, y-axis, x = 2, y = −2
This video shows reflection over y = x, y = − x. A reflection that results in an overlapping shape.
This video shows reflection over the x-axis, y-axis, x = −3, y = 5, y = x, and y = − x.
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.
How to define a rotation and what information is necessary to perform a rotation.
The following videos show clockwise and anticlockwise rotation of 0˚, 90˚, 180˚ and 270˚ about the origin (0, 0). The pattern of the coordinates are also explored.
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).
How to define a composition of transformations; how to perform a composition of translations and a composition of reflections.
A glide reflection is a composition of transformations. In a glide reflection, a translation is first performed on the figure, then it is reflected over a line. Therefore, the only required information is the translation rule and a line to reflect over. A common example of glide reflections is footsteps in the sand.
How to describe a glide reflection and identify the information needed to perform a glide reflection.
How to Apply Composition of Transformations
Looking at composition of transformations - combining transformations in a series. Specifically looking at glide reflections, two reflections in parallel lines (a translation), and two reflections in intersecting lines (a rotation).
Two reflections in parallel lines is the same as a translation
Two reflections in intersecting lines is the same as a rotation
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