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Transformation in Geometry




 

A series of free, online High School Geometry Video Lessons.

In these lessons, we will learn

  • Geometry Transformations and Isometries
  • Geometry Translation
  • Geometry Reflection
  • Geometry Rotation
  • Composition of Transformations
Related topics:
Videos, worksheets, and activities to help Geometry students.

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.

How to determine the types of transformations, the definition of isometry, and how to say and write a transformation's new image.



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 + __).
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.


 

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).
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, = 5, y = x, and 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.
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.
Glide Reflection
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.


 

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.


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