InterMath Dictionary
Transformations
Tranformations
Examples of Transformations Tranformations are movements of figures. Some transformations affect the length of the figure, e.g., dilation. Other transformations, called isometries, preserve lengths, e.g., translations, reflections and rotations .
Examples of Isometries In the word isometry, iso means same and metry means measure. In other words, isometries preserve lengths. Place a Pokemon card on a flat surface (a plane). If you slide the card (translation) along the surface, flip the card over (reflection), or if you spin the card (rotation), the lengths of the sides of the card remain the same. So, translations, reflections and rotations are isometries.
Isometries are also called rigid motions.
Click on this link to take a look at transformations through wallpaper patterns.
External Links to Extend Basic Understanding
Harcourt's Interactive Math Glossary: Read another definition of transformation.
Related Investigations on InterMath to Challenge Teachers
Tessellating a Plane: Determine which polygons tessellate the plane and why others do not. Constructing a Square: Use a dynamic software tool to construct a square in different ways. Transformations: Explore transformations of quadrilaterals. Soma Cubes: Create your own set of soma cubes pieces and explore how they can fit together. Absolute Value Functions and Transformations: Describe how the graph of an absolute function changes when you modify a, b, and c.
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Translation
In a translation, each point is moved the same distance and in the same direction. For example, in the figure to the left, point A is moved 2 units down and 3 units to the right to form the point A'. Also, point B is moved 2 units down and 3 units to the right to form the point B'. Since the two figures have the same size and shape, a translation is an isometry.
Check for Understanding
Choose each answer from the menus.
Harcourt's Interactive Math Glossary: See an animated definition of translation.
Tessellating a Plane: Determine which polygons tessellate the plane and why others do not. Absolute Value Functions and Transformations: Describe how the graph of an absolute function changes when you modify a, b, and c. Twice Reflected over Parallel Lines: Find an equation that relates the distance between points reflected over parallel lines.
Reflection
Let's reflect the object on the left over the line on the right. This line is called the reflection line.
The object on the right is the reflection (mirror) image of the original object.
After reflecting the image, connect a point, A, and its image, A'. Then point P lies on the reflection line and is the midpoint of the segment AA'. Also, the reflection line is perpendicular to the segment AA'.
A JavaSketchPad page is provided for you to explore reflections. Move any red point and observe what happens.
Harcourt's Interactive Math Glossary: See an animated definition of reflection.
Reflecting Coordinates: What line can you reflect the point (x,y) around to end up at (y,x)? (-x,y)? (x,-y)? (-y,-x)? Explain how you found your answers. Bouncing Barney: Follow Barney's walking pattern to determine how many times he will reach a wall before returning to his starting point. Symmetry Lines: Determine the number of symmetry lines a polygon has. All Swimmed Out: Determine the shortest path in a noncollinear route using triangles. Locus of Reflection: Trace the path of a reflected point to determine a pattern and the reason the pattern occurs.
Rotation
The original image is the square labeled ABCD. The image after rotating the square 45 degrees (couterclockwise) about the center is the square labeled A'B'C'D'.
The original image is the square labeled ABCD. The image after rotating the square 45 degrees (couterclockwise) about the point P is the square labeled A'B'C'D'. Note that point P is outside the square.
The angle of rotation is the angle APA'. This is the same as the angle CPC'.
A JavaSketchPad page is provided for you to explore rotations. Move any red point and observe what happens. Try various positions for point P (i.e., inside and outside the figure).
Harcourt's Interactive Math Glossary: See an animated definition of rotation.
Rotating Coordinates: State the location of (x,y) if you rotate 90 degrees counterclockwise? 180 degrees counterclockwise? 270 degrees counter clockwise? Explain how you found your answer. Twice Reflected over Intersecting Lines: Find an equation which relates the angles formed by intersecting lines and points reflected over these lines. Vertex Angles in a Regular Polygon: Investigate how the vertex angle of a regular polygon relates to its number of sides. Transformations: Explore transformations of quadrilaterals.
Dilation
The original image is the polygon labeled CDEFG. The image after dilation is the polygon labeled C'D'E'F'G'. The scale factor can be determined by the ratio:
r = length C'D'/length CD.
If this r < 1 then the image is smaller than the original polygon. If this r > 1 then the image is larger than the original polygon. If this r = 1 then the image is the same as the original polygon.
A JavaSketchPad page is provided for you to explore dilations. Move any red point and observe what happens. Try various positions for point P (i.e., inside and outside the figure).
Harcourt's Interactive Math Glossary: See an animated definition of dilation.
Dilating Coordinates: What happens when you multiply both coordinates of a point by the same number and plot this new point? Transformations: Explore transformations of quadrilaterals.
Symmetry
Assymmetrical
The figure G is not symmetrical.
Notice that there is no line that we can draw through the figure G that will 'cut' the figure so that the two sides are mirror images. There is always some part of the G that appears on one side of the line and not on the other side of the line.
Consider the square below:
Four different axes (or lines of symmetry) are shown. You can prove this by printing this page and cutting out the squares and folding along the dotted lines.
The square also has rotational symmetry. If you cut out the square below and place it on a flat surface. Place a pen or pencil on the point in the center of the square. If you rotate that square 90, 180, or 270 degrees clockwise or counterclockwise, the square will look exactly the same as it did before you rotated it.
JavaSketchPad pages are provided for you to explore symmetry with quadrilaterals and pentagons. Move the red points until the original figure and the reflection image are superimposed (i.e., when they figure and image are on top of each other). When you have accomplished this the reflection line is now the axis or line of symmetry for the figure.
Harcourt's Interactive Math Glossary: See animated definitions of line of symmetry and line symmetry.
Symmetry at ThinkQuest.org: Read another definition of symmetry, and see examples of symmetry in everyday life.
Quadrilateral Conjectures: Make conjectures about properties of quadrilaterals and provide examples or counterexamples. Symmetry Lines: Determine the number of symmetry lines a polygon has. Symmetry Lines II: Find polygons that have line symmetry. Many polygons do not have line symmetry. SAM: Explore Symmetry with Altitudes and Medians Symmetry of Polynomial Functions: Describe the symmetry of a function.
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