HSCE: G3.1.1 Define reflection, rotation, translation, and glide reflection and find the image of a figure under a given isometry.

Clarifying Examples and Activities:

A transformation that creates an image that is congruent to the original figure is called a rigid transformation or isometry, also called symmetry transformations. Three types of rigid transformations are translation, rotation, and reflection. An example of a non-rigid transformation is a size transformation.

Coordinate rules for geometric transformations:

Transformation / Coordinate Rule
Translation with components
h=horizontal k=vertical /
Reflection over y-axis /
Reflection over x-axis /
Reflection over y=x /
Reflection over y—x /
90 degree counterclockwise rotation /
180 degree rotation /
90 degree counterclockwise rotation /
Glide Reflection parallel to and over the x axis through a distance of h units /
Glide Reflection parallel to and over the y axis through a distance of k units /
Size Transformation of k (dilation) /

Examples 1:

ISOMETRIES" OR "RIGID TRANSFORMATIONS" IN SKETCHPAD

http://mathforum.org/workshops/unioncity/2000/rigidtrans.html

Example 2:

Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2), translate the pre-image 5 units right and 3 units down. Then, sketch the image.

Web Resources:

Rotations; translations, or slides; and reflections, or flips, are geometric transformations that change an object's position or orientation but not its shape or size. The interactive figures in this four-part example allow a user to manipulate a shape and observe its behavior under a particular transformation or composition of transformations.

http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm

Transformations Applet: Coordinate mapping rules and lots of practice.

http://enlvm.usu.edu/ma/nav/activity.jsp?sid=__shared&cid=emready@transformations&lid=3

HSCE G3.1.2: Given two figures that are images of each other under an isometry, find the isometry and describe it completely.

Example 1:

Completely describe the transformation that is applied to form the two images. .

Example 2:

Form a six point star start with a pre-image that is an equilateral triangle. What transformation is used to form the second triangle (image) such that the two triangles together form the six point star?

Example 3:

When constructing a new building an architect will draw a sketch of the outside of the building taking into consideration the concept of “symmetry” when viewing the building from the front. Look at the picture given below. The windows of the building are symmetric, i.e. equally spaced about a vertical line of symmetry drawn through the middle of the building. What form of transformation might the architect use when drawing the front view of the building?

Example 4:

Which transformation is defined by the two images in the picture.

A.  Reflection

B.  Translation

C.  Rotation

D.  Glide Reflection

E.  Dilation

dy@transformations&lid=3

Enrichment:

Define the transformation matrices. Use matrix operations to perform geometric transformations.

Web Resource:

CPMP-Tools is a suite of both general purpose and custom software tools designed to support student investigation and problem solving in the 2nd edition Core-Plus Mathematics texts. Geometry tools include an interactive drawing tool for constructing, measuring, manipulating, and transforming geometric figures, a simple object-oriented programming language for creating animation effects, and a set of custom tools for studying geometric models. This free domain software can be downloaded at:

http://www.wmich.edu/cpmp/CPMP-Tools/index.html

HSCE G3.1.3: Find the image of a figure under the composition of two or more isometries, and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure.

Example 1:

A glide reflection is a translation followed by a reflection over a line parallel to the translation vector. Find the image of the point (3, 2) under a translation of (x, y) à (x + 3, y) then reflected over the x-axis.

Example 2:

The triangle ABC is translated with horizontal component of -2 and vertical component of -3 then reflected over the line y = 1. What are the new coordinates of the image under this glide reflection? Sketch the image. Explain.

Example 3:

The triangle with vertices at A(2, 4), B(2, 1), and C(6, 1) is reflected over the y-axis followed by a second reflection over the x-axis. The new image is a ______of the pre-image.

A.  Glide Reflection

B.  Rotation

C.  Translation

D.  Dilation

E.  Reflection

Web resource:

Lesson 3 in the 9-12 geometry lesson SYMMETRIES II contains illustrations of the composition of reflections over parallel lines and intersecting lines.

http://illuminations.nctm.org.

Enrichment Activity:

Programming Animation on the graphing calculator:

Multiple rotations of a triangle in 30-degree increments (or any small angle value) will produce neat pictures. The utilization of programming on the TI-84 can “speed” up the animation process.

Explore this problem: A triangle with all of its vertices in the first quadrant is rotated 30 degrees 24 times about the origin. How many triangles are in the final picture? Why not 24 triangles? See program below.

TI-84 Program

pROGRAM TITLE:ROTTRI30

:ClrDraw

:ClrHome

:AxesOff

:GridOff

:PlotsOff

:FnOff

:-15Xmin

:15Xmax

:1Xscl

:-10àYmin

:10Ymax

:1Yscl

:[[1,5,8][2,1,6]][A]

:[[√(3)/2,-.5][.5,√(3)/2]] [B]

:For(C,1,24)

[B]*[A] [A]

:Line([A](1,1),[A](2,1),[A](1,2),[A](2,2))

:Line([A](1,1),[A](2,1),[A](1,3),[A](2,3))

:Line([A](1,3),[A](2,3),[A](1,2),[A](2,2))

:End

Explain how does the now-next statement [B]*[A] [A] creates the animation of the shape [A]?

Extension:

Teachers may link this program to all student calculators. Students may experiment with making changes in the program, observe and document results. Students can create original animation programs.