Examine the result of
reflecting a shape successively through two different lines.
Each of the compositions in the interactive figure shows the results
of successive reflections of a shape over two different lines. In
composition 1 the reflection lines are perpendicular, in composition 2
they are parallel, and in composition 3 they intersect but are not
necessarily perpendicular. Drag the red shape to observe the behavior of
its image shown in black. To select a shape click on the shape from the
icons at the top. To select a different composition click on the icons
on the left. Change the shape of the red square or red triangle by
dragging from an edge or vertex while pressing the Control key. Change
their orientation by dragging from a vertex. Resize the circle by
dragging from any point on the circumference.
Each of the compositions in
the interactive figure below shows the results of successive reflections over
two different lines. In composition 1 the lines of reflection are perpendicular,
in composition 2 they are parallel, and in composition 3 they intersect but are
not necessarily perpendicular. Your task is to explore each of these
compositions and then determine what single transformation, if any, would
produce the same effect. First, consider the red triangle in the interactive
figure below. Drag it and observe the behavior of its image after two successive
reflections when the lines of reflection are perpendicular. Now choose a
different shape and observe the behavior of its image. Change the shape of the
red square or red triangle by dragging it by an edge or a vertex while pressing
the "Control" key. Change the orientation by dragging it by a vertex. Which
single transformation, if any, would have the same effect on the original figure
as the double reflection has? Now try answering the same question using another
Using dynamic geometry software, students can consider what happens
when reflections are composed. Teachers can then ask students to make
conjectures about which single transformation, if any, would have the
same effect on the original figure as the composition has. The tools
made available by the software allow students to test their conjectures.
In these activities, the final image that results from reflecting a
figure using one line, then reflecting the image using a second line,
will be either a translation of the original figure (if the lines are
parallel) or a rotation (if the lines intersect). A challenging test of
students' understanding of transformations is to give them two congruent
shapes and ask them to specify a transformation or a composition of
transformations that will map one to the other.
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