Videos and lessons with examples and solutions to help High School students
when given a geometric figure and a rotation, reflection, or translation, draw
the transformed figure using, e.g., graph paper, tracing paper, or
geometry software. Specify a sequence of transformations that will
carry a given figure onto another.

Common Core: HSG-CO.A.5

Composite Transformation Theorems

There is a connection between the three transformations:
reflections, translations and rotations.

Parallel Lines Theorem: A
composition of reflections across two parallel lines is a
translation. When a figure is reflected in two parallel lines, the
final image is a translation in the direction perpendicular to the
parallel lines and twice the distance between them.

Intersecting Lines Theorem: A
composition of reflections across two intersecting lines is a
rotation. When a figure is reflected in two intersecting lines,
then the final image is a rotation of the figure about the point
of intersection of the reflecting lines through an angle twice the
angle between the intersecting lines.

Composition of Transformations (1)

Looking at composition of transformations - combining
transformations in a series. Specifically looking at glide
reflections: translation followed by reflection.

Composition of Transformations (2)

Looking at composition of transformations - combining transformations in a series. Specifically looking at glide reflections, reflection followed by rotation.

Composition of Transformations (3)

Composition Theorem: The composition of two or more isometries is an isometry.

Reflections in Parallel lines Theorem: If lines k and m are parallel then the reflection in line k followed by a reflection in line m is the same as a translation.

Composition of Transformations (4)

Reflection in Intersecting lines Theorem: If lines k and m intersect at point, P, then a reflection in k followed by a reflection in m is the same as a rotation about the point, P.

Reflecting in Parallel Lines

Reflect the blue triangle in the blue-green line to form the green image. Reflect the green triangle in the green-red line to form the red image. The red triangle, being reflected twice, is oriented like the blue triangle.

Ignoring the green triangle, the result of the two reflections is to translate the blue triangle to the red triangle through twice the distance between the two mirror lines.

Reflecting in Parallel Lines from the Wolfram
Demonstrations Project by George Beck

Reflecting in Intersecting Lines

Reflect the blue triangle in the blue-green line to form the green image. Reflect the green triangle in the green-red line to form the red image. The red triangle, being reflected twice, is oriented like the blue triangle.

Ignoring the green triangle, the result of the two reflections is to rotate the blue triangle to the red triangle about the intersection of the two mirror lines through twice the angle between them.

Reflecting in Intersecting Lines from the Wolfram Demonstrations Project by George
Beck

Transformational Geometry (Translations, Rotations, Reflections).

Defining transformations to match polygons.

Apply Composition of Transformations

Glide Reflection: Transformation followed by reflection

Translation followed by rotation

Composition of Transformations

Reflections in Parallel Lines Theorem

Reflections in Intersecting Lines Theorem

Composite Transformations in Geometric Figures

This video discusses composite transformations and work through some examples of how to do a composite transformation.

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