The following vocabulary terms will
appear throughout the lessons in the section on Transformational
Geometry.
Image: An image is the resulting
point or set of points under a transformation. For example, if the
reflection of point P in line l is P', then P'
is called the image of point P under the reflection. Such a
transformation is denoted rl (P) = P'.
Isometry:
An isometry is a transformation of the plane that
preserves length. For example, if the sides of an original
pre-image triangle measure 3, 4, and 5, and the sides of its image after
a transformation measure 3, 4, and 5, the transformation preserved
length.
A
direct isometry preserves
orientation or order - the letters on the diagram go in the same
clockwise or counterclockwise direction on the figure and its image.
A
non-direct or opposite isometry
changes the order (such as clockwise changes to counterclockwise).
Invariant:
A figure or property that remains unchanged under a transformation of
the plane is referred to as invariant. No variations have
occurred.
Opposite
Transformation: An opposite
transformation is a transformation that changes the orientation of a
figure. Reflections and glide reflections are opposite
transformations.
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For example, the original
image, triangle ABC, has a clockwise orientation - the
letters A, B and C are read in a clockwise direction.
After the reflection in the x-axis, the image
triangle A'B'C' has a counterclockwise orientation - the
letters A', B', and C' are read in a counterclockwise
direction.
A reflection is an opposite transformation. |
Orientation: Orientation refers
to the arrangement of points, relative to one another, after a
transformation has occurred. For example, the reference made to
the direction traversed (clockwise or counterclockwise) when traveling
around a geometric figure.
(Also see the diagram shown under "Opposite Transformations".)
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Counterclockwise
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Clockwise
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Position
vector: A position vector is a
coordinate vector whose initial point is the origin. Any vector
can be expressed as an equivalent position vector by translating the
vector so that it originates at the origin.
Transformation: A transformation
of the plane is a one-to-one mapping of points in the plane to points in
the plane.
Transformational Geometry: Transformational
Geometry is a method for studying geometry that illustrates congruence
and similarity by the use of transformations.
Transformational Proof: A
transformational proof is a proof that employs the use of
transformations.
Vector: A
quantity that has both magnitude and direction; represented
geometrically by a directed line segment.
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