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Transformations Preserving Congruence
Reflection,
Translation, Rotation |
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Remember:
A transformation that
preserves distance is an isometry.
A direct isometry also preserves
orientation or order.
An indirect or opposite isometry
changes the order (such as from clockwise to
counterclockwise lettering). |
Definition: A
reflection over a line, k, (notation rk) is a
transformation in which each point of the original figure (pre-image)
has an image that is the same distance from
the line of reflection as the original point but is on the opposite
side.
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The line of reflection is the
perpendicular bisector of the segment joining every point and its
image.
Reflection:
-- image is congruent to original figure.
-- opposite isometry |
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A line reflection creates a figure that is
congruent
to the original figure and is an isometry. Since when naming
the figure in a reflection requires changing the order, it is an indirect, or opposite isometry.
Definition: A
translation is a transformation, Ta,b, that
slides every point of a figure the same distance in the same
direction.
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Translation:
-- image is congruent to original figure.
-- direct isometry |
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A translation creates a figure that is
congruent to the original figure and
preserves distance and orientation – it is a
direct isometry.
Definition: A
rotation is a transformation, Rdegrees ,
that moves every point around a fixed point (usually the origin).
Rotations > 0 are counterclockwise. Rotations < 0 are
clockwise.
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A rotation of 180º is also called a
point reflection in the origin.
Rotation:
-- image is congruent to original figure.
-- direct isometry |
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A rotation creates a figure that is
congruent to the original figure and
preserves distance and orientation – it is a
direct isometry.
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