Describing Rigid Motions and Predicting the Effects. Adapted from Walch Education. Rigid motions are transformations that don’t affect an object’s shape and size. This means that corresponding sides and corresponding angle measures are preserved. - PowerPoint PPT Presentation
Describing Rigid Motions and Predicting the Effects
Adapted from Walch EducationDescribing Rigid Motions and
Predicting the Effects
1.4.1: Describing Rigid Motions and Predicting the Effects2Key
ConceptsRigid motions are transformations that dont affect an
objects shape and size. This means that corresponding sides and
corresponding angle measures are preserved.When angle measures and
sides are preserved they are congruent, which means they have the
same shape and size.The congruency symbol ( ) is used to show that
two figures are congruent.
1.4.1: Describing Rigid Motions and Predicting the Effects3Key
Concepts, continuedThe figure before the transformation is called
the preimage.The figure after the transformation is the
image.Corresponding sides are the sides of two figures that lie in
the same position relative to the figure. In transformations, the
corresponding sides are the preimage and image sides, so and are
corresponding sides and so on.
1.4.1: Describing Rigid Motions and Predicting the Effects4Key
Concepts, continuedCorresponding angles are the angles of two
figures that lie in the same position relative to the figure. In
transformations, the corresponding vertices are the preimage and
image vertices, so A and A are corresponding vertices and so
on.Transformations that are rigid motions are translations,
reflections, and rotations.Transformations that are not rigid
motions are dilations, vertical stretches or compressions, and
horizontal stretches or compressions.1.4.1: Describing Rigid
Motions and Predicting the Effects5TranslationsA translation is
sometimes called a slide.In a translation, the figure is moved
horizontally and/or vertically.The orientation of the figure
remains the same.Connecting the corresponding vertices of the
preimage and image will result in a set of parallel lines.1.4.1:
Describing Rigid Motions and Predicting the Effects6Translating a
Figure Given the Horizontal and Vertical ShiftPlace your pencil on
a vertex and count over horizontally the number of units the figure
is to be translated.Without lifting your pencil, count vertically
the number of units the figure is to be translated.Mark the image
vertex on the coordinate plane.Repeat this process for all vertices
of the figure.Connect the image vertices.1.4.1: Describing Rigid
Motions and Predicting the Effects7ReflectionsA reflection creates
a mirror image of the original figure over a reflection line.A
reflection line can pass through the figure, be on the figure, or
be outside the figure.Reflections are sometimes called flips.The
orientation of the figure is changed in a reflection.1.4.1:
Describing Rigid Motions and Predicting the Effects8Reflections,
continuedIn a reflection, the corresponding vertices of the
preimage and image are equidistant from the line of reflection,
meaning the distance from each vertex to the line of reflection is
the same.The line of reflection is the perpendicular bisector of
the segments that connect the corresponding vertices of the
preimage and the image.1.4.1: Describing Rigid Motions and
Predicting the Effects9Reflecting a Figure over a Given Reflection
LineDraw the reflection line on the same coordinate plane as the
figure.If the reflection line is vertical, count the number of
horizontal units one vertex is from the line and count the same
number of units on the opposite side of the line. Place the image
vertex there. Repeat this process for all vertices.If the
reflection line is horizontal, count the number of vertical units
one vertex is from the line and count the same number of units on
the opposite side of the line. Place the image vertex there. Repeat
this process for all vertices.(continued)If the reflection line is
diagonal, draw lines from each vertex that are perpendicular to the
reflection line extending beyond the line of reflection. Copy each
segment from the vertex to the line of reflection onto the
perpendicular line on the other side of the reflection line and
mark the image vertices.Connect the image vertices.1.4.1:
Describing Rigid Motions and Predicting the Effects10RotationsA
rotation moves all points of a figure along a circular arc about a
point. Rotations are sometimes called turns.In a rotation, the
orientation is changed.The point of rotation can lie on, inside, or
outside the figure, and is the fixed location that the object is
turned around.The angle of rotation is the measure of the angle
created by the preimage vertex to the point of rotation to the
image vertex. All of these angles are congruent when a figure is
rotated.1.4.1: Describing Rigid Motions and Predicting the
Effects11Rotations, continuedRotating a figure clockwise moves the
figure in a circular arc about the point of rotation in the same
direction that the hands move on a clock.Rotating a figure
counterclockwise moves the figure in a circular arc about the point
of rotation in the opposite direction that the hands move on a
clock.1.4.1: Describing Rigid Motions and Predicting the
Effects12Rotating a Figure Given a Point and Angle of RotationDraw
a line from one vertex to the point of rotation.Measure the angle
of rotation using a protractor.Draw a ray from the point of
rotation extending outward that creates the angle of rotation.Copy
the segment connecting the point of rotation to the vertex (created
in step 1) onto the ray created in step 3.Mark the endpoint of the
copied segment that is not the point of rotation with the letter of
the corresponding vertex, followed by a prime mark ( ). This is the
first vertex of the rotated figure.Repeat the process for each
vertex of the figure.Connect the vertices that have prime marks.
This is the rotated figure.1.4.1: Describing Rigid Motions and
Predicting the Effects13PracticeDescribe the transformation that
has taken place in the diagram to the right.
1.4.1: Describing Rigid Motions and Predicting the
Effects14Solving the problem:Examine the orientation of the figures
to determine if the orientation has changed or stayed the same.
ArmPreimage orientationImage orientationShorterPointing upward
from the corner of the figure with a negative slope at the end of
the armPointing downward from the corner of the figure with a
positive slope at the end of the armLongerPointing to the left from
the corner of the figure with a positive slope at the end of the
armPointing to the left from the corner of the figure with a
negative slope at the end of the arm1.4.1: Describing Rigid Motions
and Predicting the Effects15The orientation of the figures has
changed. In the preimage, the outer right angle is in the bottom
right-hand corner of the figure, with the shorter arm extending
upward. In the image, the outer right angle is on the top
right-hand side of the figure, with the shorter arm extending
down.PreimageImage
1.4.1: Describing Rigid Motions and Predicting the
Effects16Compare the slopes of the segments at the end of the
longer arm. The slope of the segment at the end of the arm is
positive in the preimage, but in the image the slope of the
corresponding arm is negative.
Preimage
Image1.4.1: Describing Rigid Motions and Predicting the
Effects17A similar reversal has occurred with the segment at the
end of the shorter arm. In the preimage, the segment at the end of
the shorter arm is negative, while in the image the slope is
positive.
Preimage
Image
1.4.1: Describing Rigid Motions and Predicting the
Effects18Determine the transformation that has taken place.
Since the orientation has changed, the transformation is either
a reflection or a rotation. Since the orientation of the image is
the mirror image of the preimage, the transformation is a
reflection. The figure has been flipped over a line.1.4.1:
Describing Rigid Motions and Predicting the Effects19Determine the
line of reflection.Connect some of the corresponding vertices of
the figure. Choose one of the segments you created and construct
the perpendicular bisector of the segment. Verify that this is the
perpendicular bisector for all segments joining the corresponding
vertices. This is the line of reflection.The line of reflection for
this figure is y = 1, as shown on the next slide.1.4.1: Describing
Rigid Motions and Predicting the Effects20
1.4.1: Describing Rigid Motions and Predicting the Effects21Your
turn.Rotate the given figure 45 counterclockwise about the
origin.
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