Adapted from Walch Education 1.4.1: Describing Rigid Motions and Predicting the Effects 2 Rigid...
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Adapted from Walch Education 1.4.1: Describing Rigid Motions and Predicting the Effects 2 Rigid motions are transformations that don’t affect an object’s
1.4.1: Describing Rigid Motions and Predicting the Effects 2
Rigid 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.
Slide 4
1.4.1: Describing Rigid Motions and Predicting the Effects 3
The 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.
Slide 5
1.4.1: Describing Rigid Motions and Predicting the Effects 4
Corresponding 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.
Slide 6
1.4.1: Describing Rigid Motions and Predicting the Effects 5 A
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.
Slide 7
1.4.1: Describing Rigid Motions and Predicting the Effects 6
Translating a Figure Given the Horizontal and Vertical Shift
1.Place your pencil on a vertex and count over horizontally the
number of units the figure is to be translated. 2.Without lifting
your pencil, count vertically the number of units the figure is to
be translated. 3.Mark the image vertex on the coordinate plane.
4.Repeat this process for all vertices of the figure. 5.Connect the
image vertices.
Slide 8
1.4.1: Describing Rigid Motions and Predicting the Effects 7 A
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.
Slide 9
1.4.1: Describing Rigid Motions and Predicting the Effects 8 In
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.
Slide 10
1.4.1: Describing Rigid Motions and Predicting the Effects 9
Reflecting a Figure over a Given Reflection Line 1.Draw the
reflection line on the same coordinate plane as the figure. 2.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. 3.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) 4.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. 5.Connect the image vertices.
Slide 11
1.4.1: Describing Rigid Motions and Predicting the Effects 10 A
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.
Slide 12
1.4.1: Describing Rigid Motions and Predicting the Effects 11
Rotating 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.
Slide 13
1.4.1: Describing Rigid Motions and Predicting the Effects 12
Rotating a Figure Given a Point and Angle of Rotation 1.Draw a line
from one vertex to the point of rotation. 2.Measure the angle of
rotation using a protractor. 3.Draw a ray from the point of
rotation extending outward that creates the angle of rotation.
4.Copy the segment connecting the point of rotation to the vertex
(created in step 1) onto the ray created in step 3. 5.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. 6.Repeat
the process for each vertex of the figure. 7.Connect the vertices
that have prime marks. This is the rotated figure.
Slide 14
1.4.1: Describing Rigid Motions and Predicting the Effects 13 *
Describe the transformation that has taken place in the diagram to
the right.
Slide 15
1.4.1: Describing Rigid Motions and Predicting the Effects 14
Examine the orientation of the figures to determine if the
orientation has changed or stayed the same. ArmPreimage
orientationImage orientation ShorterPointing upward from the corner
of the figure with a negative slope at the end of the arm Pointing
downward from the corner of the figure with a positive slope at the
end of the arm LongerPointing to the left from the corner of the
figure with a positive slope at the end of the arm Pointing to the
left from the corner of the figure with a negative slope at the end
of the arm
Slide 16
1.4.1: Describing Rigid Motions and Predicting the Effects 15 *
The 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
Slide 17
1.4.1: Describing Rigid Motions and Predicting the Effects 16
Compare 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 Image
Slide 18
1.4.1: Describing Rigid Motions and Predicting the Effects 17 A
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
Slide 19
1.4.1: Describing Rigid Motions and Predicting the Effects 18
Determine 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.
Slide 20
1.4.1: Describing Rigid Motions and Predicting the Effects 19
Determine 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.
Slide 21
1.4.1: Describing Rigid Motions and Predicting the Effects
20
Slide 22
1.4.1: Describing Rigid Motions and Predicting the Effects 21
Rotate the given figure 45 counterclockwise about the origin.