Chapter 7 Organizer - Transformations

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Chapter 7 Organizer - Transformations
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Rigid Motion in a Plane
Figures in a plane can be reflected, rotated, or
translated to produce new figures. The new
figure is called the Image The original figure
is called the Preimage The operation which MAPS
one to the other is called the Transformation.
Reflection (Flip) Rotation
(Turn) Translation (Slide)
Isometry a transformation that preserves
length, angle measure, parallel lines, and
distance between lines.
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Naming Transformations
Name or describe the transformation Is the first
triangle congruent to its image? What is the
line of reflection?
Name or describe the transformation Is the first
triangle congruent to its image? Is the
transformation an isometry?
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Reflections
P P
P P
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Various Reflections in a Coordinate Plane
Pre-image Points on Same side of line
Pre-image points on Opposite side of the line
Pre-image points One point on the line
Pre-image points One point on the line __
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Symmetry and Finding Lines of Symmetry
A figure has a line of symmetry, if it can be
mapped onto itself by a reflection in the line
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Symmetry In the World Around Us
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Rotations
A rotation is a transformation in which a figure
is turned on a FIXED POINT The fixed point is
Center of Rotation. Rays drawn from the center
of rotation to a point and its image form an
angle which is called the Angle of Rotation.
The angle of rotation for Point P to P Is equal
to the angle of rotation for Q to Q
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Looking at Different Cases for the Rotational
Theorem
Theorem 7.2 A rotation is an isometry.
Q
R Q
R P
P, Q, R are collinear
P and R are the same point
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Proving a Rotation is an Isometry
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Rotations in a Coordinate Plane
Figure ABCD A(2, -2) B(4, 1) C(5, 1) D(5,-1)
Figure ABCD A(2,2) B(-1,4) C(-1,5) D(1,5)
Figure ABCD is a 90o rotation of ABCD. What do
you notice about the coordinates?
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Rotational Symmetry
A figure in a plane is said to have rotational
symmetry if the figure can be mapped onto itself
by a rotation of 180o or less.
P Q
P Q

P

Q
0o Rotation 45o
Rotation 90o
Rotation
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Rotational Symmetry All Around US
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Translations and Vectors
Translations in a Coordinate Plane - When
creating a Translation, the value of the distance
between all x coordinates will be equal, and the
value of the distance between all y coordinates
will be equal
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Translation Using Vectors
A Vector is a quantity that has both direction
and magnitude (or size). Vectors have an initial
point (starting point), and terminal point
(ending point). - There is both a horizontal and
vertical component They are Notated as
follows PQ which is read vector
PQ. The component form combines the horizontal
vertical components The component of PQ is
5, 3
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What is the Vector of JK? MN? TS?
K

N J
M
T
S
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Translation Using Vectors
Using a Vector of 4, 2, Translate the Triangle
whose vertices are A (-2, 1) B (-2, -4) C (1,
-4)
A (2, 3) B (2, -2) C (5, -2)
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Finding Vectors
Using the coordinates of the vertices of a
diagram, we can calculate the vector of the
translation. How?
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Glide Reflections and Compositions
A Glide Reflection is a Transformation that maps
every point P onto a point P by the
following 1. A Translation maps P onto P 2.
A Reflection in a line (k), parallel to the
translation, maps P onto P
When two or more translations are combined to
produce a single transformation, the result is
called a Composition. Because a glide reflection
is a composition of a translation and a
reflection, they are also Isometries.
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Describing a Composition
Compositions are described by the order of the
translations used to create them

• 1. A Translation along a vector of -7,0
• 2. Followed by a Reflection around the x-axis.

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So Whats Going on Here?
1. A Reflection across the y-axis
2. Followed by a 90o Counterclockwise rotation
about the origin
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Frieze Patterns
A Frieze Pattern, or Border Pattern is a pattern
that extends to both the left and right in such a
way that the pattern can be mapped onto itself by
a horizontal translation. - The pattern can also
be mapped by a combination of other
transformations
Originated in Greek Architecture around 500 600
BC. Popular Architectural Ornamentation Style
from Ancient Greece, to Japanese art and
architecture, to Modern Times Today - Prevalent
in Wall Paper Borders Sweaters and other
designs. Can even be seen in Patterns of
Repetitive Musical Notes
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