Chapter 7 Organizer - Transformations. Rigid Motion in a Plane Figures in a plane can be reflected,...
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Chapter 7 Organizer - Transformations
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Transcript of Chapter 7 Organizer - Transformations. Rigid Motion in a Plane Figures in a plane can be reflected,...
Chapter 7 Organizer - Transformations
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.
Naming Transformations
Name or describe the transformation
Is the first triangle congruent to it’s image?
What is the line of reflection?
Name or describe the transformation
Is the first triangle congruent to it’s image?
Is the transformation an isometry?
Reflections
P
P’
* H(2,2)
* H(2,-2)
P P’
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 & _|_
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
Symmetry In the World Around Us
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.
Q
R R’
Q’ P
The angle of rotation for Point P to P’Is equal to the angle of rotation for Q to Q’
Looking at Different Cases for the Rotational Theorem
Theorem 7.2: A rotation is an isometry.
Q
R Q’
R’
P
Q Q’
R R’ P
Q
Q’ R
R’
P
P, Q, R are non-collinear
P, Q, R are collinear P and R are the same point
Proving a Rotation is an Isometry
Q
Q’ R
R’
P
P, Q, R are non-collinear
Given: A rotation around P, maps P to P’ and Q to Q’Prove: QR = Q’R’
_ _~
PQ = P’Q’, PR = P’R’ Def. of rotationm< QPQ’ = m<RPR’ Def of Rotationm< QPQ’ = m<QPR + m<RPQ’ Angel + Post.m<RPR’ = m<RPQ’ + m<Q’PR’ Angle + Postm<QPR + m<RPQ’ = m<RPQ’ + m<Q’PR” Substitution Prop of =m<RPQ’ = m<RPQ’ Reflexive Prop of =m<QPR = m<Q’PR’ Subtraction Prop of =/\ QPR = /\ Q’PR’ SAS congruence PostulateQR = Q’R’ CPCTC~
~
Rotations in a Coordinate Plane
Figure A’B’C’D’A(2, -2)B(4, 1)C(5, 1)D(5,-1)
Figure ABCDA(2,2)B(-1,4)C(-1,5)D(1,5)
Figure ABCD is a 90o rotationof ABCD. What do you notice about the coordinates?
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
Translations and Vectors
A Translation is a transformation that maps every two points P and Q in a plane, to points P’ and Q’ so the following properties are true:
- PP’ = QQ’- PP’ || QQ’
P’
P
Q’
Q
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
A (-1, -3) A’ (-3, 2)B (2, -3) B’ (0, 2)C (-1, 0) C’ (-3, 5)
What relationship do you see between x to x’ and y to y’?
C’
A’ B’
C
A B
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}
Q
P
3
5
K
N
J
M
T
S
What is the Vector of
JK?
MN?
TS?
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)
Finding Vectors
Using the coordinates of the vertices of a diagram, we can calculate the vector of the translation. How?
A’ B’
A B
C’ D’
C D
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”
A’ B’ A B
C’ D’ C D
C” D”
A” B”
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.
A’ B’ A B
C’ D’ C D
C” D”
A” B”
Describing a Composition
Compositions are described by the order of the translations used to create them:
What would the result be if we were to reverse the steps of the composition?
1. A Translation along a vector of {-7,0}
2. Followed by a Reflection around the x-axis.
B’ A’ A B
D’ C’ C D
A”
C”
D”
B”
So What’s Going on Here?
1. A Reflection across the y-axis
2. Followed by a 90o Counterclockwise rotation about the origin
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
