Unit 3 Transformations
Ex. of composition of rigid motion
(Rl o T<-2, 5>) (ABC) *Read from right to left 1. Translate ^ABC left 2 units and up 5 units 2. Reflect ^A'B'C' across line l *This notation uses a small open circle to indicate a composition of rigid motion on ^ABC
Rule for reflecting when y=-x
(x,y)------> (y, -x)
Rule for reflecting when y=x
(x,y)------> (y,x)
Rule for reflecting when x=c
(x,y)-----> (-x + 2c, y)
r(270, O) rule
(x,y)----->(y, -x)
r(90, O) rule
(x,y)----> (-y,x)
Rule for reflecting when y=c
(x,y)----> (x, -y + 2c)
r(180,O) rule
(x,y)---->(-x,-y)
Rotational Symmetry
-A figure that maps onto itself when it is rotated about its center by an angle measuring less than 360 degrees has rotational symmetry -A figure with 180 degrees rotational symmetry has point symmetry
What are the properties of a reflection?
-If a point A is on line m, then the point and its image are the same point (that is A'=A) -If a point B is not on line m, line m is the perpendicular bisector of BB'
What are the properties of a rotation?
-The image of P is P' (that is P'=P) -For a preimage point A, PA=PA' and the measure of angle APA'= x.
Reflecting Points Across the x-axis and y-axis
-When any point P(x,y) on the coordinate plane is reflected across the x-axis, its image is P'(x, -y) -When any point P(x, y) on the coordinate plane is reflected across the y-axis, its image is P'(-x, y)
Since a rotation is a rigid motion, what is true about rotations?
-length and angle measure are preserved -any rotation is a composition of reflections across two intersecting lines *Note that a rotation is counterclockwise for a positive angle measure
Theorem 3-1
A translation is a composition of reflections across two parallel lines -Both reflection lines are perpendicular to the line containing a preimage point and its corresponding image points -The distance between the preimage and the image is twice the distance between the two reflection lines
Corollary to Theorem 3-4
Any rigid motion can be expressed as a composition of reflections. -If M is a rigid motion.... Then M=Rl or M= Rl o Rm or M=Rl o Rm o Rn
Theorem 3-4
Any rigid motion is either a translation, reflection, rotation, or glide reflection
Theorem 3-2
Any rotation is a composition of reflections across two lines that intersect at the center of rotation. The angle of rotation is twice the angle formed by the lines of reflection
Cosine
Cosine Adjacent/Hypotenuse
What are the properties of a translation?
If T(x,y)(^ABC)=A'B'C' then, -Line AA' is parallel to line BB' which is parallel to line CC' -Line AA' is congruent to line BB' which is congruent to line CC' -^ABC and ^A'B'C' have the same orientation
What is true since a transformation is a rigid motion?
Length and angle measure are preserved
What is true since a reflection is a rigid motion?
Length and angle measures are preserved
What are the notations for reflections, translations, and rotations?
R-reflection T-Translation r-rotation
How can the reflection of triangle ABC across a line be written? (^is the symbol for triangle)
Rm(^ABC)=A'B'C'
Translation
a transformation in a plane that maps all points of a preimage the same distance and in the same direction
Rigid motion
a transformation that preserves length and angle measure
Reflection
a transformation that reflects each point in the preimage across a line of reflection
Composition of rigid motion
a transformation with two or more rigid motions in which the second rigid motion is preformed on the image of the first rigid motion
Rotation
r(x degrees, P) is a transformation that rotates the center of rotation by an angle measure of x degrees, called the angle of rotation.
glide reflection
the composition of a reflection followed by a translation in a direction parallel to the line of reflection
Reflection Symmetry
the type of symmetry for which a reflection maps the figure onto itself. The line of reflection for a reflection for a reflection is called the line of symmetry
Point Symmetry
the type of symmetry for which there is rotation of 180 degrees that maps a figure onto itself. A parallelogram has 180 degrees rotational symmetry, or point symmetry
Rotations in the Coordinate Plane
Rules can be used to rotate a figure 90, 180, and 270 degrees about the origin O in the coordinate plain
Rule for reflecting over the x-axis
Rx (x,y)-----> (x, -y)
Rule for reflecting over the y-axis
Ry (x,y)------> (-x,y)
Sine
Sine (pronounced Sign) Opposite/Hypotenuse
What do we use to find angle measures and side lengths for right triangles?
SohCahToa
How can the translation of ^ABC by x units along the x-axis and by y units along the y-axis be written?
T(x,y) (^ABC)= ^A'B'C'
Tangent
Tangent Opposite/Adjacent
Theorem 3-3
The composition of two or more rigid motions is a rigid motion
Image
The result of a transformation.
Preimage
The set of points that a transformation acts on.
Theta
Variable that stands for an angle in geometry
