Transformations at the Beach - River Dell Regional School District

Report
Transformations at
the Beach
By Daniella Kay, Lucy Almberg,
Greg Lubin, and James Kim
Table Of Contents
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Reflections- Slide 3
Dilations- Slide 13
Translations- Slide 19
Tessellations- Slide 24
Rotations- Slide 26
Reflections
a transformation which uses a line that acts
as a mirror to reflect an image in that line.
Reflections at the Beach
● When building a sand castle, you reflect
the pail onto the sand.
● When you are riding in a boat, you can
see the reflection of the boat in the water
Key Vocabulary
● Line of Reflection- the line which acts like
a mirror in a reflection
● Line of Symmetry- a line in a figure where
the figure can be mapped onto itself by a
reflection in the line
Determining Lines of Symmetry
● All regular polygons have the same
number of lines of symmetry as they do
sides
Lines of Symmetry at the Beach
Beach towels are
rectangles and
have 1 line of
have 2 lines of symmetry
Sunglasses
symmetry
Reflecting a Preimage over a
Line
● Reflections are an isometry so the size of
the preimage stays the same when reflected
● Each point of the image will be the same
distance from the line of reflection as the
preimage, just on the opposite
side
Rules for Reflecting Over
Specific Lines
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Rx-axis (x,y) → (x, -y)
Ry-axis (x,y) → (-x,y)
Ry=x (x,y) → (y,x)
Ry= -x (x,y) → (-y, -x)
A(-2,4)
D(-2,1)
B(4, 4)
C(4, 1)
Rx-axis
D’(-2,-1)
C’(4,-1)
A’(-2,-4)
B’(4,-4)
Finding Equations for Lines of
Reflection
● Find the midpoints between the points of
the image and preimage
● Use this two points and the slope formula
to find the slope of the line of reflection
● Plug the slope into the equation y=mx+b
with one of the points to solve for b, which
is the y-intercept of the line of reflection
Finding Minimum Distance
● Find point C on the x-axis so AC+BC is a
minimum distance
● To find point C, reflect either point A or B
over the x-axis
● Draw a line from the new point (A’ or B’)
to the point that was not reflected
● The point where the line intersects the xaxis is Point C.
Let’s Try It!
Johnny owns a surf shop on an island with two beaches.
He wants his surf shop to be an equal distance from
each beach. Use minimum distance to help Johnny
decide where on the road (line m) he should put his
shop.
A
B
m
Dilations
A dilation with the center C and scale factor K is a
transformation that maps every point P in the plane to a
point P’ so that the following properties are true.
1. If P is not the center point C, then the image P’ lies on
CP. The scale factor k is a positive number such that CP’
k=
CP
and k =1
2. If P is the center point C, then P = P’
Dilations
● A dilation is not an isometry, the distance
is not preserved.
● Dilations are the same shape but not the
same size
● All sides must increase by the same scale
factor
Dilations
Key words● reduction- when the scale factor is less
than 1 but greater than 0, the dilation is a
reduction
● enlargement- when the scale factor is
greater than 1, the dilation is a
enlargement
What is the scale factor?
Is it an enlargement or reduction?
8 ft
4 ft
3 ft
6 ft
answer= 2 enlargement
Scale Factor= 2
What are the coordinates of the image?
pre-image
Is this a dilation?
4 ft
12 ft
answer- no
7 ft
3 ft
Translations
Vocabulary:
Translation - A transformation that maps every two
points so that the preimage and image of the segment
made by connecting them for both points are parallel,
collinear and congruent
Initial point - the starting point of the the vector of a
translation
Terminal point - the ending point of a vector at the
image
Component form - A form of a vector that combines the
horizontal and vertical components
Vectors and Coordinate Notation
A vector can be used to describe a translation by mapping out the distance of
points from the starting point to the terminal point with horizontal and
vertical distance.
So if point A was (1,4) and the image of point A was (3,6) the vector form of
the translation would be <2,2> because the point moved two units to the
right on the x-axis and 2 units up on the y-axis.
Coordinate notation is another way of describing a translation. For example,
if we use point A (1,4) and point A’ (3,6), the way to describe that translation
with coordinate notation would be (x + 2, y+2).
Matrices
Matrices are a different way to describe transformations,
points, and shapes in a coordinate plane.
For point A (4,5), in matrix form it would be [ 4 ] for
describing a translation [ 2 ].
5
2
The matrix for the translation means two units right and
two units up. To figure out the image of the translation
you would add them.
So [ 4 ] + [ 2 ] would be [ 6 ]
5
2
7
How to See if a Transformation is a Translation
All translations are isometries, that means that the preimage and the image
of the polygon, or segment that is translated will be congruent.
Not a Translation
Is a Translation
The segments that are made when connecting the preimage to the image
for the points in a translation are always parallel and congruent to each
other.
Real World Applications
In the real world, vectors and coordinate notation are used to describe
movement with coordinates.
For example:
The boat at the shore was at 39 degrees and 55 minutes North and 74
degrees and 4 minutes West (-74.07, 39.92) and the boat at the ocean was
at
39 degrees and 53 minutes North and 74 and 3 minutes West (-74.05,
39.88)
Can you write in vector form and coordinate notation the translation?
Tesselations
Vocabulary:
Frieze pattern (or border pattern) - a pattern that extends to the left and right
in a way so that a pattern can be mapped onto itself with horizontal
translation.
Types of Frieze patterns:
T: Translation
TR: Translation and 180 degree rotation
TG Translation and horizontal glide reflection
TV: Translation and vertical line reflection
TRVG: Translation, 180 degree rotation, vertical line reflection and horizontal
glide reflection
TRHVG: Translation, 180 degree rotation, horizontal glide reflection, vertical
line reflection
Frieze Patterns
A frieze pattern with horizontal translation:
A frieze pattern with horizontal translation
and 180 degree rotation:
Rotations
A rotation is a transformation in
which a figure is turned about a
fixed point
Rotations at the Beach
● When you put an umbrella up by turning it in the
sand, the top piece of the umbrella rotates.
● When you are swimming in the ocean and do a
flip, you are rotating 360°.
Rotations
Key Vocabulary:
● Center of rotation- the fixed point around which the preimage
rotates
● Angle of rotation- Rays drawn from the center of rotation to a
point and its image from the angle at which the figure rotated
● Rotational Symmetry- when a figure can be mapped onto itself by
a clockwise rotation of 180º or less
● Theorem 7.2- A rotation is an isometry
Determining if a figure has rotational symmetry
● All regular figures have rotational symmetry
● If a figure is not regular, it could still have rotational
symmetry. In order to find out if it has rotational
symmetry, find the number of lines of symmetry in that
figure and determine if you would be able to rotate it
less than 360º so that the image is the same as the
preimage.
Do these figures have rotational symmetry?
Yes
No
Determining the Angle of Rotation
● For a regular figure, divide 360 by the number
of sides the figure has.
● For an irregular figure, you can divide 360 by
the number of lines of symmetry it has.
The starfish has 5 sides
360/5=72
The angle of rotation of the starfish is 72º.
Drawing a rotation image using a protractor
Rotate ABC 140° counter clockwise about point K. A
1. Draw a segment from vertex A to
K
point K.
B
2. Measure a 140° angle using KA as one side and mark
it.
3. Measure the length of KA.
4. Draw a segment the same length as KA, going from
point K towards the 140° mark and make that point A’.
5. Erase all the lines you drew but keep point A’.
6. Repeat steps 1-5 for vertices B and C and then draw
C
Using a Protractor Cont’d
°
140
A’
3cm
3cm
A
K
B
C
Rotating a Figure 90º,180º,and 270º
Just use these equations to rotate the figure:
Clockwise:
Counter Clockwise:
R - (x,y) = (y, -x)
R - (x,y) = (-x, -y)
R - (x,y) = (-y, x)
90
180
270
R - (x,y) = (-y, x)
R - (x,y) = (-x, -y)
R - (x,y) = (y,-x)
90
180
270
Find the new Coordinates!
Rotate the figure 180º clockwise.
Coordinates of
the Preimage:
A (5,3)
B (2,3)
C ( 5,-1)
D (2, -1)
Coordinates of
the Image:
A’ (-5,-3)
B’ (-2,-3)
C’ (-5,1)
D’ (-2,1)
Finding the Angle of Rotation when a
preimage is reflected over two lines
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If lines k and m intersect at point P, then a reflection
in k followed by a reflection in m is a rotation about
point P.
The angle of rotation is 2x°, where x° is the measure of
the acute or right angle formed by k and m.
Lets try it!
What is the angle of rotation?
Image
Final Image
160°
x=80
2x=160
80°
Preimage
T
R
A
N
S
F
O
C
V
R
O
O
C
S
A
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