Rotations - River Dell Regional School District

```Kayla Sampayo, Sean Carey, Jotaro Kurachi, Joanna Ha, and Sara Manzelli
Translations: Joanna
Rotations: Kayla
Reflections: Sara
Dilations: Jotaro
Tessellations: Sean
To put it simply, a translation is like a slide…
Translation
Definition
PREIMAGE
• It’s when you two images, the preimage
and the image, in different locations but
look exactly the same. You do it without
rotating or flipping the image. It is an
isometry. It’s like how a monkey moves
from one tree to another.
IMAGE
Examples
P
P’
• 1) PP’ = QQ’
• 2) PP’ || QQ’, or PP’ and QQ’
are collinear.
Q
Q’
Double reflections
• It’s when you reflect an image twice, then you’ll get a
image that is transformed.
reflecting
translation
Component form and vector
Example
• The component form of a vector
combines the horizontal and
vertical components.
DEFINITION OF VECTOR:
• A VECTOR IS A QUANTITY THAT HAS BOTH
DIRECTION AND MAGNITUDE OR SIZE.
COMPONENT FORM
COMPONENT FORM
Definition of component form:
How to find vector component form
• If you have line JK, and you want
to find the component form,
you create a right triangle with
the line.
K
Then count how much from j to k
it moves from the x and y axis.
form.
4
3
2
1
J
-6 -5 -4 -3 -2 -1
COMPONENT FORM
OF JK IS <-6, 4>
PROBLEMS!
Find the component form of vector AB.
B
A
<-9, 5>
Equation to translation
The equation for translation is:
(x,y)  (x+a, x+b)
• a = horizontal change
• B = vertical change
So basically what your doing is
adding the component form to a
coordinate pair.
Example:
Find the new vertices after the transformation of
parallelogram ABCD. The component form is <2, -4>.
A
D
B
C
A (6,7); B (7,2); C (10,2); D
(9,7)
A’ – (6 + 2, 7 + -4) = (8, 3)
B’ – (7 + 2, 2 + -4) = (9, -2)
C’ – (10 + 2, 2 + -4) = (12, -2)
D’ – (9 + 2, 7 + -4) = (11, 3)
Practice problem!
Find the coordinates of this shape after a translation. Find the component form
using vector bc and the new coordinates. Then graph it. Use the equation!
A
C
B
• Component form: <3,
2>
• A’ = (3, 9)
• B’ = (4, 6)
• C’ = (7, 8)
Matrices
• Another way to get the coordinates of an
image after translation is by using
matrices.
• Matrices is a group of numbers.
• You but the x value on the top and the y
value on the bottom in between two big
brackets.
• You get a second set of brackets and put
in the component form and repeat the x
and y depending on how many vertices
there are. For example: If a triangle has
three vertices, then you would repeat
the component form three times.
• Then you add the two brackets together
according to their positions.
If the component form was <5, -7>…
Use matrices to find the translation of figure
EFG.
Use vector FG for the component form.
G
E
F
E’ = (3, 6)
F’ = (7, 4)
G’ = (8, 7)
PRACTICE!
Using matrices, find the coordinates after a translation, using the
vector CD.
A
B
D
• Component form: <-1,
6>
• A’ = (5, 13)
• B’ = (4, 7)
• C’ = (7, 7)
• D’ = (6, 13)
C
A word problem
A group of scientist researchers are planning to move their research center in a rainforest located in
South America. They put the building on the coordinates A(1,2), B(2,6), C(4,7), and D(5,1). Find the
component form of vector CD and find the transition from their original building to the new
building. Use the equation or matrices to solve.
B
Component form: <-1, 5>
A’ = (1 + -1, 2 + 5)  (0,7)
B’ = (2 + -1, 6 + 5)  (1,11)
C’ = (4 + -1, 7 + 5)  (3,12)
D’ = (5 + -1, 1 + 5)  (4,6)
C
A
D
Translations are everywhere in the world. In the rainforest, translations
are seen all the time when animals are walking, flying, swimming,
climbing, and crawling! This Jaguar is stalking its prey!
WORD SCRAMBLE
CRTOVE
7
CNNETPOMOMORF
13
6
14 4
NTLRAINSAOT
3 10
______
SICMTARE
5
8
DBLEUONTRFELEOTIC
1
11
GEPMIEAR
12 2
GEMIA
9
________
________ _____
___________
______ ___________
________
_____
______________
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A rotation is a transformation in which a
figure is turned about a fixed point.
better understand rotations are below:
• Center of rotation- the fixed point of a rotation
• Angle of rotation- rays drawn from the center of
rotation to a point
• Rotational symmetry- a figure in a plane that can
be mapped onto itself by a clockwise rotation of
180° or less
Here is an example of a rotation.
The center of rotation is point P
One way to rotate a figure
You can use these equations to plot the new point of the figure that is
rotating 90°, 180°, or 270°
• R90 (x,y)= (-y,x)
• R180 (x,y)=(-x,-y)
• R270 (x,y)= (y,-x)
• R-90 (x,y)= (y,-x)
Remember that R-90 refers to a rotation of
90° clockwise
Another way to rotate
• First, you must identity where the center of rotation is.
• Once you locate it, draw a line from one of the vertices to the
center of rotation. Keep in mind that the center of rotation can
be inside a figure, outside the figure, or one of the vertices
• Use a protractor and mark where the angle of rotation is. Make
sure you measure the angle counterclockwise unless the
problem says otherwise.
• Measure the length of the line you drew in second step.
• Draw a line from the center of rotation that is the same
measurement as the line you drew in step two. The new line
should meet with the mark you made from step 4.
• The endpoint of the line in step 5 is the prime version of the
vertex you selected in step 2
• Lastly, repeat the steps for each vertex and connect the dots to
form the final figure.
This diagram illustrates the steps to find
one of the point on the final figure.
In order to decide whether a figure has
rotational symmetry…
You have to see whether the figure can be mapped onto itself by
a clockwise rotation of 180° or less.
For example, both of these shapes have rotational symmetry. If
the purple polygon is rotated 180° or 60°, it is perfectly aligned
with the original image. If the other figure is turned 180° it will
also map onto them original image.
Theorem 7.3 states…
• If L1 and L2 intersect at point P, then a reflection in L1 followed by a
reflection in L2 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 L1 and L2.
Important
Concept for
Rotation
Sample Problems
Which of these figures
rotational symmetry?
What is the angle of rotation?
When the parrot changes course
it rotates around his right wing.
This wing acts as the point of
rotation. The parrot rotates 90°
clockwise.
Rotation Activity!
Rotate the sloth 180° clockwise on the tree branch
It should look something like this:
• A reflection is a transformation which
uses a line that acts like a mirror, with
an image reflected in the line.
• The line which acts like a mirror in a
reflection is called the line of
reflection.
• A figure in the plane has a line of
symmetry if the figure can be
mapped onto itself by a reflection in
the line.
• A reflection is an isometry, the
original figure and the image are
congruent
Reflections
When reflecting a point over the x-axis, y-axis, y=x, or y=-x
How do you reflect a point
over a line such as x= -2?
The reflected point will be the same
distance away from the line of
reflection.
If easily seen, count how far away
the point is from the line, it will be
the same distance away from the
line when reflected across it (1 to
the left, it will then be 1 to the right)
Reflect the triangle over the y-axis
• Find the new points by
using the formula:
(x,y) = (-x,y)
• The new points should
be:
(1,2) (1,6) (4,2)
• Plot the new points to
form the image.
7
(-1,6) 6
5
4
3
(-4,2)
-4
-3
-2
(-1,2)
2
-1
1
0
1
2
3
4
A figure has a line of symmetry if
it can be mapped onto itself by a
reflection.
The number of lines of symmetry in a regular polygon will be equal to
the number of vertices
• How many lines of symmetry do these shapes have?
Regular Pentagon has 5 lines of symmetry
Rectangle has 2 lines of symmetry
Isosceles Trapezoid has 1 line of symmetry
There are examples of reflections everywhere in the
world around us. The rainforest is home to many
butterflies. These butterflies all have a line of symmetry.
Using your knowledge of reflections and lines of symmetry, decorate your own butterfly!
Dilations
• A dilation is a transformation where the original image and the after image are
•
•
•
•
the same shape but differ in size.
The scale factor cannot be less than zero.
The scale factor is represented as k.
A reduction is when the scale factor is greater than zero but less than 1.
An enlargement is when the scale factor is greater than 1.
Here’s some
important
vocabulary!
This would be a reduction, since the afterimage is smaller than the pre-image.
When the center is the origin, you can find the coordinates of the afterimage
by multiplying the coordinates of the pre-image by the scale factor
The scale factor is 2.
For example, the
coordinates of the
pre-image are:
(1,1) (3,1) (2,3)
Therefore, the
coordinates of the
afterimage would be:
(2,2) (6,2) (4,6)
You can also use matrices to solve for the coordinates of the afterimage.
You put the numbers in a matrix, then you multiply them by the scale factor
.
x2
Now you try!
This is an enlargement because the
image is larger than the pre-image.
The scale factor is 3
When a tree is cut down, you can see the tree rings. Tree rings are used to determine how
old the tree is. These rings are dilations of each other. Each outer ring is an enlargement of
its inner rings. Believe or not, there are dilations all around us, even inside trees!
What is a
Tessellation?
A tessellation is a repeating pattern of figures that
completely covers a plane without any overlaps or
gaps
You can make a tessellation only by using isometrics
(Translation, rotations, and reflections)
Triangles and Quadrilaterals can always tessellate
D
C
What is an easy way to tell if
a figure can be tessellated?
Do you remember
the equation
(n-2)180/n?
If 360 is divisible by
will tessellate.
Tessellation Vocab
• Vector- a quantity that has both direction and magnitude, it is represented by an arrow
drawn between two points.
• Transformation- The operation that maps or moves a pre-image onto an image
• Regular Polygon- a polygon that is equiangular and equilateral
• Semi regular tessellation- more than one kind of regular polygon is used and the same
arrangement of polygons meets at any vertex of the tessellation.
Here are two examples of
semi regular tessellations:
Try to Tessellate!
Can you tessellate this caterpillar?
Your tessellation should look like this:
Tessellations in the Rainforest
There are tessellation all around us in
our world and especially in the rain
forest! A perfect example of a real
world example of a tessellation is a
turtle shell, you may not realize but a
turtle shell is a tessellation!
Bibliography: Websites
• http://office.microsoft.com/en-us/powerpoint-help/animate-text-or-objectsHA010021497.aspx
• http://www.mathsisfun.com/algebra/vectors.html
• http://www.mathsisfun.com/algebra/matrix-introduction.html
• http://www.tessellations.org/tessellations-all-around-us.shtml
• http://puzzlemaker.discoveryeducation.com/WordSearchSetupForm.asp?campaign=fly
out_teachers_puzzle_wordcross
• http://www.virtualnerd.com/middle-math/integers-coordinateplane/transformations/reflection-definition
• http://mathbydesign.thinkport.org/images/pdfs/TransformationsReflections_LessonPla
n.pdf
• http://www.regentsprep.org/Regents/math/geometry/GT1/reflect.htm
Bibliography: Pictures
• http://www.clker.com/cliparts/T/b/y/u/y/A/rain-drop-md.png
• http://imgs.tuts.dragoart.com/how-to-draw-a-rainforest_1_000000004065_5.jpg
• http://www.canvas101.co.uk/images/gallery-art/Animal/AN0072%20Large%20Red%20and%20Blue%20Macaw%20Parrot%20Flying%20.jpg
• http://www.education.vic.gov.au/images/content/studentlearning/mathscontinuum/RotationalExamples.gif
• http://o.quizlet.com/jS0zgaFgt4ZTZSacC2ABzQ_m.png