Around the World with
Transformations
Table of Contents
Rotations: Slides 3-13 (Jake Campbell)
Translations: Slides 14-19 (Alison Sorkenn)
Dilations: Slides 20-29 (Eric Park)
Reflections: Slides 30-40 (Angela Hong)
Tessellations: Slides 41-47 (Conor McCorry)
Rotations
Rotations
●
●
●
●
A rotation is when a figure is turned about a fixed point
Rotations are isometries (Segment and angle measures remain congruent)
This point is called the center of rotation
The amount of degrees that the figure is rotated about that point is the angle
of rotation
● Unless noted, the angle of rotation is always counter-clockwise
Rotations on the Coordinate Plane
● To rotate points counterclockwise around the origin
when given coordinates, use the
equations below
● 90°- (x,y) = (-y,x)
● 180°- (x,y) = (-x,-y)
● 270° - (x,y) = (y,-x)
● For clockwise rotations the
equations are
● 90°- (x,y) = (y,-x)
● 180°- (x,y) = (-x,-y)
● 270° - (x,y) = (-y,x)
Activity:
Rotate ΔABC 180° about the origin
A= (1,1)
B= (4,1)
C= (5,3)
A′= (-1,-1)
B′= (-4,-1)
C′= (-5,-3)
Now plot the points of the image
and preimage.
C
A
B’
C’
A’
B
Finding Angles of Rotation
● Reflecting over two
intersecting lines is the
same as rotating
●
To find the angle of
rotation, find the angle
of where the lines meet
and multiply it by 2
In image on the left, what
is the angle of rotation?
32°
Answer: 64°
Rotational Symmetry
● If a figure can be rotated onto itself in 180° or less, it has rotational symmetry
● All regular polygons have rotational symmetry
● You can find the angle of rotation for regular polygons by by using the formula
360/n (number of sides)
● To determine if an irregular shape has rotational symmetry, you have to eyeball it
● If the irregular shape has rotational symmetry, figure out how many ways the
shape can map on to itself with rotations less than or equal to 360°
● The number of times a shape can map onto itself is known as the order
● To find the angle of rotational symmetry, use the equation 360/o (order)
Real-Life Examples of Rotational
Symmetry
This Hindu (the primary
religion of Bangladesh)
swastika has 90° rotational
symmetry
The flag of Bangladesh has
180° rotational symmetry
Does this flower,
native to Bangladesh,
have rotational
symmetry?
If so, what is the angle
of rotation? (Assume
all petals are
congruent)
Answer: The flower has 72° rotational
symmetry. Since it can be rotated onto itself 5
times within a 360° rotation, its order is 5.
When we use the equation 360/o, we get 72°.
Rotating Around a Point Using a Ruler and Protractor
(Original Artwork)
1. Rotate triangle ABC 90°
clockwise around point D
2. Draw a line segment from point
C to the point of rotation (P)
3. Using a protractor, create the
angle of rotation (90°) using
point P as the vertex. Make sure
that the angle is in the direction
that you’re rotating (clockwise)
5. Measure the length of CP
4. Finish drawing in the
angle
6. Construct point C’
the length that you
previously obtained
along the angle that
you drew in.
7. Erase angle CPC’ and
repeat steps 1-6 for point A
8. Repeat steps 1-6 for
point B. The final, rotated
triangle should look like
this.
Translations
Key Words to Know
•When an object is
translated, all points are
moved the same distance
and in the same direction.
•Translation is a transformation
that maps original points to
their final points
•Vector is a quantity that has
both direction and
magnitude
•A translation is an isometry,
which means that the lengths
stay the same.
<6,-2>
6 units right, 2 units down
● Reflecting an object twice
over parallel lines results
in a translation.
Translating Objects
•When an object is translated, it can
be moved to any quadrant, as long as
it stays in the same direction and is the
y
same size.
x
•Original objects are
known as pre-images
and translated objects
are the images of the
original.
Rules to Use for Translations
•There are three different ways to write out translations.
•Coordinate Notation:
(x,y) → (x+a,y+b)
-represents how the coordinates are translated; adding/subtracting x and y
•Vector in Component Form: <6,-2>
-shows movement of the coordinates
•Matrices:
x
y
-rows and columns showing x and y coordinates
-use matrix addition to find translated points
Finding Image of Pre-Image
•If you are given a pre-image with points on
a graph and a rule, you will have to figure
out the points of the image.
•Ex: A(-8,7) B(-8,1) C(-1,7) D(-1,1)
Pre-Image
A
Image
A’
C’
B’
D’
C
•Rule= (x,y) → (x+10,y+2)
•Find the coordinates of the image and the
vector that describes the new image.
•The x-coordinate will increase by 10 units
and the y-coordinate will increase by 2
units.
● Final Points: A(2,9) B(2,3) C(9,9) D(9,3)
B
D
Word Problem
•The Brazil soccer team is
playing a game and needs
to score a goal. One of the
players with the ball sees
an open player and
decides to pass it to him.
Write the vector in
component form that
describes the translating
of the ball from player to
player.
Dilations
Dilations
Dilation- with a center of C and a scale factor of K, is a transformation that maps every point from the
pre-image to their corresponding points of the dilated image
1.
If p is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive
number such that k = CP’/CP, and K =/=1
2.
If P is the center point C, then P = P’
Reduction- A dilation is a reduction if: 0 < K < 1
Enlargement- A dilation is an enlargement if: K > 1
Finding the scale factor of an image
Put the value of the dilated image over the value of the preimage
Dilated image/ Preimage
10/5 (Then reduce the fraction)
2/1
The scale factor of this dilation is 2
Preimage
5 inches
Dilated image
10 inches
Dilating Images with the origin as the center
Matrix- rectangular arrangement of numbers in rows and columns
Scalar multiplication- multiplying a matrix by a real number
Multiply the x and y coordinates of the primage by the scale factor to get the coordinates of the dilated image
Ex:
(Scale factor 3)
Coordinates of Preimage: (0,4) (5,-1) (-2,2)
X [ 0 5 -2 ]
Y [4 -1 2 ]
[ (3)0 (3)5 (3)-2 ]
[ (3)4 (3)-1 (3)2 ]
Coordinates of Dilated image: (0,12)(15,-3)(-6,6)
[ 0 15 -6]
[ 12 -3 6]
Dilating Images with the origin as the center
Use scalar multiplication to find the coordinates of the dilated image
Given: Scale factor = 2. (0,4) (2,3) (3,1)
Answer:
[ (2)0 (2)2 (2)3]
[ (2)4 (2)3 (2)1]
[ 0 4 6] (0,8) (4,6) (6,2)
[ 8 6 2]
Dilating Images with the origin as the center
Plot the points on the graph from the previous slide:
Dilating Images with the origin as the center
Now you try!
Dilate the coordinates (1,3) (2,4) (5,1). (First graph these points). With a scale factor of 3 and
the center as the origin
Answers:
A’=(3,9)
B’=(6,12)
C’=(15,3)
Dilating Images with the origin as the center
The British soccer team Manchester United uses three
people to pass the ball between. They each travel back
the same distance as the others. If the original
coordinates of the three people are: (1,4)(3,6)(4,2). And
the coordinates of the three people after traveling back
an equal amount: (4,16)(12,24)(16,8). Find the scale
factor.
Answer:
Dilated coordinate over
preimage coordinate: 4/1= 4
is the scale factor
Challenge yourself
Imagine you are a toy modeler. You are trying to make a replica of the building Big Ben.
Big Ben is approximately 316 feet. The height of your model is 66 inches. The width of Big
Ben is 50 feet. Find the width of your model.
Answer:
First convert feet to inches (You want your
toy model in inches). (3792 inches, and 600
inches for width)
Then put the dilated image over the
preimage. 66/3792 = 0.017 which is your
scale factor.
Then multiply 50 by your scale factor.
(which gets you
is the answer
0.87inches) Which
Dilating an image with a center that is not the origin
When the center of the dilation is not the origin, you can’t simply multiply the
x and y coordinates of the preimage by the scale factor. Use this equation
when dilating an image on a coordinate plane. (This equation does work
when the center is the origin, but that just adds extra work)
K(x-a)+a
K(y-b)+b
Where:
(x,y) are the coordinates of a point on a preimage
(a,b) coordinates of the center of the dilation
K is the scale factor
Reflections
● A reflection is a transformation in which an image is reflected over a line
● The line of reflection is the line over which the image is reflected
● A figure has a line of symmetry if it can be mapped onto itself by a
reflection over the line
○ Reflections are isometries, meaning that the lengths of the preimage
(original) and image (transformed figure) are preserved
An easier way to think about reflections is through the
comparison of mirrors.
● The object you hold up to a mirror is the preimage, and
its reflection in the mirror is the image
● The line of reflection acts as the mirror itself, because it
is what reflects the image onto a new space.
MATH is
reflected
over the line
y = -x
Reflections
Above, the image reflects
itself over the line x = 0
The Louvre Pyramid Zis
reflected in the water at night
Determining Lines of Symmetry
A figure has a line of symmetry if one side of the shape
can be reflected over the line and be congruent to the
other. In other words, imagine “folding” the figure over
the line, like a piece of paper.
NO
This is not a line of
symmetry.
YES
This is a line of
symmetry.
Determining Lines of Symmetry
Triangles can have 3, 1,
or 0 lines of symmetry
Equilateral
Isosceles
Right
Hexagons can have different lines of
symmetry depending on their shape
Irregular
Regular
Irregular
In a regular polygon, the number of lines of symmetry is equal to the number of
sides the polygon has, because regular polygons have equal sides and angles.
3 Sides = 3 Lines
4 Sides = 4 Lines
5 Sides = 5 Lines
6 Sides = 6 Lines
The Palace of Versailles is an example of a
symmetrical landmark.
The line of symmetry runs vertically through
the middle of the palace and splits the
figure into two congruent halves.
The Eiffel Tower also has a
vertical line of symmetry.
Reflecting a Preimage With Coordinates
When reflecting a preimage over a line on a coordinate plane, you can either…
1. Pick a vertex on the preimage, and
count the number of units it takes to
reach the line of reflection by using
“rise/run”. Make sure you are
counting box-by-box!
2. Count that same number of units
from the line of reflection in the
opposite direction of the preimage
and plot the new point.
3. Repeat steps 1-2 for all the points of
the preimage.
4. Connect the plotted points to form
the new image.
Use These Rules:
OR
Reflect over x-axis
(x, y)  (x, - y)
Reflect over y-axis
(x, y)  (- x, y)
Reflect over y = x
(x, y)  (y, x)
Reflect over x = y
(x, y)  (- y, - x)
Write An Equation for a Line of
Reflection
For lines…
1. Choose a point on the preimage and
locate the corresponding point on the
image
2. Use midpoint formula to find the
midpoint of the two points
3. Find the perpendicular slope of the
two points by taking the opposite
recipricol of the regular slope
4. Use the perpendicular slope as “m” to
create a slope-intercept equation by
plugging in the midpoint coordinates
as “x” and “y”
For polygons…
1. Choose a point on the preimage and
locate the corresponding point on the
image
2. Use midpoint formula to find the
midpoint of the two points
3. Repeat steps 1-2
4. Find the slope of the two plotted
points and use it to create an equation
by plugging in one of the midpoint
coordinates as “x” and “y”
Finding Minimum Distance
Find point C on line m so that AC + BC is of minimum distance.
A
B
m
1. Reflect one of the points over the line
2. Use distance formula or measure with
a ruler to find the distance between
the reflected and the other unreflected
point (A’ and B, or A and B’)
3. The distance calculated is the
minimum distance
B
A
A’
B
A’
Minimum Distance = 7 cm
Now You Try!
Jocelyn’s town is installing new water pipes on Chestnut
Avenue. Her dad is in charge of the construction project, and
asks her to help. The pipes must stretch from House A to
House D, but they must also reach the pump of the reservoir
on the other side of the street in order to draw water. What is
the least amount of piping Jocelyn’s dad can use?
A
For this problem:
Assume 1 cm = 10 yards
B
C
RESERVOIR
D
Now You Try: Answer
1. The pipes must go from House A to House
D. All other Houses in between are
irrelevent to the problem.
2. Reflect one of the houses over the street,
and use a ruler to measure the distance
between House A’ and House D. Distance
formula cannot be used in this particular
problem because no coordinates were
given.
A
D
A
D
A’
3. The distance measured is 5 cm, and since
1 cm = 10 yards, the least amount of piping
Jocelyn’s dad must use is 50 yards.
* Not to Scale *
Minimum Distance = 5 cm
1 cm = 10 yards  5 cm = 50 yards
Tessellations
Classification of Frieze Pattern
Key Vocabulary T
A frieze pattern or
border pattern is a
pattern that extends to
the left and right in such
a way that the pattern
can be mapped onto
itself by a horizontal
translation.
Tessellations are
isometries, because the
lengths of the preimage
and image are
unchanged.
Translation
TR
Translation 180 degree rotation
TG
Translation and horizontal glide
reflection
TV
Translation and vertical line reflection
THG
Translation, 180 degree rotation, and
horizontal glide reflection
TRVG Translation, 180 degree rotation,
vertical line reflection, and horizontal glide
reflection
TRHVG Translation, 180 degree rotation,
horizontal line reflection, vertical line reflection,
and horizontal line reflection
Determining what Tessellates
Using the equation
180(n-2)/n, you can
determine what objects
tessellate
n=number of sides
If the answer you get is a
factor of 360, then the
object will tessellate
Determine if a dodecagon tessellates
180(12-2)/12
180(10)/12
1800/12
150
360/150=2.4
This is an
example of a
tesselation.
Determine the Pattern
TV
1.
2.
T or THG
1. Name the Transformation that maps
A onto B
2. Name the Transformation that maps
B onto E
A
B
E
C
F
D
3. Name the Transformation that maps
A onto E
A1. T
A2. TG
A3. THG
Works Cited
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
http://www.regentsprep.org/Regents/math/geometry/GT4/ROTATEPIC3.gif
http://deepbrazil.com/2010/10/22/brazilian-animated-gifs/
http://mathbits.com/MathBits/StudentResources/GraphPaper/GraphPaper.htm
http://ef004.k12.sd.us/ch9notes.htm
http://www.regentsprep.org/Regents/math/geometry/GT2/PracT.htm
http://www.regentsprep.org/Regents/math/geometry/GT2/Trans.htm
http://www.mathportal.org/linear-algebra/matrices/matrix-operations.php
http://endtheneglect.org/2010/07/announcing-the-winner-of-our-brazil-soccer-jersey-giveaway/
http://www.mathsisfun.com/geometry/translation.html
https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3Abcbfb8debbaad8102081a82ec6135e11bdd5fc8760196bfef6b45654%2BIMAGE%2BIMAGE.1
http://www.mathsisfun.com/geometry/reflection.html
http://upload.wikimedia.org/wikipedia/commons/4/49/Rotation_illustration.png
http://commons.wikimedia.org/wiki/File:Flag_of_South_Africa.svg
http://designblog.rietveldacademie.nl/?p=17009
Modified version of: http://cimg1.ck12.org/datastreams/fd%3A2fc873cb473d2fbe6a33d203b304a984a1f0662b1ec24fc093172546%2BIMAGE%2BIMAGE.1
http://cimg2.ck12.org/datastreams/f-d%3Aadb5c35674c07f6d68b1ca752a45b25e35804f9993f8838532b4f2d9%2BIMAGE%2BIMAGE.1
Modified Version of: http://sirjohnalexandermacdonald.org/wp-content/uploads/2013/12/bangladesh-flag.jpg
•
•
•
•
•
•
http://upload.wikimedia.org/wikipedia/commons/thumb/6/63/HinduSwastika.svg/220px-HinduSwastika.svg.png
Modified Version of: http://www.free-world-maps.com/map-images/source-physical-world-map-b.jpg
Modified version of: http://www.photo.com.bd/main.php?g2_view=core.DownloadItem&g2_itemId=8488&g2_serialNumber=2
https://sites.google.com/site/wcodavisgeometry/about-me/gsp-activity-10-tessellations-using-only-translations
http://www.bigbenfacts.com/images/big%20ben%20london.jpg
http://wallpaprezt.com/wp-content/uploads/2014/03/Manchester-United-Logo-Full-HD-Wallpaper-4.jpg
•
http://www.regentsprep.org/Regents/math/geometry/GT3/Dtigers.gif