Transformations
3-6, 3-7, & 3-8
Transformation
•Movements of a figure in a
plane
•May be a SLIDE, FLIP, or
TURN
•a change in the position,
shape, or size of a figure.
Image
The figure you get after a translation
A
A’
Slide
C
Original
B
C’
Image
B’
To identify the image of point A, use prime notation Al.
You read Al as “A prime”.
The symbol ‘ is read “prime”.
ABC has been moved to
A’B’C’.
A’B’C’ is the image of
ABC.
Translation
• a transformation that moves each
point of a figure the same distance
and in the same direction.
AKA - SLIDE
A
C
A’
B
C’
B’
Writing a Rule for
a Translation
Finding the amount
of movement LEFT
and RIGHT and UP
and DOWN
Writing a Rule
9
8
B
Right 4 (positive change in x)
7
Down 3
(negative
change in y)
6
5
B’
4
A
C
3
2
1
0
A’
1
2
3
4
5
6
7
C’
8
9
Writing a Rule
Can be written as:
R4, D3
(Right 4, Down 3)
Rule: (x,y) (x+4, y-3)
Translations
Example 1: If triangle ABC below is translated 6 units to the right and 3
units down, what are the coordinates of point Al.
A (-5, 1)
B (-1, 4)
C (-2, 2)
Rule (x+6, y-3)
-First write the rule and then translate each point.
Al
Bl
= (1, -2)
= (5, 1)
Cl = (4, -1)
-Now graph both triangles and see if your image points are correct.
B
A
C
B’
C’
A’
Example 2: Triangle JKL has vertices J (0, 2), K (3, 4), L (5, 1). Translate
the triangle 4 units to the left and 5 units up. What are the new coordinates
of Jl?
-First graph the triangle and then translate each point.
Jl
l = (-1, 9)
K
= (-4, 7)
Ll = (1, 6)
-You can use arrow notation to
describe a translation.
K’
For example: (x, y)
(x – 4, y + 5)
shows the ordered pair (x, y) and
describes a translation to the left 4
unit and up 5 units.
J’
L’
J
K
L
You try some:
Graph each point and its image after the given translation.
a.) A (1, 3) left 2 units
b.) B (-4, 4) down 6 units
Al (-1, 3)
Bl (-4, -2)
B
Bl
Al
A
Example 3: Write a rule that describes the translation below
Point A (2, -1)
Al (-2, 2)
Point B (4, -1)
Bl (0, 2)
Point C (4, -4)
Cl (0, -1)
Point D (2, -4)
Dl (-2, -1)
Rule (x, y)
(x – 4, y + 3)
Example 4: Write a rule that describes each translation below.
a.) 3 units left and 5 units up
Rule (x, y)
(x – 3, y + 5)
b.) 2 units right and 1 unit down
Rule (x, y)
(x + 2, y – 1)
Reflection
Another name for a
FLIP
A
C
A’
B
B’
C’
Reflection
Used to create
SYMMETRY on the
coordinate plane
Symmetry
When one
side of a
figure is
a
MIRROR
IMAGE of
the
other
Line of Reflection
The line you
reflect a
figure
across
Ex: X or Y
axis
X - axis
In the diagram to the left you will
notice that triangle ABC is reflected
over the y-axis and all of the points are
the same distance away from the yaxis.
Therefore triangle AlBlCl is a reflection
of triangle ABC
Example 1: Draw all lines of reflection for the figures below. This is a
line where if you were to fold the two figures over it they would line up.
How many does each figure have?
a.)
1
b.)
6
Example 2: Graph the reflection of each point below over each line of
reflection.
a.) A (3, 2) is reflected over the x-axis
B
b.) B (-2, 1) is reflected over the y-axis
A
Bl
Al
Example 3: Graph the triangle with vertices A(4, 3), B (3, 1), and C (1, 2).
Reflect it over the x-axis. Name the new coordinates.
C
A
B
C’ (1,-2)
B’ (3,-1)
A’ (4,-3)
Symmetry of the Alphabet
• Sort the letters of the alphabet into
groups according to their symmetries
• Divide letters into two categories:
• symmetrical
• not symmetrical
Symmetry of the Alphabet
• Symmetrical: A, B, C, D, E, H, I, K, M,
N, O, S, T, U, V, W, X, Y, Z
• Not Symmetrical: F, G, J, L, P, Q, R
Rotation
Another name for a
TURN
C’
B
A’
C
A
B’
Rotation
A transformation
that turns about a
fixed point
Center of Rotation
The fixed point
C’
B’
B
C
A’
A
(0,0)
Rotating a Figure
Measuring the
degrees of rotation
C’
B’
B
A’
C
A
90 degrees
Rotations in a Coordinate Plane
In a coordinate plane, sketch the quadrilateral whose vertices are A(2, –2),
B(4, 1), C(5, 1), and D(5, –1). Then, rotate ABCD 90º counterclockwise about
the origin and name the coordinates of the new vertices. Describe any
patterns you see in the coordinates.
SOLUTION
Plot the points, as shown in blue. Use a
protractor, a compass, and a straightedge to find
the rotated vertices. The coordinates of the
preimage and image are listed below.
In the list, the x-coordinate of the image
is the opposite of the y-coordinate of the
preimage. The y-coordinate of the image
is the x-coordinate of the preimage.
This transformation can be
described as (x, y)
(–y, x).
Figure ABCD
A(2, –2)
B(4, 1)
C(5, 1)
D(5, –1)
Figure A'B'C'D'
A '(2, 2)
B '(–1, 4)
C '(–1, 5)
D '(1, 5)
Rotational
symmetry can
be found in
many objects
that rotate
about a
centerpoint.
A.
Determine the angle of
rotation for each hubcap.
Explain how you found the
angle.
B.
Some of the hubcaps also
have reflectional symmetry.
Sketch all the lines of
symmetry for each hubcap.
Hubcap 1
A. Determine the
angle of rotation
for each hubcap.
Explain how you
found the angle.
B. Some of the
hubcaps also have
reflectional
symmetry. Sketch
all the lines of
symmetry for each
hubcap.
Hubcap 1
There are 5
lines of
symmetry in
this design.
360 degrees
divided by 5 =
Hubcap 1
The angle of
rotation is
72º.
72º
Hubcap 2
There are NO
lines of
symmetry in
this design.
Hubcap 2
The angle of
rotation is
120º.
(360 / 3)
There are NO
lines of
symmetry in
this design.
120º
Hubcap 3
A.
Determine the angle of
rotation for each hubcap.
Explain how you found the
angle.
B.
Some of the hubcaps also
have reflectional symmetry.
Sketch all the lines of
symmetry for each hubcap.
Hubcap 3
There are 10
lines of
symmetry in
this design.
360 / 10 = 36
However to make it look
exactly the same you need
to rotate it 2 angles.
36 x 2 = 72
Hubcap 3
A.The angle of
rotation is
36º.
B.There are 10
lines of
symmetry in
this design.
36º
Hubcap 4
A.
Determine the angle of
rotation for each hubcap.
Explain how you found the
angle.
B.
Some of the hubcaps also
have reflectional symmetry.
Sketch all the lines of
symmetry for each hubcap.
Hubcap 4
A. .
B.There are 9
lines of
symmetry in
this design.
Hubcap 4
A.The angle of
rotation is
40º.
B.There are 9
lines of
symmetry in
this design.
40º
Think About it:
Is there a way
to determine
the angle of
rotation for a
particular
design without
actually
measuring it?
When there are
lines of symmetry
360 ÷ number of
lines of symmetry
= angle of rotation
When there are no
lines of symmetry:
360 ÷ number of
possible rotations
around the circle.
5 lines of
symmetry
3 points to
rotate it to
Homework
• Pg 138 #8, 12, 18, & 22
• Pg 143 #8, 10, 16, & 18
• Pg 148 #6, 8, 10
Tessellation
A design that
covers a plane
with NO GAPS and
NO OVERLAPS
Tessellation
Formed by a
combination of
TRANSLATIONS,
REFLECTIONS, and
ROTATIONS
Pure Tessellation
A tessellation
that uses only
ONE shape
Pure Tessellation
Pure Tessellation
Semiregular
Tessellation
A design that
covers a plane
using more than
one shape
Semiregular Tessellation
Semiregular Tessellation
Semiregular Tessellation
Semiregular Tessellation
Tessellation
Used famously in
artwork by M.C.
Escher
Group Activity
• Choose a letter (other than R) with
no symmetries
• On a piece of paper perform the
following tasks on the chosen letter:
• rotation
• translation
• Reflection