Transcript File
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 A l . You read A l 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 A’ C B C’ B’
Writing a Rule for a Translation
Finding the amount of movement LEFT and RIGHT and UP and DOWN
Writing a Rule 6 5 4
A
3 2 1 0 9 8 7 1
B
2
Right 4 (positive change in x) C
3
A’
4
B’
5 6
Down 3 (negative change in y)
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 A l .
A (-5, 1) B (-1, 4) C (-2, 2) Rule (x+6, y-3)
-First write the rule and then translate each point.
A l = (1, -2) B l = (5, 1) C l = (4, -1) -Now graph both triangles and see if your image points are correct.
A B 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 J l ?
-First graph the triangle and then translate each point.
J l = (-4, 7) -You can use arrow notation to describe a translation.
K l = (-1, 9) L l = (1, 6)
K’ J’ L’
For example: (x, y) unit and up 5 units.
(x – 4, y + 5) shows the ordered pair (x, y) and describes a translation to the left 4
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
A l (-1, 3) B l (-4, -2) B A l A B l
Example 3: Write a rule that describes the translation below Point A (2, -1) Point B (4, -1) Point C (4, -4) Point D (2, -4) A l (-2, 2) B l (0, 2) C l (0, -1) D l (-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 b.) 2 units right and 1 unit down
Rule (x, y) (x – 3, y + 5) Rule (x, y) (x + 2, y – 1)
Reflection
Another name for a FLIP
A A’ C 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
1 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 y axis.
Therefore triangle A l B l C l of triangle ABC is a reflection 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.) 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 (-2, 1) is reflected over the y-axis
B B l A A l
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’ (1,-2) C A B 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’ B A’ C A
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 C A 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 patterns you see in the coordinates.
ABCD
90º counterclockwise about the origin and name the coordinates of the new vertices. Describe any 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
72º The
angle of rotation is 72º .
Hubcap 2
There are NO lines of symmetry in this design.
Hubcap 2
120º The
angle of rotation is 120º . (360 / 3)
There are NO lines of symmetry in this design.
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
36º A.The
36º.
angle of rotation is
B.There are 10 lines of symmetry in this design.
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
40º A.The
40º.
angle of rotation is
B.There are 9 lines of symmetry in this design.
Think About it: Is there a way to determine the angle of rotation actually for a particular design without 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