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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. B. C’. B’. Image. Original.

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Transformations 3 6 3 7 3 8

Transformations3-6, 3-7, & 3-8


Transformation
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
Image

The figure you get after a translation

A

A’

Slide

C

B

C’

B’

Image

Original

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
Translation

  • a transformation that moves each point of a figure the same distance and in the same direction.

    AKA - SLIDE

A

A’

C

C’

B

B’


Writing a rule for a translation
Writing a Rule for a Translation

Finding the amount of movement LEFT and RIGHT and UP and DOWN


Writing a rule
Writing a Rule

Right 4 (positive change in x)

B

Down 3

(negative

change in y)

B’

C

A

A’

C’


Writing a rule1
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.

Cl = (4, -1)

Al = (1, -2)

Bl = (5, 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.

Kl = (-1, 9)

Ll = (1, 6)

Jl = (-4, 7)

-You can use arrow notation to describe a translation.

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.

K’

J’

L’

K

J

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

Al

A

Bl


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 b.) 2 units right and 1 unit down

Rule (x, y) (x – 3, y + 5)

Rule (x, y) (x + 2, y – 1)


Reflection
Reflection

Another name for a FLIP

A

A’

C

B

B’

C’


Reflection1
Reflection

Used to create SYMMETRY on the coordinate plane


Symmetry
Symmetry

When one side of a figure is a MIRROR IMAGE of the other


Line of reflection
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 y-axis.

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.) b.)

1

6


Example 2: is reflected over the y-axis and all of the points are the same distance away from the y-axis. 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

A

B

Bl

Al


Example 3: is reflected over the y-axis and all of the points are the same distance away from the y-axis. 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
Symmetry of the Alphabet is reflected over the y-axis and all of the points are the same distance away from the y-axis.

  • Sort the letters of the alphabet into groups according to their symmetries

  • Divide letters into two categories:

    • symmetrical

    • not symmetrical


Symmetry of the alphabet1
Symmetry of the Alphabet is reflected over the y-axis and all of the points are the same distance away from the y-axis.

  • 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
Rotation is reflected over the y-axis and all of the points are the same distance away from the y-axis.

Another name for a TURN

C’

B’

B

A’

C

A


Rotation1
Rotation is reflected over the y-axis and all of the points are the same distance away from the y-axis.

A transformation that turns about a fixed point


Center of rotation
Center of Rotation is reflected over the y-axis and all of the points are the same distance away from the y-axis.

The fixed point

C’

B’

B

A’

C

A

(0,0)


Rotating a figure
Rotating a Figure is reflected over the y-axis and all of the points are the same distance away from the y-axis.

Measuring the degrees of rotation

C’

B’

B

90 degrees

A’

C

A


Rotations in a coordinate plane
Rotations in a Coordinate Plane is reflected over the y-axis and all of the points are the same distance away from the y-axis.

This transformation can be described as (x, y) (–y, x).

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.

Figure A'B'C'D'

Figure ABCD

A(2, –2)

A'(2, 2)

B(4, 1)

B'(–1, 4)

C(5, 1)

C'(–1, 5)

D(5, –1)

D'(1, 5)


Rotational symmetry can be found in many objects that rotate about a centerpoint
Rotational symmetry can be found in many objects that rotate about a centerpoint.

Determine the angle of rotation for each hubcap. Explain how you found the angle.

Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.


Hubcap 1
Hubcap 1 about a

Determine the angle of rotation for each hubcap. Explain how you found the angle.

Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.


Hubcap 11
Hubcap 1 about a

There are 5 lines of symmetry in this design.

360 degrees divided by 5 =


Hubcap 12
Hubcap 1 about a

72º

The angle of rotation is 72º.


Hubcap 2
Hubcap 2 about a

There are NO lines of symmetry in this design.


Hubcap 21
Hubcap 2 about a

120º

The angle of rotation is 120º.

(360 / 3)

There are NO lines of symmetry in this design.


Hubcap 3
Hubcap 3 about a

Determine the angle of rotation for each hubcap. Explain how you found the angle.

Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.


Hubcap 31
Hubcap 3 about a

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 32
Hubcap 3 about a

36º

The angle of rotation is 36º.

There are 10 lines of symmetry in this design.


Hubcap 4
Hubcap 4 about a

Determine the angle of rotation for each hubcap. Explain how you found the angle.

Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.


Hubcap 41
Hubcap 4 about a

.

There are 9 lines of symmetry in this design.


Hubcap 42
Hubcap 4 about a

40º

The angle of rotation is 40º.

There are 9 lines of symmetry in this design.


Think About it: about a 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 rotationWhen 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
Homework symmetry = angle of rotation

  • Pg 138 #8, 12, 18, & 22

  • Pg 143 #8, 10, 16, & 18

  • Pg 148 #6, 8, 10


Tessellation
Tessellation symmetry = angle of rotation

A design that covers a plane with NO GAPS and NO OVERLAPS


Tessellation1
Tessellation symmetry = angle of rotation

Formed by a combination of TRANSLATIONS, REFLECTIONS, and ROTATIONS


Pure tessellation
Pure Tessellation symmetry = angle of rotation

A tessellation that uses only ONE shape


Pure tessellation1
Pure Tessellation symmetry = angle of rotation


Pure tessellation2
Pure Tessellation symmetry = angle of rotation


Semiregular tessellation
Semiregular Tessellation symmetry = angle of rotation

A design that covers a plane using more than one shape


Semiregular tessellation1
Semiregular Tessellation symmetry = angle of rotation


Semiregular tessellation2
Semiregular Tessellation symmetry = angle of rotation


Semiregular tessellation3
Semiregular Tessellation symmetry = angle of rotation


Semiregular tessellation4
Semiregular Tessellation symmetry = angle of rotation


Tessellation2
Tessellation symmetry = angle of rotation

Used famously in artwork by M.C. Escher


Group activity
Group Activity symmetry = angle of rotation

  • 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


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