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Transformations 3-6, 3-7, & 3-8

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

- 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.

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.

- 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’

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

Right 4 (positive change in x)

B

Down 3

(negative

change in y)

B’

C

A

A’

C’

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 unitsb.) 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 upb.) 2 units right and 1 unit down

Rule (x, y) (x – 3, y + 5)

Rule (x, y) (x + 2, y – 1)

Another name for a FLIP

A

A’

C

B

B’

C’

Used to create SYMMETRY on the coordinate plane

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

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: 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: 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)

- Sort the letters of the alphabet into groups according to their symmetries
- Divide letters into two categories:
- symmetrical
- not symmetrical

- 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

Another name for a TURN

C’

B’

B

A’

C

A

A transformation that turns about a fixed point

The fixed point

C’

B’

B

A’

C

A

(0,0)

Measuring the degrees of rotation

C’

B’

B

90 degrees

A’

C

A

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)

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.

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.

There are 5 lines of symmetry in this design.

360 degrees divided by 5 =

72º

The angle of rotation is 72º.

There are NO lines of symmetry in this design.

120º

The angle of rotation is 120º.

(360 / 3)

There are NO lines of symmetry in this design.

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.

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

36º

The angle of rotation is 36º.

There are 10 lines of symmetry in this design.

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

.

There are 9 lines of symmetry in this design.

40º

The angle of rotation is 40º.

There are 9 lines of symmetry in this design.

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

- Pg 138 #8, 12, 18, & 22
- Pg 143 #8, 10, 16, & 18
- Pg 148 #6, 8, 10

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

Formed by a combination of TRANSLATIONS, REFLECTIONS, and ROTATIONS

A tessellation that uses only ONE shape

A design that covers a plane using more than one shape

Used famously in artwork by M.C. Escher

- 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