Dec. 14

HW 18: Transformations

Aim: Working with Dilation, Reflection, Translations, and Rotations. Review from 7th Accelerated.

Materials you will need for this homework:•pencil •ruler

There are 4 types of transformation:

1. Dilation 2. Reflection 3. Rotation4. Translation

A transformation in geometry is defined as when an object (shape) undergoes a change in position or size.

Definitions:

2. Dilation

A dilation is a type of transformation that changes the size of the image but the image is the same shape. To make an image smaller or larger you multiply by the scale factor. The orientation of the image is the same as the original figure.

Try out the example on the website

A reflection is a kind of transformation. It is basically a “flip” of a shape over the line of reflection to create a mirror image of the shape. The image is congruent to the original shape. Orientation of the image is different from the original figure.

3. Reflection

-6 -3 3 6

6

3

-3

-6

y

x

A B

C

D2  A(1, -3) A’ ________  B(3, -3) B’________ C(2, -1) C’________  

Ex1. Dilate triangle ABC, with vertices A(1, -3), B(3, -3), and C(2, -1) with a scale factor of 2. Then write the new coordinates as A’B’C’ in the table below.

A’ B’

C’

Rule for Dilation:

Multiply each coordinate (x, y) by the scale factor.

D2

Notice that the base and height of the triangles are in the ratio of 1:2

(2, -6)

(6, -6)

(6, -2)

-6 -3 3 6

6

3

-3

-6

y

x

Ex2. Dilate rectangle BRAT, with vertices B(-6, 6), R(3, 6), A(3, 3), and T(-6, 3) with a scale factor of . Then write the new coordinates as B’R’A’T’ in the table below.

B(-6, 6) B’ ________  R(3, 6) R’_________ A(3, 3) A’_________ T(-6, 3) T’_________

B

R

AT

B’ R’

A’T’

D

Notice that the lengths and widths are in the ratio of 1:3

(-2, 2)

(1, 2)

(1, 1)

(-2, 1)

Ex3. Reflect Trapezoid MATH over the x-axis and label it M’A’T’H’

Rover x

M(1, 6) M’_________

A(3, 6) A’_________ T(5, 2) T’_________

H(1, 2) H’_________

y

x-6

-4

-2

2 4 6

6

4

2

-2

-4

-6

M A

TH

Rule (Reflecting over the x-axis): (x,y) (x, -y)

M’ A’

T’H’

When reflecting over the x-axis, keep the x coordinate the same and negate the y coordinate.

Rx

(1, -6)

(3, -6)

(5, -2)

(1, -2)

Rover y

M(1, 6) M”_________

A(3, 6) A” _________ T(5, 2) T”_________

H(1, 2) H”_________

Rule (Reflecting over the y-axis):

(x,y) (-x, y)

Ex4. Reflect Trapezoid MATH over the y-axis and label it M’’A’’T’’H’’y

-6

-4

-2

2 4 6

6

4

2

-2

-4

-6

M A

TH

M”A”

T” H”

x

When reflecting over the y-axis, negate the x coordinate and keep the y coordinate the same.

Ry(-1, 6)

(-3, 6)

(-5, 2)

(-1, 2)

4.  A translation is another type of transformation. It is the same as sliding/shifting an object. The notation for translate is T(a, b) where a and b represent how much you slide in the x and the y directions, respectively. The shape still looks exactly the same, just in a different place.

5. Another type of transformation is a rotation. A rotation turns a figure 90 degrees or 180 degrees clockwise or counterclockwise. The image is congruent to the original figure. Orientation of the image is different from the original figure.

-6 -4 -2 2 4 6

6

4

2

-2

-4

-6

y

x

T5, -3 S(-6, 6) _________

U(-6, 2) _________

N(-2, 2) _________

Notation for example problem:T5, -3 means 5 units right and 3 units down and label the new coordinates.

Example 4Translate Triangle SUN 5 units right and 3 units down and label the new coordinates.

S

U N

S’

U’ N’

Translations

+5 -3

S’(-1, 3)

There’s 2 ways to translate a figure. Using the graph or the notation.

+5 -3

+5 -3

U’(-1, 1)

N’(3, -1)

-6 -3 3 6

6

3

-3

-6

y

x

Example 5a.) Translate Trapezoid BIRD 2 units left and 6 units up and label the new figure B’I’R’D’.b.) Write the translation above using proper notation.

BI

R D

T B(6, -6) __________

I(1, -6) __________

R(2, -2) __________ D(4, -2) __________

B’I’

R’ D’

-2, 6

-2 +6

B’(4, 0)

-2 +6

-2 +6

I’(-1, 0)

R’(0, 4)

-2 +6

D’(2, 4)

Example 6.What type of transformation is shown in the diagram? Explain your answer.

The type of transformation that is shown in the diagram is a translation.The object is being translated 7 units to the right and 5 units down.

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

A Rotation of 90° counterclockwise about (0,0)

xxxxx

What did you noticed about the points in the original triangle and the new triangle?

(2, 1) ______

(4, 2) ______

(3, 5) ______

(-1, 2)

(-2, 4)

(-5, 3)

Rule for rotation of 90°____________________________________________________________________________________________

Notation: (x, y) (-y, x)Negate the y-coordinate then flip the x and y coordinates.

R90  N(1, 2) ________  H(5, 2) _________ O(6, 6) _________  T(2, 6) _________

Ex7. Rotate 90 counterclockwise Parallelogram NHOT with vertices: N(1, 2), H(5, 2), O(6, 6), T(2, 6)

-6 -3 3 6

6

3

-3

-6

y

N

T O

HN’

T’

O’

H’

x

N’(-2, 1)

H’(-2, 5)

O’(-6, 6)

T’(-6, 2)

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

A Rotation of 180° about (0,0)

xxxxx

xxx

What did you noticed about the points in the original triangle and the new triangle?

(2, 1) ______

(4, 2) ______

(3, 5) ______

(-2, -1)

(-4, -2)

(-3, -5)

Rule for rotation of 180°________________________________________________________________________________

Notation: (x, y) (-x, -y)Negate the x-coordinate and y-coordinate.

R180  B(6, -6) ________  I(1, -6) _________ R(2, -2) _________  D(4, -2) _________

Ex8. Rotate 180 counterclockwise Trapezoid BIRD.

-6 -3 3 6

6

3

-3

-6

y

x

D

IR

R

D’

I’ R’

R’

B’(-6, 6)

I’(-1, 6)

R’(-2, 2)

D’(-4, 2)

Ex9. What transformation is shown on the grid? Explain your answer.

-6 -3 3 6

6

3

-3

-6

y

x

H

Y

E

E’

H’

Y’

The transformation that is shown on the grid is a rotation of 90 degrees clockwise.