Geometry Workbook 3:

Transformations

Student Name __________________________________________

STANDARDS:

G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular line, parallel lines, and line segment.

G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

SKILLS:

I will know the definitions of the isometric transformations (reflect, rotate, & translate).

I will be able to describe rotations, reflections and translations.

I will be able to determine and apply the properties of the isometric transformations.

I will be able to identify which transformation has taken place based on the properties found

between the pre-image and image.

I will be able to identify the orientation relationship between the pre-image and image.

I will be able to perform a reflection, a rotation, and a translation.

I will be able to perform a sequence of transformations.

I will be able to determine the sequence of transformations performed between a given pre-

image and image.

I will be able to identify a transformation by its coordinate rule and then apply it to transform the

shape.

I will be able to demonstrate how some composite transformations are not commutative.

Notes:

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G.CO.A.4 STUDENT GUIDED PRACTICE WS #1/#2 – geometrycommoncore.com 1

THE ISOMETRIC TRANSFORMATIONS

THE REFLECTION

The line of reflection is the perpendicular bisector of the

segment joining every point and its image.

' ' 'mr ABC A B C

CHARACTERISTICS

DISTANCES FROM PRE-IMAGE TO IMAGE

ORIENTATION

SPECIAL POINTS

THE ROTATION

A rotation is an isometric transformation that turns a figure about a fixed point called the center of rotation (notation: Rcenter, degree).

An object and its rotation are the same shape and size, but the figures may be turned in different directions.

, ' ' 'OR ABC A B C

CHARACTERISTICS

DISTANCES FROM PRE-IMAGE TO IMAGE

ORIENTATION

SPECIAL POINTS

m

B'

C'

A'A

C

B

θ

B'A'

C'C

A

B

O

G.CO.A.4 STUDENT GUIDED PRACTICE WS #1/#2 – geometrycommoncore.com 2

ROTATION DIRECTION

EQUIVALENT ROTATIONS

Coterminal angle = initial angle + 360n

SPECIAL ROTATION – ROTATION OF 180

A rotation of 180 maps A to A’ such that:

a) mAOA’ = 180 (from definition of rotation)

b) OA = OA’ (from definition of rotation)

c) 𝑂𝐴⃗⃗⃗⃗ ⃗ and 𝑂𝐴′⃗⃗⃗⃗⃗⃗ ⃗ are opposite rays. (They form a line.)

𝐴𝑂⃡⃗⃗⃗ ⃗ is the same line as 𝐴𝐴′⃡⃗ ⃗⃗ ⃗⃗

THE TRANSLATION

The pre-image and image have the same shape and size.

, ' ' 'x yT ABC A B C

A'

A

O

B'

A'

C'C

A

B

G.CO.A.4 STUDENT GUIDED PRACTICE WS #1/#2 – geometrycommoncore.com 3

DEFINITION

A translation is an isometric transformation that maps every two points A and B in the plane to points A’ and B’, so that the following properties are true; 1. AA’ = BB’ (a fixed distance).

2. '|| 'AA BB (a fixed direction).

CHARACTERISTICS

DISTANCES FROM PRE-IMAGE TO IMAGE

ORIENTATION

SPECIAL POINTS

SPECIAL TRANSLATION PROPERTY – TRANSLATING AN ANGLE ALONG ONE OF ITS RAYS

A translation of ABC by vector 𝐵𝐴⃗⃗⃗⃗ ⃗ maps all points such that

1. ABC A’B’C’ (Isometry) 2. B, A, B’ and A’ are collinear (translation on angle ray) Because the two angles are equal and formed on the same ray, then:

BC || ' 'B C

This is a key property to translations – All segments that are translated are parallel to each other.

B'

A'A

B

B C

A

A'

C'

B C

A = B'

G.CO.A.4 WORKSHEET #1 – geometrycommoncore.com NAME: _____________________ 1

1. Which transformation has taken place?

a) ___________________

b) ___________________

c) ___________________

d) ___________________

2. Complete the chart.

Relationship between pre-image and image

ROTATION REFLECTION TRANSLATION

Distances

SAME OR DIFFERENT SAME OR DIFFERENT SAME OR DIFFERENT

Orientation

SAME OR DIFFERENT SAME OR DIFFERENT SAME OR DIFFERENT

Special Points

3. Given that ABC was mapped to A’B’C’ using a single transformation.

a) Why couldn’t this mapping have resulted by a single translation? b) What transformation must have mapped these two triangles? Explain your answer.

4. Given that ABC was mapped to A’B’C’ using a single transformation.

a) Why couldn’t this mapping have resulted by a single reflection? b) What transformation must have mapped these two triangles? Explain your answer.

B'

D'C'

C

D

B

D'

C'C

D

B

B'B'

D'

C'

C

D

B

D'

F'

E'E

F

D

C'

B'

A'A

B

C

A'

C'

B'B

C

A

G.CO.A.4 WORKSHEET #1 – geometrycommoncore.com 2

5. ABC is congruent to A’B’C’. A student tries to determine which of these single transformations

mapped ABC onto A’B’C’. She concludes that a reflection had to be involved and more than one transformation had to map these two triangles.

a) How can she conclude that a reflection was involved? b) How can she conclude that this wasn’t just a single reflection?

6. Determine the location of Point A,

a) after a reflection A = A’, where was point A? __________________________________

b) after a rotation of 27 A = A’, where was point A? _________________________________

7. After a reflection AA’ = 24 cm, how far was A away from the line of reflection? ______________

8. If after a reflection A = A’ and BB’ = 6 cm, what is the relationship between BAB’ and the line of

reflection? Draw a diagram.

9. The distance from point A to the line of reflection is 10 cm, and the distance from point B to the line of

reflection is also 10 cm. Jeffrey concludes that B is the image of A under a reflection. What do you think of

this conclusion?

10 . BC was translated by the arrow, making ' 'BC B C and

|| ' 'BC B C .

a) What other segments in the diagram are congruent? _____________ b) What other segments in the diagram are parallel? _______________

C'

B'

A'

A

B

C

C'

C

B

B'

G.CO.A.5 GUIDED PRACTICE WS #1 – geometrycommoncore.com 1

1. Reflect the following over the given line of reflection.

2. Determine the pre-image coordinates, then reflect it, and determine the image coordinates.

a) A = (-2, 5) ( )x axisr A A ‘ = ( ____ , ____)

b) B = (5, 7) ( )y axisr B B ‘ = ( ____ , ____)

c) C = (3, -4) ( )mr C C ‘ = ( ____ , ____)

d) D = (4, 4) ( )x axisr D D ‘ = ( ____ , ____)

e) A = (-2, 5) ( )nr A A ‘ = ( ____ , ____)

f) B = (5, 7) ( )mr B B ‘ = ( ____ , ____)

g) C = (3, -4) ( )y axisr C C ‘ = ( ____ , ____)

3. Determine the name of the point that meets the given conditions.

a) ( )mr A ________ b) ( )nr C ________

c) ( )rr D ________ d) (______)pr B

e) ( )nr H ________ f) ( )mr B ________

g) ( )pr G ________ h) (______)pr C

n

m

A

B

C

D

r

p

n

m

A

C

E

G

BD

FH

G.CO.A.5 WORKSHEET #1 – geometrycommoncore.com NAME: _________________________ 1

1. Use the grid or patty paper to reflect the following figures over their respective line m. Label the image.

G.CO.A.5 WORKSHEET #1 – geometrycommoncore.com 2

2. Determine the line of reflection for the following pre-image and images.

a) b) c)

d) e) f)

3. Determine the pre-image coordinates, then reflect it, and determine the image coordinates.

a) A = ( ____ , ____) ( )x axisr A A ‘ = ( ____ , ____)

b) B = ( ____ , ____) ( )y axisr B B ‘ = ( ____ , ____)

c) C = ( ____ , ____) ( )mr C C ‘ = ( ____ , ____)

d) D = ( ____ , ____) ( )x axisr D D ‘ = ( ____ , ____)

e) E = ( ____ , ____) ( )x axisr E E ‘ = ( ____ , ____)

f) F = ( ____ , ____) ( )nr F F ‘ = ( ____ , ____)

g) G = ( ____ , ____) ( )y axisr G G ‘ = ( ____ , ____)

n

m

A

B

C

D

E

F

G

G.CO.A.5 WORKSHEET #1 – geometrycommoncore.com 3

4. Determine the name of the point that meets the given conditions.

a) ( )mr A ________ b) ( )hr C ________

c) ( )hr D ________ d) (______)gr B

e) ( )nr D ________ f) ( )nr B ________

g) ( )mr D ________ h) (______)mr C

5. Determine the name of the point that meets the given conditions.

a) ( )mr A ________ b) ( )FC

r A ________

c) ( )hr B ________ d) (______)AD

r B

e) ( )nr D ________ f) ( )nr B ________

g) ( )FC

r D ________ h) (______)BE

r C

h

g

nm

CB

A D

h

n

m

A

F E

D

B C

G.CO.A.5 GUIDED PRACTICE WS #2 – geometrycommoncore.com 1

1. Rotate the following.

2. Perform the rotation and then determine the image coordinates.

,90 ( )OR ABC

a) A = (-2, 5) ,90 ( )OR ABC A ‘ = ( ____ , ____)

b) B = (5, 7) ,90 ( )OR ABC B ‘ = ( ____ , ____)

c) C = (3, 4) ,90 ( )OR ABC C ‘ = ( ____ , ____)

, 90 ( )OR DEF

d) D = (6, 5) , 90 ( )OR DEF D ‘ = ( ____ , ____)

e) E = (8, -2) , 90 ( )OR DEF E ‘ = ( ____ , ____)

f) F = (6, -2) , 90 ( )OR DEF F ‘ = ( ____ , ____)

3. Determine the name of the point that meets the given conditions.

a) ,90 ( )OR A ________ b) ,180 ( )OR B ________

c) ,270 ( )OR D ________ d) , 180 (______)OR H

e) , 90 ( )OR E ________ f) , 90 ( )OR F ________

g) ,90 ( )FR E ________ h) ,180 ( )BR A ________

(notice new center of rotation) (notice new center of rotation)

AC

B

D

EF

O

OA

C

E

G

BD

FH

G.CO.A.5 WORKSHEET #2 – geometrycommoncore.com NAME: ___________________________ 1

1. Use the grid or patty paper to rotate the following figures. Label the image.

G.CO.A.5 WORKSHEET #2 – geometrycommoncore.com 2

2. Circle the center of rotation for the following pre-image and images.

a) A rotation of 90 b) A rotation of 180 c) A rotation of 180

3. Determine the pre-image coordinates, then rotate it, and determine the image coordinates. Patty paper

will help determine the image coordinates. (Patty paper might be helpful here.)

,90 ( )OR ABC

a) A = ( ____ , ____) ,90 ( )OR ABC A ‘ = ( ____ , ____)

b) B = ( ____ , ____) ,90 ( )OR ABC B ‘ = ( ____ , ____)

c) C = ( ____ , ____) ,90 ( )OR ABC C ‘ = ( ____ , ____)

,90 ( )OR DFE

d) D = ( ____ , ____) ,90 ( )OR DFE D ‘ = ( ____ , ____)

e) E = ( ____ , ____) ,90 ( )OR DFE E ‘ = ( ____ , ____)

f) F = ( ____ , ____) ,90 ( )OR DFE F ‘ = ( ____ , ____)

4. Determine the name of the point that meets the given conditions.

a) ,60 ( )GR A ________ b) ,180 ( )GR B ________

c) ,300 ( )GR D ________ d) , 120 (______)GR B

e) ,240 ( )GR E ________ f) , 240 ( )GR F ________

g) ,60 ( )AR B ________ h) ,120 ( )CR D ________

A = A'

B

B'

E'B'

B

E

D = D'

B'

E'

B

E

A

C

B

D

E

F

O

G.CO.A.5 GUIDED PRACTICE WS #3 – geometrycommoncore.com 1

1. Translate the following.

2. Determine the translation rule from the pre-image and image.

Coordinate Notation Vector Notation

a) A (-4, 1) A’ (-1, 3) T (x, y) ------------- > (_________ , _________) T< ____ , ____> (x, y)

(b) A (7, 1) A’ (3, 0) T (x, y) ------------- > (_________ , _________) T< ____ , ____> (x, y)

3. Given a translation rule, determine the missing point.

a) T (x, y) ------- > (x - 6, y – 1) A (-1, 2) A’ (_______ , ______)

b) T (x, y) ------- > (x – 1, y + 1) A (4, 3) A’ (_______ , ______)

c) T (x, y) ------- > (x, y – 4) A (_______ , ______) A’ (8, -5)

d) 1, 2 ( )T A A (9, -5) A’ (_______ , ______)

e) 5,0 ( )T A A (_______ , ______) A’ (-1, -7)

f) 4, 3 ( )T A A (0, -7) A’ (_______ , ______)

4. T (x,y) ------- > (x – 3, y + 5) followed by T (x, y) ------- > (x – 7, y – 1) results in a single translation of all

points. What would that translation be?

T<-3,-1>( LMN)

T<6,0>( EFG)T<2,-3>( JKH)

E

H

FGK

J

M

N

L

G.CO.A.5 WORKSHEET #3 – geometrycommoncore.com NAME: ___________________________ 1

1. Use the grid or patty paper to translate the following figures. Label the image.

2. Determine the translation coordinate rule from the vector.

a) T (x, y) ----> (_______,________)

b) T (x, y) ----> (_______,________)

c) T (x, y) ----> (_______,________)

3. Determine the translation rule from the pre-image and image.

a) A (3, 5) A’ (-1, 3) T (x, y) ------------- > (___________ , ___________)

b) A (-4, 11) A’ (3, 0) T (x, y) ------------- > (___________ , ___________)

c) A (0, -8) A’ (-1, -3) T (x, y) ------------- > (___________ , ___________)

d) A (8, 3) A’ (11, 3) T (x, y) ------------- > (___________ , ___________)

4. Given a translation rule, determine the missing point.

a) T (x, y) ------- > (x + 3, y – 5) A (-4, 7) A’ (_______ , ______)

b) T (x, y) ------- > (x – 7, y – 1) A (9, 1) A’ (_______ , ______)

c) T (x, y) ------- > (x + 1, y + 6) A (_______ , ______) A’ (4, -1)

G.CO.A.5 WORKSHEET #3 – geometrycommoncore.com 2

d) T (x, y) ------- > (x, y + 4) A (8, -4) A’ (_______ , ______)

e) T (x, y) ------- > (x + 3, y + 1) A (_______ , ______) A’ (-4, 1)

f) T (x, y) ------- > (x – 8, y – 5) A (_______ , ______) A’ (-3, -3)

5. Convert between vector component form and coordinate form.

a) 5,2 ( )T A T (x, y) ------- > (______________ , ______________)

b) 0, 12 ( )T A T (x, y) ------- > (______________ , ______________)

c) 1.5, 7 ( )T A T (x, y) ------- > (______________ , ______________)

6. Write the coordinate rule that matches the description.

a) 4 down and 3 right T (x, y) ------- > (______________ , ______________)

b) left 7 and down 2 T (x, y) ------- > (______________ , ______________)

c) right 1 T (x, y) ------- > (______________ , ______________)

7. What is the resultant translation of Point A after mapping T (x, y) followed by R (x, y)?

a) A (-4, 8) T (x, y) ------- > (x + 3, y – 7) A’ (_____ , _____) R (x, y) ------- > (x – 8, y – 2) A’’ (_____ , _____)

b) A (2, 0) T (x, y) ------- > (x - 1, y) A’ (_____ , _____) R (x, y) ------- > (x – 3, y + 3) A’’ (_____ , _____)

c) A (5, -11) T (x, y) ------- > (x +7, y – 11) A’ (_____ , _____)R (x, y) ------- > (x – 9, y + 9) A’’ (_____ , _____)

8. Can you find a shortcut to doing two translations?

9. What is the pre-image of A’(-5, 4) mapped by translation T (x, y) ------ > (x – 5, y + 11)?

G.CO.A.5 STUDENT NOTES & PRACTICE WS #8 – geometrycommoncore.com 1

COMPOSITE FUNCTIONS

Composite function is the term for when a sequence of transformations takes place. So if we reflect point A

over the x axis, we get A’, and then if we rotate A’ 90 about the origin, we get A’’. The first thing to notice is

that as we perform transformations in a succession, we continue to label with the primes to show what is the

pre-image of what. A is the pre-image of A’, A’ is the pre-image of A’’, A’’ is the pre-image of A’’’, and so on.

DOES THE ORDER WE DO THEM IN MATTER?

A translation of <3, -5> followed by a reflection over the y axis.

A reflection over the y axis followed by a translation of <3, -5>.

Notice that the two composite transformations result in different locations,

NYTS (Now You Try Some)

1. A rotation of 90 about O followed by a translation of <3, -5>

2. A translation of <3, -5> followed by

a rotation of 90 about O.

3. Why do you think order alters the resultant location of A’’B’’C’’?

A''

A

A' A''

A'A

A

B

C

E

F

O

A

B

C

E

F

O

G.CO.A.5 WORKSHEET #8 – geometrycommoncore.com NAME: _________________________ 1

COMPOSITE TRANSFORMATIONS

DOES ORDER MATTER – Use the composite transformation to plot A’B’C’ and A’’B’’C’’

1a) A reflection over the y axis followed by a translation (x – 3, y + 5) .

b) A translation (x – 3, y + 5) followed by a reflection over the y axis.

c) Did doing the transformations in a different order matter? Explain why?

2a) A rotation of 90o about the origin followed by a reflection over the x axis.

b) A reflection over the x axis followed by a rotation of 90o about the origin.

c) Did doing the transformations in a different order matter? Explain why?

A (6,-1)

B (5,-7)

C (3,-4)

A (6,-1)

B (5,-7)

C (3,-4)

A (1,8)

B (4,7)

C (1,3)

A (1,8)

B (4,7)

C (1,3)