Translation: slide

Reflection: mirror

Rotation: turn

Dialation: enlarge or reduce

Geometric Transformations:

Pre-Image: original figure

Image: after transformation. Use prime notation

Notation:

A

A’

B

B’

C

C ’

Isometry

AKA: congruence transformation

a transformation in which an original figure and its image are congruent.

Theorems about isometries

FUNDAMENTAL THEOREM OF ISOMETRIESAny any two congruent figures in a plane can be

mapped onto one another by at most 3 reflections

ISOMETRY CLASSIFICATION THEOREMThere are only 4 isometries. They are:

TRANSLATION:

moves all points in a plane

a given direction

a fixed distance

TRANSLATION VECTOR:

DirectionMagnitude

PRE-IMAGE

IMAGE

Translate by the vector <x, y>

x moves horizontaly moves vertical

Translate by <3, 4>

Different notationT(x, y) -> (x+3, y+4)

Translations PRESERVE:

SizeShape

Orientation

Reflectionover a line (mirror)

line l is a line of reflection

C'

D'

E'

A'

B'

B

A

E

D

C

Properties of reflections

PRESERVE• Size (area, length, perimeter…)• Shape

CHANGE orientation (flipped)

Reflect x-axis: (a, b) -> (a,-b)Change sign y-coordinate

Reflect y-axis: (a, b) -> (-a, b)Change sign on x coordinate

6

4

2

-2

-4

-10 -5 5 10

A: (2, 5)

6

4

2

-2

-4

-10 -5 5 10

A': (2, -5)

A: (2, 5)

X-axis reflection

Y-axis reflection

6

4

2

-2

-4

-10 -5 5 10

A': (-2, 5) A: (2, 5)

PARTNER SWAP:Part I: (Live under my rules)• Use sketchpad to graph & label any three points

• Graph & Reflect them over the line y = x– Graph->Plot new function->x->OK– Construct two points on the line and connect them– Mark this line segment as your mirror.

• WRITE a conjecture about how (a, b) will be changed after reflecting over y = x. Explain.

• Repeat by reflecting over the line y = -x. Write a conjecture.

Starter:

1. Find one vector which would accomplish the same thing as translating (3, -1) by <3, 8> then applying the transformation T(x, y)->(x-4, y+9)

2. Find coordinates of (7, 6) reflected over:a.) the y-axisb.) the x-axisc.) the line y = xd.) the line x = -3

3. HW Check & Peer edit

Rotations have:

Center of rotation

Angle of rotation:

CENTER of rotation

Example: Rotate Triangle ABC

60 degrees clockwise about “its center”

C

A

B

C''

A''

B''

C

A

B

mA''FA = 60.00

C''

A''

B''

C

A

B

F

•Find the image of A after a 120 degree rotation

•Find the image of A after a 180 degree rotation

•Find the image of A after a 240 degree rotation

•Find the image of A after a 300 degree rotation

•Find the image of A after a 360 degree rotation

Rotated 90 degrees counterclockwise

C

A

B

F

C'

A'

B'

C

A

B

F

mC'FC = 90.00

C'

A'

B'

C

A

B

F

ROTATIONS PRESERVE

SIZE– Length of sides– Measure of angles– Area– Perimeter

SHAPE

ORIENTATION

PARTNER SWAP:Part II: (Live under new

rules)

• Use sketchpad to graph & label any three points. Connect them and construct triangle interior.

• Rotate your pre-image about the origin 90• Rotate the pre-image about the origin 180• Rotate the pre-image about the origin 270• Rotate the pre-image about the origin 360

WRITE A CONJECTURE: What are the coordinates of (a, b) after a 90, 180, and 270 degree rotation about the origin?

Rotations on a coordinate plane about

the origin90 (a, b) -> (-b, a)

180 (a, b) -> (-a, -b)

270 (a, b) -> (b, -a)

360 (a, b) -> (a, b)

DEBRIEFING:Find the coordinates of (2, 5)

• Reflected over the x-axis

• Reflected over the y-axis

• Reflected over the line x = 3

• Reflected over the line y = -2

• Reflected over the line y = x

• Rotated about the origin 180

• Rotated about the origin 270

• Rotated about the origin 360

Review the rules for coordinate geometry

transformations

• Which two transformations would accomplish the same thing as a 90 degree rotation about the origin?

• Use sketchpad to justify your answer

Coordinate Geometry rules

Reflectionsx axis (a, b) -> (a, -b)y axis (a, b) -> (-a, b)y=x (a, b) -> (b, a)

Rotations about the origin

90 (a, b) -> (-b, a)180 (a, b) -> (-a, -b)270 (a, b) -> (b, -a)360 (a, b) -> (a, b)

GLIDE REFLECTIONS

You can combine different Geometric Transformations…

Practice: Reflect over y = x then translate by the vector <2, -3>

After Reflection…

After Reflection and translation…

Santucci’s Starter:Complete the following transformations on (6, 1) and list

coordinates of the image:

a. Reflect over the x-axisb. Reflect over the y-axisc. Rotate 90 about the origind. Rotate 180 about the origine. Rotate 270 about the origin

EXPLAIN in writing: what two transformations would

accomplish the same thing as a 90 degree rotation about the origin?

Starter:Find the coordinates of pre-image (3, 4)

after the following transformations (do without graphing…)

• reflect over y-axis• reflect over x-axis• reflect over y=x• reflect over y=-x• translate <-2, 6>• rotate 90 about origin• rotate 180 about origin• rotate 270 about origin• rotate 360 about origin

PAIRS Sketchpad Exploration:

1. Rotate (3, 4) 90 degrees about the point (1, 6). What two transformations will produce the same result?

2. Try it again by rotating (3, 4) 90 degrees about (-2, 5).

3. Rotate (2, -6) 90 degrees about (1, 7)

4. Describe OR LIST STEPS FOR how you can find the image of any point after a 90 rotation about (a, b).

5. Try it again with a 180 rotation about (a,b). How can you find the image?

6. Try it again with a 270 rotation about (a,b). How can you find the image?

Starter HW Peer edit Practice 12-5

1. Reflectional symmetry2. Reflectional symmetry3. Both rotational and Reflectional symmetry4. Reflectional symmetry5. See key6. See key7. No lines of symmetry8. Line symmetry (5 lines) and 72 degree rotational symmetry9. Line symmetry (1 line)10. Line symmetry (4 lines) and 90 degree rotational symmetry11. Line symmetry (8 lines) and 45 degree rotational symmetry12. 180 degree rotational symmetry13. Line symmetry (1 line)14. Line symmetry (8 lines) and 45 degree rotational symmetry15. 180 degree rotational symmetry16. Line symmetry (1 line) #17-21 see key

SymmetryLine Symmetry

If a figure can be reflected onto itself over a line.

Rotational SymmetryIf a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees

What kinds of symmetry do each of the following have?

What kinds of symmetry do each of the following have?

Rotational (180) Point Symmetry

Rotational (90, 180, 270)Point Symmetry

Rotational (60, 120, 180, 240, 300)Point Symmetry

Isometry Wrap Up…

1. Sketchpad Activitiy # 6 Symmetry in Regular Polygons

2. Dilations Exploration

NOTE: TEST WILL BE END OF NEXT WEEK!!!

Dilations• Plot any 5 points to make a convex polygon and fill in its interior red.

• Mark the origin as center.

• Make the polygon larger by a scale factor of 2 and fill it in green.

• Make the polygon smaller by a scale factor of 1/3. Fill it in red.

• Measure your coordinates and Explain how you can find coordinates of a dilation image.

• Try marking a new center and dilating a few points. What is the “center” of a dilation? How does it change the measurements?

Tessellations web-quest

VISIT: http://www.tessellations.org/tess-what.htm

Explore & read information underTessellations:What are theyThe beginningsSymmetry & MC EscherThe galleriesSolid Stuff

Answer the following questions:1. What is symmetry and list the types discussed.2. What are the Polya’ symmetries?3. How many Polya’ symmetries are there?4. What are the Rhomboid possibilities?5. What is the difference between a periodic and aperiodic

tiling?

TO-DO

• Complete Tessellations Sketchpad explorations, # 8, 9

• Read rubric and write questions. Begin design

INDIRECT PROOFIf ~q then ~p

1. Assume that the conclusion is FALSE.2. Reason to a contradiction.

If n>6 then the regular polygon will not tessellate.

ASSUME: The polygon tessellatesSHOW: n can not be >6

Indirect proof

Regular polygons with n>6 sides will not tessellate

Proof:Assume a polygon with n>6 sides will tessellate.

This means that n*one interior <measure will equal 360

• IF n = 3 there are 6 angles about center point• IF n = 4 there are 4 angles about center point• IF n = 6 there are 3 angles about center point •Therefore, if n>6 then there must be fewer

than 3 angles about the center point. In other words, there must be 2 or fewer. If there are 2 angles about the center point then each angle must measure 180 to sum to 360

•But no regular polygon exists whose interior angle measures 180 (int. < sum must be LESS than 180). Therefore, the polygon can not tessellate.

Santucci’s StarterDetermine if the following will tessellate & provide proof:

– Isosceles triangle

– Kite

– Regular pentagon

– Regular hexagon

– Regular heptagon

– Regular octagon

– Regular nonagon

– Regular decagon

Review practice1. Find the image of A(-1, 4) reflected over the

x-axis then over the y-axis (two intersecting lines). What one transformation would accomplish the same result?

2. Find the image of B(6, -2) reflected over x=3 then over x=-5 (two parallel lines). What one transformation would accomplish the same result?

3. List all the rotational symmetries of a regular decagon.

4. Draw a regular octagon with all its lines of symmetry (on sketchpad).

Problem

from

HSPA te

st

Coordinate Transformations

MOAT gameGroups of “3”

Write answer on white board and send one “runner” to stand facing the class with representatives from all other groups (hold board face down). When MOAT is called flip answer so all members seated can see answer.

1st group correct = +3 points2nd group correct = +2 points3rd group correct = +1 points

Group with HIGHEST # points +3 on quizGroup with 2nd highest # points +2 on quizGroup with 3rd highest # points +1 on quiz

HW Answers p. 650

10. H11. M12. C13. Segment BC14. A15. Segment LM16. I17. K34. a.) B(-2, 5)b.) C(-5, -2)c.) D(2, -5)d.) Square: 4 congruent sides & angles

12-44. F translate twice the distance6. Translate T across m twice the

distance between l and m8. V rotated 14510-17. Peer edit18.opp; reflection20.same; translation22.same; 270 rotation 24.opp; reflection26.Glide <-2, -2>, reflect over y = x – 128. Glide <0, 4>, reflect over y = 0 (x-

axis)