 # Geometric Transformations:

## Presentation on theme: "Geometric Transformations:"— Presentation transcript:

Geometric Transformations:
Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce

Notation: Pre-Image: original figure Image: after transformation. Use prime notation A’ C C ’ B B’ A

Isometry AKA: congruence transformation
a transformation in which an original figure and its image are congruent.

FUNDAMENTAL THEOREM OF ISOMETRIES Any any two congruent figures in a plane can be mapped onto one another by at most 3 reflections ISOMETRY CLASSIFICATION THEOREM There are only 4 isometries. They are:

TRANSLATION: moves all points in a plane a given direction a fixed distance

TRANSLATION VECTOR: Direction Magnitude
PRE-IMAGE IMAGE

Translate by the vector <x, y>

x moves horizontal y moves vertical
Translate by <3, 4>

Different notation T(x, y) -> (x+3, y+4)

Translations PRESERVE: Size Shape Orientation

Reflection over a line (mirror)

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

X-axis reflection

Y-axis reflection

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: 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) Find coordinates of (7, 6) reflected over: a.) the y-axis b.) the x-axis c.) the line y = x d.) the line x = -3 3. HW Check & Peer edit

Rotations have: Center of rotation Angle of rotation:

Example: Rotate Triangle ABC 60 degrees clockwise about “its center”
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

ROTATIONS PRESERVE SIZE SHAPE ORIENTATION 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 origin
90 (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

Coordinate Geometry rules
Reflections x 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-axis b. Reflect over the y-axis c. Rotate 90 about the origin d. Rotate 180 about the origin e. 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

Rotate (3, 4) 90 degrees about the point (1, 6). What two transformations will produce the same result? Try it again by rotating (3, 4) 90 degrees about (-2, 5). Rotate (2, -6) 90 degrees about (1, 7) Describe OR LIST STEPS FOR how you can find the image of any point after a 90 rotation about (a, b). Try it again with a 180 rotation about (a,b). How can you find the image? Try it again with a 270 rotation about (a,b). How can you find the image?

Starter HW Peer edit Practice 12-5
Reflectional symmetry Both rotational and Reflectional symmetry See key No lines of symmetry Line symmetry (5 lines) and 72 degree rotational symmetry Line symmetry (1 line) Line symmetry (4 lines) and 90 degree rotational symmetry Line symmetry (8 lines) and 45 degree rotational symmetry 180 degree rotational symmetry Line symmetry (1 line) #17-21 see key

Symmetry Line Symmetry Rotational Symmetry
If a figure can be reflected onto itself over a line. Rotational Symmetry If 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

NOTE: TEST WILL BE END OF NEXT WEEK!!!
Isometry Wrap Up… Sketchpad Activitiy # 6 Symmetry in Regular Polygons 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: Explore & read information underTessellations: What are they The beginnings Symmetry & MC Escher The galleries Solid 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 PROOF If ~q then ~p
Assume that the conclusion is FALSE. Reason to a contradiction. If n>6 then the regular polygon will not tessellate. ASSUME: The polygon tessellates SHOW: n can not be >6

Indirect proof Regular polygons with n>6 sides will not tessellate
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 Starter Determine if the following will tessellate & provide proof: Isosceles triangle Kite Regular pentagon Regular hexagon Regular heptagon Regular octagon Regular nonagon Regular decagon

Review practice 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? 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? List all the rotational symmetries of a regular decagon. Draw a regular octagon with all its lines of symmetry (on sketchpad).

Problem from HSPA test

Coordinate Transformations MOAT game
Groups 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 points 2nd group correct = +2 points 3rd group correct = +1 points Group with HIGHEST # points +3 on quiz Group with 2nd highest # points +2 on quiz Group with 3rd highest # points +1 on quiz

HW Answers p. 650 12-4 10. H 4. F translate twice the distance 11. M
13. Segment BC 14. A 15. Segment LM 16. I 17. K 34. a.) B(-2, 5) b.) C(-5, -2) c.) D(2, -5) d.) Square: 4 congruent sides & angles 12-4 4. F translate twice the distance Translate T across m twice the distance between l and m V rotated 145 Peer edit opp; reflection same; translation same; 270 rotation Glide <-2, -2>, reflect over y = x – 1 28. Glide <0, 4>, reflect over y = 0 (x-axis)