Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce
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Transcript of Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce
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)