Geometric Transformations:
Translation:
slide
Reflection:
mirror
Rotation:
turn
Dialation:
enlarge or reduce
Notation:
Pre-Image:
Image:
original figure
after transformation.
Use prime notation
A’
C
B
A
C’
B’
Isometry
AKA: congruence transformation
a transformation in which an original
figure and its image are congruent.
Theorems about
isometries
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)
D'
B'
E'
B
A'
C'
D
A
C
line l is a line of reflection
E
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
A: (2, 5)
4
2
-10
-5
5
-2
-4
10
X-axis reflection
6
A: (2, 5)
4
2
-10
-5
5
-2
-4
A': (2, -5)
10
Y-axis reflection
A': (-2, 5)
6
A: (2, 5)
4
2
-10
-5
5
-2
-4
10
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.)
b.)
c.)
d.)
the y-axis
the x-axis
the line y = x
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”
A''
A''
m A''FA = 60.00
C CC
AA
A
F
C'' C''
B''
B''
BB
B
•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
m C'FC = 90.00
C'
C'
C
C
C
A
A
A
B'B'
F
F
B
B
B
A'
A'
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 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
• 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
Reflections
x axis
y axis
y=x
(a, b)
(a, b)
(a, b)
->
->
->
(a, -b)
(-a, b)
(b, a)
(a,
(a,
(a,
(a,
->
->
->
->
(-b, a)
(-a, -b)
(b, -a)
(a, b)
Rotations about the origin
90
180
270
360
b)
b)
b)
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.
b.
c.
d.
e.
Reflect over the x-axis
Reflect over the y-axis
Rotate 90 about the origin
Rotate 180 about the origin
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Practice 12-5
Reflectional symmetry
Reflectional symmetry
Both rotational and Reflectional symmetry
Reflectional symmetry
See key
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)
Line symmetry (8 lines) and 45 degree rotational symmetry
180 degree rotational symmetry
Line symmetry (1 line)
#17-21 see key
Symmetry
Line 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
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 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
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 tessellates
SHOW: 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
IF n = 4
IF n = 6
there are 6 angles about center point
there are 4 angles about center point
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
1.
Find the image of A(-1, 4) reflected over the xaxis 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).
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
10. H
11. M
12. C
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
6. Translate T across m twice the distance
between l and m
8. V rotated 145
10-17. Peer edit
18. opp; reflection
20. same; translation
22. same; 270 rotation
24. opp; reflection
26. Glide <-2, -2>, reflect over y = x – 1
28. Glide <0, 4>, reflect over y = 0 (x-axis)