Rotations and Rotational Symmetry in Geometry

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S
ECTION
 7.6
R
OTATIONS
 
AND
 R
OTATIONAL
S
YMMETRY
 
R
OTATIONS
 
There are two types of rotations:
CW – Clockwise
CCW – Counter Clockwise
Objects are usually rotated around the following
angles:  90, 180, 270
 
CW
 
CCW
 
Quarter Circle:
 
Half Circle:
 
3 Quarters Circle
 
Full Circle:
W
INDMILL
 M
ETHOD
:
 
When asked to rotate a point about the origin by 90, 180,
or 270 degrees create a windmill using the X and Y axis
ie: rotate the point (3,4) about the origin by 90°, 180°, &
270°cw
With the windmill, we can find the coordinates of the
point after the rotation
 
Practice: Given the point P(-2,4)
i) Rotate it 90° ccw about the origin
ii) Rotate it 180° ccw about the point (1,1)
 
E
X
: G
IVEN
 
THE
 
FOLLOWING
 O
BJECT
, 
ROTATE
 
IT
 90
 (
CW
) 
ABOUT
THE
 
ORIGIN
 
AND
 
FIND
 
THE
 
COORDINATES
 
OF
 
THE
 
VERTICES
:
 
When the object is rotated
90°, objects pointing up
will point to the right
 
After you rotate the object
find the coordinates of
each vertex
P
RACTICE
: G
IVEN
 
THE
 
FOLLOWING
 
OBJECTS
, 
FIND
 
THE
COORDINATES
 
OF
 
THE
 
VERTEX
 
AFTER
 
EACH
 
ROTATION
:
180 degree CW rotation
 about the origin
90 degree CCW rotation
 about point “I”
R
OTATIONAL
 S
YMMETRY
 
Objects can be rotated such that it will look the same
A square has rotational symmetry because you can
rotate it 90 degrees  and look the same
 
Order
 of rotation: How many different angles it can be
rotated so that it looks the same
Angles need
 to be less than or equal to 360 degrees
Q: How many order of rotations does a square have?
 
4 order of operation
 
90 degrees
 
180 degrees
 
270 degrees
 
360 degrees
E
X
: W
HICH
 
OF
 
THE
 
FOLLOWING
 
HAVE
 
ROTATIONAL
SYMMETRY
?  I
F
 Y
ES
, 
INDICATE
 
THE
 
ORDER
 
OF
 
ROTATION
:
 
5 order of operation
 
3 order of operation
 
2 order of operation
 
7 order of operation
 
1 order of operation
 
4 order of operation
H
OMEWORK
:
P366 # 4 – 6, 8 – 10, 13 – 14
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Delve into the concepts of rotations, rotational symmetry, and the Windmill Method in geometry. Learn about different types of rotations, how to rotate points around the origin, practice rotations with given points, and understand rotational symmetry in objects. Discover the principles of rotating objects at various angles and the order of rotations for symmetrical outcomes.

  • Geometry
  • Rotations
  • Symmetry
  • Windmill Method
  • Math

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  1. SECTION 7.6 ROTATIONS AND ROTATIONAL SYMMETRY

  2. ROTATIONS 12 11 1 There are two types of rotations: CW Clockwise 10 2 CW CCW 3 9 4 CCW Counter Clockwise 8 5 7 6 Objects are usually rotated around the following angles: 90, 180, 270 0 36 Quarter Circle: Half Circle: 3 Quarters Circle 270 90 Full Circle: 180

  3. WINDMILL METHOD: When asked to rotate a point about the origin by 90, 180, or 270 degrees create a windmill using the X and Y axis ie: rotate the point (3,4) about the origin by 90 , 180 , & 270 cw With the windmill, we can find the coordinates of the point after the rotation y 5 4 ( ) 4,3 3 2 1 x -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 ( ) -3 4,3 -4 ( ) 3, 4 -5

  4. Practice: Given the point P(-2,4) i) Rotate it 90 ccw about the origin ii) Rotate it 180 ccw about the point (1,1) y y 5 5 4 4 3 3 2 2 1 1 x x -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -1 ( ) -2 -2 4, 2 ( ) 4, 2 -3 -3 -4 -4 -5 -5

  5. EX: GIVENTHEFOLLOWING OBJECT, ROTATEIT 90 (CW) ABOUT THEORIGINANDFINDTHECOORDINATESOFTHEVERTICES: y When the object is rotated 90 , objects pointing up will point to the right 5 B A 4 3 After you rotate the object find the coordinates of each vertex 2 1 C ' A x ( ( ) ' ' ' A B C -5 -4 -3 -2 -1 0 1 2 3 4 5 4,0 -1 ) 4, 2 ( 0, 2 -2 ' C ' B ) -3 -4 -5

  6. PRACTICE: GIVENTHEFOLLOWINGOBJECTS, FINDTHE COORDINATESOFTHEVERTEXAFTEREACHROTATION: 180 degree CW rotation about the origin 90 degree CCW rotation about point I ( ) ' G 0,6 y y B 5 5 ' B ( A B 4 4 ) 2,4 G 3 3 I 2 2 1 1 C x x -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -1 -2 -2 -3 -3 ' B ) ' A( -4 ( 0, 4 -4 ) 2, 4 -5 -5

  7. ROTATIONAL SYMMETRY Objects can be rotated such that it will look the same A square has rotational symmetry because you can rotate it 90 degrees and look the same Order of rotation: How many different angles it can be rotated so that it looks the same Angles need to be less than or equal to 360 degrees Q: How many order of rotations does a square have? 4 order of operation 90 degrees 180 degrees 270 degrees 360 degrees

  8. EX: WHICHOFTHEFOLLOWINGHAVEROTATIONAL SYMMETRY? IF YES, INDICATETHEORDEROFROTATION: 5 order of operation 2 order of operation 3 order of operation 7 order of operation 4 order of operation 1 order of operation

  9. HOMEWORK: P366 # 4 6, 8 10, 13 14

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