What is similar about these objects?
What do we need
to pay attention to
when objects are
rotated?
8-10 Transformations
What am I learning today?
Rotations
What will I do to show that I
learned it?
Determine coordinates resulting from a
rotation.
Course 2
How do you determine the angle
of rotation?
A full turn is a 360° rotation.
A quarter turn is a 90°
rotation.
A half turn is a 180°
rotation.
360°
90°
270°
A three quarter turn is a
270° rotation.
What are they rotating around?
180°
8-10 Rotations
QUESTION
What do I need to know to
complete a rotation?
Course 2
8-10
Rotations
To rotate:
- the direction – CW or CCW
- the degrees – 90o, 180o, 270o
- the center or point of rotation –
origin, vertex, or point inside
the object
Course 2
8-10 Rotations
QUESTION
How are objects rotated
around the origin on a
coordinate plane?
Course 2
8-10 Rotations
To Rotate 180o around origin:
1. Keep your x- and y-values the
same.
.
2. Move to the opposite quadrant.
I to III
III to I
II to IV
IV to II
.
3. Put the appropriate signs
based on the quadrant.
Course 2
8-10
Rotations
Example:
Start: A (-4,3) in quadrant II
Rotate 180o clockwise
Finish: Quadrant IV
x is positive and
y is negative.
A’ (4,-3)
Course 2
8-10
Rotations
To Rotate 90o or 270o around origin:
1. x- and y-value switch places.
x becomes y and y becomes x.
.
2. Find the quadrant. Move one
for 90o or three for 270o. Pay
attention to the direction.
.
3. Put the appropriate signs
based on the quadrant.
Course 2
8-10
Rotations
Example:
Start: A (-4,3) in quadrant II
Rotate 270o clockwise
Finish: Quadrant III
x is negative and
y is negative.
A’ (-3,-4)
Course 2
8-10 Rotations
Rotations Around the Origin
Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0).
Rotate ∆ABC 90° counterclockwise about the origin.
C’
Graph the pre-image coordinates.
B’
B
3
A’
–3
Course 2
y
A
Cx
Remember: A 90 degree rotation x and y
change places, then pay attention to the
characteristics of the quadrants.
The coordinates of the image of
triangle A’B’C’ are A’(0, 1), B’(-3,3),
C (0.5).
8-10 Rotations
Rotations Around the Origin
Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0).
Rotate ∆ABC 180° counterclockwise about the origin.
y
Graph the pre-image coordinates.
B
3
A
C’
A’
B’
Course 2
–3
Cx
Remember: A 180 degree rotation only
changes the signs, so pay attention to the
characteristics of the quadrants.
The coordinates of the image of
triangle ABC are A’(-1, 0), B’(-3,-3),
C’(-5, 0).
8-10 Rotations
Rotations Around the Origin
Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0).
Rotate ∆ABC 270° counterclockwise about the origin.
y
Graph the pre-image coordinates.
B
Remember: A 270 degree rotation x and y
change places, then pay attention to the
characteristics of the quadrants.
3
Cx
A’
A
C’
B’
–3
Course 2
The coordinates of the image of
triangle A’B’C’ are A’(0,-1), B’(3,-3),
C’(0,-5).
8-10 Rotations
QUESTION
How are the coordinates
determined from a rotation
around a vertex?
Course 2
8-10 Rotations
To Rotate around a vertex:
1. Coordinates of the center of rotation
stay the same.
2. Corresponding sides create an
angle equal to the degree of
rotation
3. Each vertex in the shape must stay
an equal distance from the center
of rotation.
.
Course 2
8-10 Rotations
Rotation around a vertex
Triangle ABC has vertices A(-2,0), B(0, 3), C(0, –3).
Rotate ∆ABC 90° clockwise about the vertex A.
The pre-image coordinates of triangle
ABC are A(-2,0), B(0,3), C(0,-3).
y
3
The coordinates of the image of triangle
ABC are A’(-2,0), B’(1,-2), C’(-5,-2).
B
A
x
Course 2
–3 C
B’
C’
-2
The corresponding sides, AB
and AB’ make a 90° angle.
Notice that vertex B is 2 units
to the right and 3 units above
vertex A, and vertex B’ is 3
units to the right and 2 units to
the below vertex A.
8-10 Rotations
Rotation around a vertex
Triangle ABC has vertices A(-2,0), B(0, 3), C(0, –3).
Rotate ∆ABC 180° clockwise about the vertex A.
The pre-image coordinates of triangle
ABC are A(-2,0), B(0,3), C(0,-3).
y
C’
3
A
x
-2
B’
Course 2
The coordinates of the image of triangle
ABC are A’(-2,0), B’(-4,-3), C’(-4,3).
B
–3 C
The corresponding sides, AB
and AB’ make a 180° angle.
Notice that vertex B is 2 units
to the right and 3 units above
vertex A, and vertex B’ is 2
units to the left and 3 units
below vertex A.
Where is the point of rotation?
A point OUTSIDE the shape
A point INSIDE the Shape
How are these rotations similar?
How are these rotations different?
K
corresponding
rotation
I
M
Practice around Origin
Using these three points: P(6,3); C(-2,- 4); D(-1,0)
Rotate P 270o CCW
P’(3, -6)
Rotate C 90o CW
C’(-4,2)
Rotate D 180o CW
D’(1,0)
Rotate P 270o CW
P’(-3,6)
Rotate C 180o CCW
C’(2,4)
Rotate D 90o CW
D’(1,0)
Don’t forget
to note:
What quadrant
are you
starting in?
Practice around the Origin
Rotate 90, 180, and 270 degrees counterclockwise
P
Q
R
Practice around vertex C
Rotate 90 degrees clockwise
A
B
D
C
D
A
B
A
C
B
Rotate 180 degrees
C
Rotate 270 degrees counterclockwise