G.CO.A.4 STUDENT NOTES & PRACTICE WS #1/#2 – geometrycommoncore.com
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THE REFLECTION
A reflection over a line m (notation rm) is an isometric
transformation in which each point of the original figure (preimage) has an image that is the same distance from the line of
reflection as the original point but is on the opposite side of the
line. Because a reflection is an isometry, the image does not change
size or shape. The line of reflection is the perpendicular bisector of
the segment joining every point and its image.
rm  ABC   A ' B ' C '
m
A'
A
B
B'
C'
C
DEFINITION
A reflection in a line m is a isometric transformation that maps every point P in the plane to a point P’, so
that the following properties are true;
a) If point P is NOT on line m, then line m is the
perpendicular bisector of PP ' .
b) If point P is ON line m, then P = P’
m
m
P = P'
P
CHARACTERISTICS
1. Distances from
pre-image to image
2. Orientation
3. Special Points
What is meant by Orientation?
P'
REFLECTION
G.CO.A.4 STUDENT NOTES & PRACTICE WS #1/#2 – geometrycommoncore.com
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THE ROTATION
A rotation is an isometric transformation that turns a figure
about a fixed point called the center of rotation. Rays drawn
from the center of rotation to a point and its image form an
angle called the angle of rotation (notation Rcenter, degree).
A'
C
θ
C'
An object and its rotation are the same shape and size, but
the figures may be turned in different directions.
A
B'
B
O
RO,  ABC   A ' B ' C '
DEFINITION
A rotation about a Point O through Ɵ degrees is an isometric transformation that maps every point P in
the plane to a point P’, so that the following properties are true;
a) If point P is NOT point O, then OP = OP’ and
mPOP’ = Ɵ.
P'
b) If point P IS point O, then P = P’. The center of
rotation is the ONLY point in the plane that is
unaffected by a rotation.
A'
P
A
θ
O
CHARACTERISTICS
4. Distances from
pre-image to image
5. Orientation
6. Special Points
θ
O = P = P'
ROTATION
G.CO.A.4 STUDENT NOTES & PRACTICE WS #1/#2 – geometrycommoncore.com
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ROTATION DIRECTION
One full rotation is 360, this would return all points in the plane to their original location. Because a rotation
can go in two directions along the same arc we need to define positive and negative rotation values.
COUNTERCLOCKWISE IS A POSITIVE DIRECTION, and clockwise is a negative direction.
Students often want to know why this ‘backwards’ relationship happens….. my belief is that the rotation
direction comes from the coordinate plane, In the coordinate plane, the initial arm of the angle is the positive
x axis, and the terminal arm opens in a counterclockwise direction from there.
7. POSITVE ROTATIONS ARE __________________ AND NEGATIVE ROTATIONS ARE _____________________.
EQUIVALENT ROTATIONS
Because angles are formed along an arc of a circle there are two ways to get to the same location, a positive
direction and a negative direction. A 60 rotation is the same as a -300 rotation. Co-terminal angles can be
calculated using the formula, a coterminal angle = initial angle + 360n, where n is an integer.
60 = 60 + (1)360 = 420
60 = 60 + (2)360 = 780
60 = 60 + (-1)360 = -300
60 = 60 + (-2)360 = -660
Conterminal angle = initial angle + 360n
Determine a Positive and Negative Coterminal Angle to the given one. (Many possible answers)
8. Given 100
9. Given 320
10. Given -40
11. Given 400




+ co-terminal = _______
+ co-terminal = _______
+ co-terminal = _______
+ co-terminal = _______
- co-terminal = ________
- co-terminal = ________
- co-terminal = ________
- co-terminal = ________
SPECIAL ROTATION – ROTATION OF 180
A rotation of 180 maps A to A’ such that:
A
a) mAOA’ = 180 (from definition of rotation)
b) OA = OA’ (from definition of rotation)
O
c) Ray OA and Ray OA ' are opposite rays. (They form a line.)
AO is the same line as AA '
A'
G.CO.A.4 STUDENT NOTES & PRACTICE WS #1/#2 – geometrycommoncore.com
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THE TRANSLATION
A translation slides an object a fixed distance in a given
direction. When working in the plane this is usually represented by
an arrow, the arrow provides both distance and direction of the
translation. When working on the coordinate plane, a vector is
used to describe the fixed distance and the given direction often
denoted by <x,y>. The x value describes the effect on the x
coordinates (right or left) and the y value describes the effect on the
y coordinates (up or down).
The pre-image and image have the same shape and size.
A'
A
B'
B
C'
C
T x, y   ABC   A ' B ' C '
DEFINITION
A translation is an isometric transformation that maps every two points A and B in the plane to points A’
and B’, so that the following properties are true;
1. AA’ = BB’ (a fixed distance).
2. AA '|| BB ' (a fixed direction).
CHARACTERISTICS
4. Distances from
pre-image to image
A'
A
B'
B
TRANSLATION
5. Orientation
6. Special Points
Important Property for Translations
All SEGMENTS THAT ARE TRANSLATED ARE PARALLEL TO EACH OTHER.