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Geometry
Glide Reflections and
Compositions
Goals
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Identify glide reflections in the plane.
Represent transformations as
compositions of simpler
transformations.
Glide Reflection
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A glide reflection is a transformation
where a translation (the glide) is
followed by a reflection.
Line of Reflection
Glide Reflection
1. A translation maps P onto P’.
2. A reflection in a line k parallel to the
direction of the translation maps P’ to
P’’.
1
Line of Reflection
2
3
Example
Find the image of ABC after a glide
reflection.
A(-4, 2), B(-2, 5), C(1, 3)
Translation: (x, y)  (x + 7, y)
Reflection: in the x-axis
Find the image of ABC after a glide
reflection.
A(-4, 2), B(-2, 5), C(1, 3)
Translation: (x, y)  (x + 7, y)
Reflection: in the x-axis
(-2, 5)
(-4, 2)
(1, 3)
Find the image of ABC after a glide
reflection.
A(-4, 2), B(-2, 5), C(1, 3)
Translation: (x, y)  (x + 7, y)
Reflection: in the x-axis
(-2, 5)
(-4, 2)
(1, 3)
(3, 2)
(5, 5)
(8, 3)
Find the image of ABC after a glide
reflection.
A(-4, 2), B(-2, 5), C(1, 3)
Translation: (x, y)  (x + 7, y)
Reflection: in the x-axis
(-2, 5)
(-4, 2)
(5, 5)
(1, 3)
(3, 2)
(3, -2)
(8, 3)
(8, -3)
(5, -5)
Find the image of ABC after a glide
reflection.
A(-4, 2), B(-2, 5), C(1, 3)
Translation: (x, y)  (x + 7, y)
Reflection: in the x-axis
(-2, 5)
(-4, 2)
(5, 5)
Glide
(1, 3)
(3, 2)
(3, -2)
(8, 3)
Reflection
(8, -3)
(5, -5)
You do it.
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Locate these four points:
M(-6, -6)
N(-5, -2)
O(-2, -1)
P(-3, -5)
O
N
Draw MNOP
M
P
You do it.

Translate by 0, 7.
O
N
M
P
You do it.

Translate by 0, 7.
O’
N’
P’
M’
O
N
M
P
You do it.

Reflect over y-axis.
O’
N’
P’
M’
M
P’’
O
N
P
O’’
N’’
M’’
Compositions
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A composition is a transformation that
consists of two or more transformations
performed one after the other.
Composition Example
1. Reflect AB in
the y-axis.
2. Reflect A’B’ in
the x-axis.
A
A’
B
B’
B’’
A’’
Try it in a different order.
1. Reflect AB in
the x-axis.
A
2. Reflect A’B’ in
the y-axis.
B
B’
A’
B’’
A’’
The order doesn’t matter.
A
A’
B
B’
A’
B’
B’’
A’’
This composition is commutative.
Commutative Property
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a+b=b+a
25 + 5 = 5 + 25
ab = ba
4  25 = 25  4
Reflect in y, reflect in x is equivalent to
reflect in x, reflect in y.
Are all compositions
commutative?
Rotate RS 90 CW.
Reflect R’S’ in x-axis.
R’
R
S
S’
S’’
R’’
Reverse the order.
Reflect RS in the x-axis.
R R’’
Rotate R’S’ 90 CW.
S S’’
S’
R’
All compositions are NOT commutative. Order matters!
Compositions & Isometries
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If each transformation in a composition
is an isometry, then the composition is
an isometry.
A Glide Reflection is an isometry.
Example
Reflect MN in the line y = 1.
Translate using vector 3, -2.
M
N
Now reverse the order:
Translate MN using 3, -2.
Reflect in the line y = 1.
Both compositions are isometries, but the composition
is not commutative.
Summary
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A Glide-Reflection is a composition of a
translation followed by a reflection.
Some compositions are commutative,
but not all.