1. 7.5 Glide Reflections and Compositions
Geometry
7.5 Glide Reflections and Compositions

2. 7.5 Glide Reflections and Compositions
Goals
Identify glide reflections in the plane.
Represent transformations as compositions of simpler transformations.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

3. 7.5 Glide Reflections and Compositions
A glide reflection is a transformation where a translation (the glide) is followed by a reflection.
Line of Reflection
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

4. 7.5 Glide Reflections and Compositions
A translation maps P onto P’.
A reflection in a line k parallel to the direction of the translation maps P’ to P’’. 1 2
Line of Reflection 3
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

5. 7.5 Glide Reflections and Compositions
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
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

6. 7.5 Glide Reflections and Compositions
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)
(1, 3)
(-4, 2)
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

7. 7.5 Glide Reflections and Compositions
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
(5, 5)
(-2, 5)
(1, 3)
(8, 3)
(-4, 2)
(3, 2)
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

8. 7.5 Glide Reflections and Compositions
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
(5, 5)
(-2, 5)
(1, 3)
(8, 3)
(-4, 2)
(3, 2)
(3, -2)
(8, -3)
(5, -5)
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

9. 7.5 Glide Reflections and Compositions
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
(5, 5)
(-2, 5)
Glide
(1, 3)
(8, 3)
Reflection
(-4, 2)
(3, 2)
(3, -2)
(8, -3)
(5, -5)
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

10. 7.5 Glide Reflections and Compositions
You do it.
Locate these four points:
M(-6, -6)
N(-5, -2)
O(-2, -1)
P(-3, -5)
Draw MNOP O N P M
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

11. 7.5 Glide Reflections and Compositions
You do it.
Translate by 0, 7. M N O P M N O P
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

12. 7.5 Glide Reflections and Compositions
You do it.
Translate by 0, 7. O’ N’ P’ M’ M N O P
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

13. 7.5 Glide Reflections and Compositions
You do it.
Reflect over y-axis. O’
O’’ N’
N’’ P’
P’’
M’’ M’ M N O P
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

14. 7.5 Glide Reflections and Compositions
A composition is a transformation that consists of two or more transformations performed one after the other.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

15. 7.5 Glide Reflections and Compositions
Composition Example
Reflect AB in the y-axis.
Reflect A’B’ in the x-axis. A A’ B’ B
B’’
A’’
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

16. Try it in a different order.
Reflect AB in the x-axis.
Reflect A’B’ in the y-axis. A B B’
B’’
A’’ A’
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

17. The order doesn’t matter.B’ B B’
B’’
A’’ A’
This composition is commutative.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

18. 7.5 Glide Reflections and Compositions
Commutative Property
a + b = b + a =
ab = ba
4  25 = 25  4
Reflect in y, reflect in x is equivalent to reflect in x, reflect in y.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

19. Are all compositions commutative?
Rotate RS 90 CW.
Reflect R’S’ in x-axis. S S’
S’’
R’’
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

20. 7.5 Glide Reflections and Compositions
Reverse the order. R
R’’
Reflect RS in the x-axis.
Rotate R’S’ 90 CW. S
S’’ S’ R’
All compositions are NOT commutative. Order matters!
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

21. Compositions & Isometries
If each transformation in a composition is an isometry, then the composition is an isometry.
A Glide Reflection is an isometry.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

22. 7.5 Glide Reflections and Compositions
Example
Reflect MN in the line y = 1.
Translate using vector 3, -2.
Now reverse the order:
Translate MN using 3, -2.
Reflect in the line y = 1. N M
Both compositions are isometries, but the composition is not commutative.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

23. 7.5 Glide Reflections and Compositions
Summary
A Glide-Reflection is a composition of a translation followed by a reflection.
Some compositions are commutative, but not all.
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions

24. 7.5 Glide Reflections and Compositions
Homework
Tue, Dec 8, 11:56 AM
7.5 Glide Reflections and Compositions