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**7.5 Glide Reflections and Compositions**

Geometry 7.5 Glide Reflections and Compositions

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**7.5 Glide Reflections and Compositions**

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

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