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9-4 Study Guide and Intervention

Compositions of Transformations

Glide ReflectionsWhen two transformations are applied to a figure, one after another,the total transformation is a composition of transformations. A glide reflection is atranslation followed by a reflection in a line parallel to the translation vector. Notice thatthe composition of isometries is another isometry.

Example: Triangle ABC has vertices A(3, 3), B(4, –2) and C(–1, –3). Graph△ABC and its image after a translation along〈–2, –1〉and a reflection in thex–axis.

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Step 1translation along 〈–2, –1〉

(x, y) → (x –2, y –1)

A(3, 3) → A′(1, 2)

B(4, –2) → B′(2, –3)

C(–1, –3) → C′(–3, –4)

Step 2reflection in the x–axis.

(x, y) → (x, –y)

A′(1, 2) → A″(1, –2)

B′(2, –3) → B″(2, 3)

C′(–3, –4) → C″(–3, 4)

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Step 3Graph △ABC and its image △A″B″C″.

Exercises

Triangle XYZ has vertices X(6, 5), Y(7, –4) and Z(5, –5).Graph △XYZ and
its image after the indicated glide reflection.

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1. Translation: along 〈1, 2〉

Reflection: in y–axis

3. Translation: along 〈2, 0〉

Reflection: in x = y

2. Translation: along 〈–3, 4〉

Reflection: in x–axis

4. Translation: along 〈–1, 3〉

Reflection: in x–axis

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9-4 Study Guide and Intervention(continued)

Composition of Transformations

Compositions of ReflectionsThe composition of two reflections in parallel lines isthe same as a translation.
The compositions of two reflections in intersecting lines is thesame as a rotation.

Example: Copy and reflect figure A in lineℓ and then line m. Then describe asingle transformation that
maps A onto A″.

Step 1 Reflect A in line ℓ. Step 2 Reflect A′ in line m.

The compositions of two parallel lines is the same as a translation.

Exercises

Copy and reflect figure P in line a and then line b. Then describe a singletransformation that maps P onto P″.

1. 2.

3. 4.

Chapter 926Glencoe Geometry