1. 9.4 Composition of Transformations
Then: You drew reflections, translations, and rotations.
Now:
1. Draw glide reflections and other compositions of isometries in the coordinate plane.
2. Draw compositions of reflections in parallel and intersecting lines.

2. 9.4 Composition of Transformations
I. Definition: The resulting transformation when a transformation is applied to a figure and then another transformation is applied to its image.
Artwork by
Escher

3. 9.4 Composition of Transformations
II. Types of Composition of Transformations
A. Glide reflection: the composition of a translation followed by a reflection in a line parallel to the translation vector.

4. 9.4 Composition of Transformations
B. Graph a Glide Reflection
1. Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3),
T(–1 , 1), and S(–4, 2). Graph BGTS and its image
after a translation along 5, 0 and a reflection
in the x-axis.
B’ (_____, _____) B” (_____, _____)
G’ (_____, _____) G” (_____, _____)
T’ (_____, _____) T” (_____, _____)
S’ (_____, _____) S” (_____, _____)

5. 9.4 Composition of Transformations
B. Graph a Glide Reflection
2. Triangle XYZ has vertices X(6, 5), Y(7, –4) and
Z(5, –5). Graph △XYZ and its image after a
translation along 〈1, 2〉 and a reflection
in the y-axis.
X’ (_____, _____) X” (_____, _____)
Y’ (_____, _____) Y” (_____, _____)
Z’ (_____, _____) Z” (_____, _____)

6. 9.4 Composition of Transformations
3. From Problem #2, XYZ  X’Y’Z’ and X’Y’Z  X”Y”Z”, so by Transitive Property of Congruence, then XYZ  X”Y”Z”. 4. Glide reflections, reflections, translations, and rotations are the only four rigid motions or isometries in a plane. 5. Theorem 9.1 Composition of Isometries The composition of two (or more) isometries is an isometry.

7. 9.4 Composition of Transformations
C. Graph Other Compositions of Isometries
1. ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4).
Graph TUV and its image after a translation
along –1 , 5 and a rotation 180° about the origin.
T’ (_____, _____) T” (_____, _____)
U’ (_____, _____) U” (_____, _____)
V’ (_____, _____) V” (_____, _____)

8. 9.4 Composition of Transformations
C. Graph Other Compositions of Isometries
2. ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0).
Graph ΔJKL and its image after a translation
along 3, 1 and a rotation 90° about the origin.
J’ (_____, _____) J” (_____, _____)
K’ (_____, _____) K” (_____, _____)
L’ (_____, _____) L” (_____, _____)

9. 9.4 Composition of Transformations
D. Composition of Two Reflections
1. Composition of two reflection in parallel lines is the same as a translation.

10. 9.4 Composition of Transformations
D. Composition of Two Reflections
2. Composition of two reflections in intersecting lines is the same as a
rotation.

11. 9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern.
1. Carpet pattern:

12. 9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern.
2. Tile pattern:

13. 9.4 Composition of Transformations
E. Describe the transformations that are combined to create each pattern.
3. Wallpaper pattern:

14. 9.4 Composition of Transformations
Assignment:
P #8-16 evens, all, all, all, 38, all
Graph: 8-16 evens, 25 & 27 (2 pieces of graph paper)