1. 9.4 : Compositions of Transformations
I can draw glide reflections and other compositions of isometries in the coordinate plane.

2. If the images to the right describe a composition of transformations, in your own words, define composition of transformations:

3. Composition of Transformations
Vocabulary!
Composition of Transformations   
Glide Reflection
a transformation is applied to a preimage and then a 2nd
transformation is applied to the image.
*2nd image is double prime (“)
The composition of a translation followed by a reflection
in a line parallel to the translation vector.

4. Example 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.

5. Composition of Isometries
Vocabulary!
Isometry
A transformation that keeps the image the same size and shape as the preimage   
Composition of Isometries
(Transformations)
The composition of two or more isometries is an isometry.
Example: the composition of a reflection and then a rotation is an isometry because neither transformation changes size or shape of figure

6. Example 2:
ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along –1 , 5 and a rotation 180° about the origin.

7. Example 3:
If PQRS is translated along <3, −2>, and reflected in 𝑦=−1about the origin, what are the coordinates of P’’Q’’R’’S’’? If P(3,2), Q(4,1), R(-1,2) and S(3,4).

8. Example 4:
Quadrilateral ABCD with A(1, 5), B(6, 2), C(-1, 3), D(-4, -2) is reflected in the line y = x and then rotated 90°. Find the coordinates of the image.

9. Vocabulary! Reflections in Parallel Lines
Reflections in Parallel Lines
Reflections in Intersecting Lines
The composition of two reflections in parallel lines
can be described by a translation vector that is
perpendicular to the two lines and twice the
distance between the two lines.
The composition of two reflections in intersecting
lines is the same as a rotation about the point
where the lines intersect and through an angle
that is twice the measure of the acute or right
angle formed by the lines.

10. Example 6:
A triangle is reflected in two parallel lines. The composition of the reflection produces a translation 22 centimeters to the right. How far apart are the parallel lines?