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

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

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.

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.

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

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.

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).

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.

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.

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?