12.1.4 Compositions of Transformations
Key Objectives • Apply theorems about isometries. • Identify and draw compositions of transformations, such as glide reflections. Key Terms • A composition of transformations is one transformation followed by another. • A glide transformation is the composition of a translation and a reflection across a line parallel to the translation vector. Theorems, Postulates, Corollaries, and Properties • Theorem A composition of two isometries is an isometry. • Theorem The composition of two reflections across two parallel lines is equivalent to a translation. • The translation vector is perpendicular to the lines. • The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. • The center of rotation is the intersection of the lines. • The angle of rotation is twice the measure of the angle formed by the lines. • Theorem Any translation or rotation is equivalent to a composition of two reflections. Example 1 Drawing Compositions of Isometries
The effect of a reflection followed by a translation is determined in this example. The line of reflection and the translation vector are given. To draw the reflection across , draw a line perpendicular to from each of the preimage vertices. Measure the distance from the preimage vertex to and mark that same distance on the line on the other side of . These are the vertices of the reflected image. To translate the image, draw a line that is parallel to the translation vector and the same length as the translation vector from each of the reflected image’s vertices. Mark these points and then join the vertices with line segments to form the final image. Label the vertices A ', B ' and C '.
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