12.1.4 Compositions of Transformations (continued)
The effect of a rotation followed by a reflection in the coordinate plane is determined in this example. The coordinates of the vertices of the preimage are given. The angle of rotation and the center of rotation are given. The line of reflection is given. To determine the vertices of the image after the rotation, remember that a 90° rotation about the origin is described by the rule ( x , y ) → ( − y , x ). Map each of the vertices of the preimage to the image after the rotation by applying the rule to the coordinates of each of the vertices. To draw the reflection across the x -axis, use the rule ( x , y ) → ( x , − y ). Map each of the vertices of the preimage to the image after the reflection by applying the rule to the coordinates of each of the vertices. Join the vertices with line segments to form the final image. Label the vertices J '', K '' and L ''.
Example 2 Art Application
A transformation followed by another transformation, called a composition of transformations, may be described by a single transformation.
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 formed is twice the measure of the angle formed by the lines.
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