12.1.4 Compositions of Transformations (continued)
A single transformation is determined for two reflections in this application example. The intersecting lines of reflection and the angle between them are given. The theorem states that the two reflections are equivalent to a rotation. The center of rotation is the point of intersection of the two intersecting lines of reflection. The measure of the angle of rotation is twice the measure of the angle between the lines of reflection, or 120°.
Example 3 Describing Transformations in Terms of Reflections
Any translation or rotation is equivalent to a composition of two reflections.
Two lines of reflection are determined that result in the given translation in this example. The preimage and image of the translation are given. To find the lines of reflection, locate the midpoint ( M ) of the translation vector between any two corresponding points on the preimage and image. Then find the midpoints of AM and MA '. The lines of reflection are perpendicular to the vector and intersect with the vector at the second set of midpoints. Two lines of reflection that result in the given rotation are determined in this example. The preimage and image of the translation are given. The center of rotation is given. To find the lines of reflection, draw line segments between the center of rotation, P , and any two corresponding points on the preimage and image. Bisect the angle formed by the two line segments. Then bisect each of the two angles formed to draw the lines of reflection.
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