AMSCO'S Geometry. New York - Rye High School

254 Transformations and the Coordinate PlaneEXAMPLE 2The transformation (x, y) → (y 3, x 2) is the composition of what twotransformations?SolutionUnder a rotation of 90° about the origin, (x, y) → (y, x).When this rotation is followed by the translationT 3,–2(y, x) (y 3, x 2).Answer T 3,22+ R 908(x,y) T 3,2(y, x) (y 3, x 2)This solution is not unique.AlternativeSolutionUnder the translation T 2,3(x, y) (x 2, y 3).When this translation is followed by the rotationR 90°(x 2, y 3) ((y 3), x 2) (y 3, x 2).Answer R 908+ T 22,23(x, y) R 90°(x 2, y 3) (y 3, x 2)ExercisesWriting About Mathematics1. Owen said that since a reflection in the x-axis followed by a reflection in the y-axis has thesame result as a reflection in the y-axis followed by a reflection in the x-axis, the compositionof line reflections is a commutative operation. Do you agree with Owen? Justify youranswer.2. Tyler said that the composition of an even number of opposite isometries is a direct isometryand that the composition of an odd number of opposite isometries is an opposite isometry.Do you agree with Tyler? Justify your answer.Developing SkillsIn 3–11, find the image of A(3, 2) under the given composition of transformations.3. r x-axis+ r y-axis4. r y-axis+ r x-axis5. R O+ r x-axis6. R 908+ R O7. T 21,4+ r y5x8. R 908+ R 9089. r y-axis+ r y5x10. R O+ T 3,2211. T 22,4+ D 2212. A reflection in the line y x followed by a rotation of 90° about the origin is equivalent towhat single transformation?13. A rotation of 90° about the origin followed by another rotation of 90° about the origin isequivalent to what single transformation?