AMSCO'S Geometry. New York - Rye High School

224 Transformations and the Coordinate PlaneEXAMPLE 1On graph paper:a. Locate A(3, 1).b. Locate A, the image of A under areflection in the y-axis, and write itscoordinates.c. Locate A, the image of A under areflection in the x-axis, and write itscoordinates.Answersy1O 1xReflection in the Line y = xIn the figure, ABC is reflected in the liney x. Its image under the reflection is ABC.From the figure, we see that:A(1, 2) → A(2, 1)B(3, 4) → B(4, 3)C(1, 5) → C(5, 1)For each point and its image under a reflectionin the line y = x, the x-coordinate of the 1image is the y-coordinate of the point; they-coordinate of the image is the x-coordinate ofthe point. Note that for a reflection in the liney x, the image of (1, 2) is (2, 1) and the imageof (2, 1) is (1, 2).1xA reflection in the line y x can be designated as r y = x. For example, if theimage of (1, 2) is (2, 1) under a reflection in y x, we can write:r y = x(1, 2) (2, 1)From these examples, we form a general rule that can be proven as a theorem.yC(1, 5)B(3, 4)B(4, 3)A(1, 2)O A(2, 1) C(5, 1)Theorem 6.4Under a reflection in the line y = x, the image of P(a, b) is P(b, a).Given A reflection in the line whose equation is y x.ProveThe image of P(a, b) is P(b, a).