AMSCO'S Geometry. New York - Rye High School

222 Transformations and the Coordinate Plane6-3 LINE REFLECTIONS IN THE COORDINATE PLANEWe can apply the definition of a line reflection to points in the coordinate plane.Reflection in the y-axisIn the figure, ABC is reflected in the y-axis.Its image under the reflection is ABC.From the figure, we see that:A(1, 2) → A(1, 2)B(3, 4) → B(3, 4)B(3, 4)B(3, 4)A(1, 2) A(1, 2)C(1, 5) → C(1, 5)1For each point and its image under areflection in the y-axis, the y-coordinate ofO 1 xthe image is the same as the y-coordinate of the point; the x-coordinate of theimage is the opposite of the x-coordinate of the point. Note that for a reflectionin the y-axis, the image of (1, 2) is (1, 2) and the image of (1, 2) is (1, 2).A reflection in the y-axis can be designated as r y-axis. For example, if the imageof (1, 2) is (1, 2) under a reflection in the y-axis, we can write:r y-axis(1, 2) (1, 2)C(1, 5)From these examples, we form a general rule that can be proven as atheorem.yC(1, 5)Theorem 6.2Under a reflection in the y-axis, the image of P(a, b) is P(a, b).GivenProveA reflection in the y-axis.The image of P(a, b) under a reflection inthe y-axis is P(a, b).yP(a, b) Q(0, b) P(a, b)OxProofBy the definition of a reflection in a line, a point P is the image of P under areflection in a given line if and only if the line is the perpendicular bisector ofPPr. Therefore, we can prove that P is the image of P under a reflection inthe y-axis by showing that the y-axis is the perpendicular bisector of PPr.(1) The y-axis is perpendicular to PPr. The line of reflection, the y-axis, is avertical line. P(a, b) and P(a, b) have the same y-coordinates. PPr is asegment of a horizontal line because two points are on the same horizontalline if and only if they have the same y-coordinates. Every vertical lineis perpendicular to every horizontal line. Therefore, the y-axis is perpendicularto PPr.