AMSCO'S Geometry. New York - Rye High School

BPAPoint Reflections in the Coordinate Plane 229We can prove that angle measure, collinearity, and midpoint are preservedusing the same proofs that we used to prove the corollaries of Theorem 6.1. (Seeexercises 10–12.)Theorem 6.5 and its corollaries can be summarized in the following statement. Under a point reflection, distance, angle measure, collinearity, and midpointare preserved.We use R Pas a symbol for the image under a reflection in point P. Forexample,R P(A) B means “The image of A under a reflection in point P is B.”R (1, 2)(A) A means “The image of A under a reflection in point (1, 2) is A.”Point SymmetryDEFINITIONA figure has point symmetry if the figure is its own image under a reflection in apoint.A circle is the most common example of a figure with point symmetry. LetP be the center of a circle, A be any point on the circle, and B be the other pointat which AP g intersect the circle. Since every point on a circle is equidistant fromthe center, PA PB, P is the midpoint of AB and B, a point on the circle, is theimage of A under a reflection in P.Other examples of figures that have point symmetry are letters such as Sand N and numbers such as 8.Point Reflection in the Coordinate PlaneIn the coordinate plane, the origin is themost common point that is used to used todefine a point reflection.In the diagram, points A(3, 5) andB(2, 4) are reflected in the origin. Thecoordinates of A, the image of A, are(3, 5) and the coordinates of B, theimage of B, are (2, 4). These examplessuggest the following theorem.yB(–2, 4)1O1A(3, 5)xB(2, 4)