AMSCO'S Geometry. New York - Rye High School

218 Transformations and the Coordinate PlaneEXAMPLE 1ConstructionIf r k(CD) CrDr, construct CrDr.1. Construct the perpendicular linefrom C to k. Let the point of intersectionbe M.2. Construct the perpendicular linefrom D to k. Let the point of intersectionbe N.3. Construct point C on CM g suchthat CM MC and point D onDN g such that DN ND.4. Draw CrDr.CCkMCMDDNDNkkkLine SymmetryBEA DkFCWe know that the altitude to the base of an isosceles triangle is also the medianand the perpendicular bisector of the base. If we imagine that isosceles triangleABC, shown at the left, is folded along the perpendicular bisector of the base sothat A falls on C, the line along which it folds, k, is a reflection line. Every pointof the triangle has as its image a point of the triangle. Points B and D are fixedpoints because they are points of the line of reflection.Thus, under the line reflection in k:1. All points of ABC are reflected so thatA → C C → A E → F F → E B → B D → D2. The sides of ABC are reflected; that is, AB S CB, a statement verifyingthat the legs of an isosceles triangle are congruent. Also, AC S CA, showingthat the base is its own image.