AMSCO'S Geometry. New York - Rye High School

496 Ratio, Proportion, and Similarity A dilation of k is a transformation of the plane such that:1. The image of point O, the center of dilation, is O.h2. When k is positive and the image of P is P, then OP hand OPr are thesame ray and OP kOP.h3. When k is negative and the image of P is P, then OP hand OPr are oppositerays and OP kOP.When k 1, the dilation is called an enlargement. When 0 k 1, thedilation is called a contraction.Recall also that in the coordinate plane, under a dilation of k with the centerat the origin:P(x, y) → P(kx, ky) or D k(x, y) (kx, ky)1For example, the image of ABC is ABC under a dilation of 2 . The verticesof ABC are A(2, 6), B(6, 4), and C(4, 0). Under a dilation of 2, the rule is1yD12(x,y) 5 A 1 2 x, 1 2 y BA(2, 6) → A(1, 3)B(6, 4) → B(3, 2)C(4, 0) → C(2, 0)ABNotice that ABC and ABCappear to be similar. We can use a generaltriangle to prove that for any dilation,1 Bthe image of a triangle is a similartriangle.O 1 C C xLet ABC be any triangle in the coordinate plane with A(a, 0), B(b, d),and C(c, e). Under a dilation of k through the origin, the image of ABCis ABC, and the coordinates of ABC, are A(ka, 0), B(kb, kd), andC(kc, ke).yC(kc, ke)AOC(c, e)B(b, d)A(a, 0) A(ka, 0)B(kb, kd)x