AMSCO'S Geometry. New York - Rye High School

Dilations in the Coordinate Plane 2476-8 DILATIONS IN THE COORDINATE PLANEIn this chapter, we have learned about transformations in the plane that areisometries, that is, transformations that preserve distance. There is anothertransformation in the plane that preserves angle measure but not distance. Thistransformation is a dilation.For example, in the coordinate plane, a dilationof 2 with center at the origin will stretcheach ray by a factor of 2. If the image of A is A,then A is a point on OA hand OA2OA.yAAOxDEFINITIONA 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 the same rayand OP kOP.h3. When k is negative and the image of P is P, then OP hand OPr are oppositerays and OP kOP.Note: In step 3, when k is negative, k is positive.In the coordinate plane, the center of dilation is usually the origin. If thecenter of dilation is not the origin, the coordinates of the center will be given.In the coordinate plane, under a dilation of k with the center at the origin:P(x, y) → P(kx, ky) or D k(x, y) (kx, ky)For example, the image of ABC is ABC1under a dilation of 2. The vertices of ABC are A(2,16), B(6, 4), and C(4, 0). Under a dilation of 2, the ruleisD12(x, y) 5 A 1 2 x, 1 2 y ByAABA(2, 6) → A(1, 3)B(6, 4) → B(3, 2)C(4, 0) → C(2, 0)1O1BC Cx