Dilating a Rectangle by Factor 2.5

Pre-Algebra · Axiom Academy

EXAMPLE Dilating a Rectangle by Factor 2.5 See how a dilation from the origin scales every coordinate, side length, and the area of a figure. Rectangle ABCD has vertices A(2,1) , B(6,1) , C(6,3) , and D(2,3) . Apply a dilation centered at the origin with scale factor k = 2.5 . Find each image vertex, the new side lengths, and the new area. Original (blue) and image (orange) rectangles Each image vertex lies on the ray from the origin through the original vertex, 2.5 times as far out. Nice work. You dilated a rectangle from the origin and tracked exactly how the coordinates, lengths, and area changed. Dilation rule: from the origin with scale factor k , each point (x,y) maps to . Lengths: every side length is multiplied by k = 2.5 (here and ). Area: the area is multiplied by k^2 = (2.5)^2 = 6.25 — so , not by 2.5 . Similarity: a dilation gives a similar figure — same shape and angles, different size. Direction: dilating from the origin pushes each vertex straight out along the ray from (0,0) . Remember: for any dilation with scale factor k , lengths scale by k but areas scale by k^2 . This holds for every two-dimensional figure.

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