Dilations

Geometry · Axiom Academy

Enlarging and shrinking shapes proportionally — every point moves toward or away from one fixed center. A dilation stretches every point of a figure along a ray from the center of dilation . Multiplying its distance from the center by the scale factor k either pushes it away from the center (enlargement) or pulls it toward the center (reduction). The shape is unchanged — only the size differs. Dilation rule (center at the origin) Multiply both coordinates by the scale factor k . Each vertex slides outward along its dashed ray from the center, landing at twice its distance — a dilation with k = 2 . 2. Understanding the Scale Factor The scale factor k decides whether the image grows, shrinks, or stays the same — and the direction every point moves relative to the center. Watch one triangle sweep through the whole range of k . The same triangle as k sweeps : enlarge, shrink, then flip through the center. The image is larger than the pre-image; points move away from the center. The image equals the pre-image; nothing moves. The image is smaller than the pre-image; points move toward the center. Scales and rotates 180°: the image lands on the opposite side of the center. E.g. k = -2 enlarges by 2 and flips through the center. 3. Visualizing & Working Dilations On the grid, each point of the image sits along the dashed ray from the center, at distance |k| times the pre-image distance. Enlargement pushes out; reduction pulls in. Scale factor k = 2 — enlargement

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