Dilations in Geometry

Definition

A dilation is a transformation that produces an image that is the same shape as the original but a different size. It is defined by a center of dilation and a scale factor $k$. A dilation is a non-rigid motion (isometry is not preserved).

Why It Matters

Dilations are the logic of scale, allowing us to expand or shrink a design without breaking its structural proportions or geometric integrity. Mastering these similarity transformations ensures that a vision remains accurate whether it is rendered as a microscopic prototype or an industrial-scale reality.

Core Concepts

• Scale Factor ($k$):

  • Enlargement: $|k| > 1$.
    • How to read: “The absolute value of k is greater than one.”
    • Meaning: Image is larger than the pre-image—magnification.
  • Reduction: $0 < |k| < 1$.
    • How to read: “The absolute value of k is greater than zero and less than one.”
    • Meaning: Image shrinks toward the center—still similar shape, smaller size.
  • Identity: $k = 1$.
    • How to read: “The scale factor k equals one.”
    • Meaning: No size change; the dilation is the identity transformation.

• Algebraic Rule (Center at Origin): $P(x, y) \to P'(kx, ky)$.

  • How to read: “The point P prime is located at the coordinates k x comma k y.”
  • Meaning: Multiply every coordinate by k from the origin—uniform scaling in x and y preserves angles and parallelism.

• Preserved Properties:

  • Angle measures remain congruent.
  • Parallelism is preserved.
  • Orientation is preserved.

• Non-Preserved Properties: Distance (lengths are scaled by $|k|$).

Connected Concepts

• Similar Triangles
• Geometric Transformations Fundamentals
• Ratio
• Proportion
• Foundations