What is a dilation?

A dilation is a transformation that changes the size of a figure, but not its shape. It's like zooming in or out on an image. Here's a breakdown of key information:

• Center of Dilation: This is a fixed point in the plane. All points of the figure are scaled relative to this center.

• Scale Factor: This is a number (k) that determines the amount of enlargement or reduction.

  • k > 1: The dilation is an enlargement (the image is larger than the pre-image).
  • 0 < k < 1: The dilation is a reduction (the image is smaller than the pre-image).
  • k = 1: The dilation is an identity transformation (the image is congruent to the pre-image).
  • k < 0: The dilation involves both a scaling and a reflection across the center of dilation.

• How it works: Each point (x, y) in the pre-image is transformed to a point (x', y') in the image according to the formula:

  (x', y') = (k(x - a) + a, k(y - b) + b)

  where (a, b) is the center of dilation. This formula essentially stretches or shrinks the distance between each point and the center of dilation by a factor of k.

• Properties preserved under dilation:

  • Shape: The shape of the figure remains the same. Angles are preserved.
  • Parallelism: Parallel lines remain parallel after dilation.
  • Collinearity: Points that lie on the same line remain collinear after dilation.
  • Ratio of distances: The ratio of distances between corresponding points is equal to the scale factor.

• Properties not preserved under dilation:

  • Size: The size (area, perimeter) changes unless k = 1.
  • Distance from the center of dilation: The distance from the center of dilation to each point is multiplied by the scale factor.

In summary, a dilation is a transformation that scales a figure proportionally, maintaining its shape while changing its size, based on a chosen center and scale factor.