Sheetworks
Create

Generate Transformations Worksheets

Perform and analyze geometric transformations: reflections, rotations, translations, and dilations on the coordinate plane.

Moving Shapes: Transformations

Geometric transformations change the position, size, or orientation of a shape on the coordinate plane. The four main types — translations, reflections, rotations, and dilations — are used in computer graphics, animation, architecture, and pattern design. Understanding transformations also deepens students' grasp of symmetry and congruence.

Components of Transformations

This section covers the four types of transformations:

  • Translations (Slides): Move every point of a shape the same distance in the same direction, described by a vector (x + a, y + b).
  • Reflections (Flips): Flip a shape across a line of reflection (such as the x-axis, y-axis, or the line y = x), creating a mirror image.
  • Rotations (Turns): Turn a shape around a fixed center point by a given angle. Common rotations are 90°, 180°, and 270°.
  • Dilations (Resizing): Enlarge or shrink a shape from a center point by a scale factor. Factor > 1 enlarges; factor between 0 and 1 shrinks.

Examples of Transformations

Translation Examples

  • Translate the point (3, 5) by the vector (2, -3): New point is (3 + 2, 5 - 3) = (5, 2).
  • Translate a triangle with vertices (1, 1), (4, 1), (2, 4) right 3 and up 2: New vertices are (4, 3), (7, 3), (5, 6).
  • A chess piece moves from square (2, 1) to (5, 4), which is a translation of (+3, +3).

Reflection Examples

  • Reflect (4, 3) across the x-axis: The x-coordinate stays the same and the y-coordinate negates, giving (4, -3).
  • Reflect (4, 3) across the y-axis: The y-coordinate stays and the x-coordinate negates, giving (-4, 3).
  • Reflect the point (2, 5) across the line y = x: Swap the coordinates to get (5, 2).

Rotation Examples

  • Rotate (3, 2) by 90° counterclockwise around the origin: Apply the rule (x, y) → (-y, x) to get (-2, 3).
  • Rotate (5, -1) by 180° around the origin: Apply (x, y) → (-x, -y) to get (-5, 1).
  • A windmill blade rotates 90° clockwise. A point at (4, 0) moves to (0, -4) using (x, y) → (y, -x).

Dilation Examples

  • Dilate (2, 3) by scale factor 3 from the origin: Multiply both coordinates by 3 to get (6, 9).
  • Dilate a triangle with vertices (4, 2), (8, 2), (6, 6) by scale factor 1/2 from the origin: New vertices are (2, 1), (4, 1), (3, 3).
  • A photo is 4 inches × 6 inches. Enlarging by a scale factor of 1.5 gives 6 inches × 9 inches.