Transformations (reflections, rotations, translations, and enlargements)

Transformations (reflections, rotations, translations, and enlargements)

Transformations in Geometry

Basic Terminology

  • A transformation refers to the movement or alteration of a shape in a specific way.
  • The original shape is known as the object while the transformed shape is the image.
  • Every transformation involves a single point called the invariant point or centre of transformation that remains fixed during the transformation.

Reflections

  • A reflection is a flip of a shape over a line, called the mirror line.
  • Every point of the object is the same distance from the mirror line as its corresponding point in the image.
  • Reflections do not change the size, area or angles of the original shape.

Rotations

  • A rotation turns a shape around a fixed point called the centre of rotation.
  • Two pieces of information are needed to define a rotation: the centre of rotation and the angle of rotation.
  • Rotation can be clockwise or anti-clockwise.
  • As with reflections, rotations do not change the size, area or angles of the original shape.

Translations

  • A translation slides a shape from one place to another without turning it.
  • Translations are described using a vector, which tells you how far and in what direction to move the shape.
  • Translations do not alter the size, orientation, or angles of the shape.

Enlargements

  • An enlargement changes the size of a shape without changing its shape.
  • Enlargements are defined by a centre of enlargement and a scale factor.
  • The scale factor tells you how much larger or smaller each dimension of the shape becomes.
  • A scale factor greater than 1 causes the image to be larger than the original object, while a scale factor between 0 and 1 reduces the size of the object.
  • A negative scale factor causes an enlargement and a reflection.
  • The area scale factor is the square of the length scale factor and the volume scale factor is the cube of the length scale factor.

This comprehensive overview of geometric transformations should provide a solid grounding for understanding these key concepts and applying them to problem-solving contexts.