Rigid motions, congruence and similarity

Builds on

Required prerequisites, all the way down: 10

You will learn to

• Describe translations, rotations by multiples of 90° and reflections in the axes using coordinates.
• Decide whether two figures are congruent or similar by finding a sequence of transformations.
• Compose transformations and identify the result; for example, two reflections in intersecting lines give a rotation.

Definition

• A rigid motion (isometry) is a transformation of the plane that preserves distances.
• Two figures are congruent if some rigid motion maps one onto the other.
• A dilation with centre O and scale factor k > 0 sends each point P to the point P′ on the ray OP with OP′ = k · OP.
• Two figures are similar if a dilation followed by a rigid motion maps one onto the other.

Theorem

• Rigid motions preserve lengths and angle measures; dilations preserve angle measures and multiply lengths by k.
• The composite of two rigid motions is a rigid motion, and every rigid motion has an inverse that is also a rigid motion.
• Under a dilation with scale factor k, areas are multiplied by k².

Common misconceptions

• Thinking similar figures must be the same size, or that any two rectangles are similar.
• Believing the order of transformations never matters; a rotation followed by a translation generally differs from the reverse order.

Leads to

Optional discoveries

• Structure-preserving maps: homomorphisms and isomorphisms Isometries are the structure-preserving maps of the plane with its distances: composites and inverses of isometries are again isometries.
• Transformation geometry and the Erlangen programme Klein's idea: a geometry is the study of the properties left unchanged by a group of transformations.