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Mathematics · 5. Coordinates, vectors and transformations

Describing geometry with coordinates and transformations

Connect visual geometry to coordinate rules, vectors and invariants of each transformation.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

What information uniquely specifies a transformation?

Curriculum coverage

Vietnam Mathematics 6–10; Cambridge IGCSE transformations and vectors.

Model scope and limitations

Standard matrix rules usually assume the origin as centre; another centre requires translating to and from the origin.

Learning objectives

  • Fully describe translations, rotations, reflections and enlargements.
  • Apply coordinate rules.
  • Identify invariant properties.

Prerequisite knowledge

  • Coordinate plane and negative numbers.

Core knowledge

Isometry

Translations, rotations and reflections preserve lengths and angles; reflection reverses orientation.

Enlargement

An enlargement centred at C with factor k maps P to P′ with vector CP′ = k·CP.

Composition

Order generally matters: A then B need not equal B then A.

Worked example

A 90° anticlockwise rotation about O maps (x,y) to (−y,x), so (3,−2) maps to (2,3).

Misconceptions to avoid

  • A rotation needs centre, angle and direction; “rotate 90°” is incomplete.
  • A translation vector does not depend on the starting point.

Virtual experiment procedure: 2D transformations

  1. Predict one key point image before running the simulation.
  2. Apply the rule to every vertex and reconnect in order.
  3. Check lengths, angles, orientation and centre distances as appropriate.

Safety and cautions

  • Do not infer from a distorted display; verify with grid and numerical coordinates.

Evidence to collect

  • A preimage–image coordinate table, transformation rule and checked invariants.
Open the virtual experiment

Diagnostic check

1. Reflection in the y-axis maps (x,y) to?

  1. (−x,y)
  2. (x,−y)
  3. (y,x)
Show answer and explanation

Correct answer: (−x,y)

The x-coordinate changes sign; y stays.

2. Which transformation preserves orientation?

  1. Translation
  2. Reflection
  3. None
Show answer and explanation

Correct answer: Translation

Translation does not reverse vertex order.

Application practice

1. Translation by (−2,3) maps (5,1) to?

  1. (3,4)
  2. (7,−2)
  3. (−10,3)
Show answer and explanation

Correct answer: (3,4)

Add coordinates: (5−2, 1+3) = (3,4).

2. A 180° rotation about origin maps (4,−1) to?

  1. (−4,1)
  2. (1,4)
  3. (4,1)
Show answer and explanation

Correct answer: (−4,1)

(x,y) → (−x,−y).

3. What does enlargement factor −2 mean?

  1. Double size on opposite side of centre
  2. Half size on same side
  3. Reflection only
Show answer and explanation

Correct answer: Double size on opposite side of centre

Magnitude 2 enlarges; the negative sign puts the image across the centre.

Academic references

  1. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer