Translate every vertex using one column vector
Add a displacement to every point · Describe a translation by a column vector · Check the same movement across corresponding vertices
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Directed columns
- Plotting coordinates
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Add a displacement to every point
- Describe a translation by a column vector
- Check the same movement across corresponding vertices
One movement for the whole figure
Translation by [a; b] sends (x,y) to (x+a,y+b). Apply it to every vertex; connect the images in the original vertex order. Lengths, angles and orientation are preserved, with no turn or reflection.
To describe a translation from two figures, subtract one original point's coordinates from its corresponding image. Verify that the same vector carries the other vertices too; one pair alone may not establish the whole figure's transformation.
Signs and notation
[4; 2] means four right and two up; [−4; −2] reverses that movement. A zero translation leaves all points fixed. A translation has no rotation center or mirror line, so state its vector rather than inventing one.
Sources for this lesson
Pause and explain
What describes a translation completely?
Worked example
Translate A(−2,0), B(−1,0), C(−2,1) by [4; 2].
Show the worked solution
- Add 4 to every x and 2 to every y.
- The images are A′(2,2), B′(3,2), C′(2,3).
Answer A′(2,2), B′(3,2), C′(2,3).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Translate P(1,−2) by [3; 4].
Give me a hint
Add the components in order.
Compare my reasoning
- x′ = 1+3 = 4.
- y′ = −2+4 = 2.
P′(4,2).
Look for these in your work
- Keep component order
- Check both signs
A(2,3) maps to A′(−1,5). Find the translation vector and the image of B(4,−2).
Give me a hint
Use image minus original.
Compare my reasoning
- Vector = [−1−2; 5−3] = [−3; 2].
- B′ = (4−3,−2+2) = (1,0).
Vector [−3; 2]; B′(1,0).
Look for these in your work
- Use matching vertices
- Apply the same vector
A(0,0) maps to (2,1), but B(1,0) maps to (4,1). Can one translation produce both images?
Give me a hint
Compare the two displacements.
Compare my reasoning
- A moves by [2; 1].
- B moves by [3; 1], a different vector.
No; the required displacements differ.
Look for these in your work
- Check the whole figure
- Reject a one-pair-only claim
Common mistakes
- Apply the same movement to every vertex.
- Match corresponding points when finding the vector.
What can you explain now?
What describes a translation completely?
Compare with the explanation
Its column vector.
The same horizontal and vertical displacement applies everywhere.
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Make it stick
What describes a translation completely?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.