Translations and column vectors
Move every point by the same directed displacement and describe that movement with a column vector.
Section Paper 2 · Extended · Non-calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Read signed coordinates.
- Add positive and negative numbers.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which component records vertical movement?
What you will learn
- Translate vertices on a coordinate grid.
- Describe a translation using horizontal then vertical components.
A displacement is different from a destination
A translation moves every point the same distance in the same direction. The column vector (3, −2) means three units right and two down; its upper component is horizontal and its lower component vertical. A point (−1, 4) therefore moves to (2, 2). The vector describes the change, rather than the final position.
Translate each polygon vertex and join corresponding vertices in the same order with straight edges. Lengths, angles, orientation and area stay unchanged. There is no centre or mirror line: a complete description gives the translation and its two components.
Find the movement from corresponding points
For A(2, 5) moving to A′(−2, 8), subtract original from image coordinates: (−2 − 2, 8 − 5) = (−4, 3). Check that another corresponding vertex has the same displacement. Using one coordinate pair from unrelated vertices can produce an incorrect translation.
To reverse a translation, negate both components. To describe a column vector in writing here, (u, v) represents u above v, rather than a coordinate position. On paper, use the conventional stacked column-vector notation. Movement instructions must include signs; four left is −4 horizontally.
Pause and explain
A(2,5) moves to A′(−2,8). What is the translation?
Worked example
Translate A(−1, 4), B(2, 4), C(0, 6) by (3, −2).
Show the worked solution
- Add three to each x-coordinate.
- Subtract two from every y-coordinate.
- The image vertices are A′(2,2), B′(5,2), C′(3,4); join in the original order.
Answer A′(2,2), B′(5,2), C′(3,4).
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 (−3,2) by (5,−4).
Give me a hint
Add each component to the matching coordinate.
Compare my reasoning
- New x = −3 + 5 = 2.
- New y = 2 − 4 = −2.
- The image is (2,−2), five right and four down.
(2,−2).
Look for these in your work
- Matched components to axes.
- Preserved movement signs.
A(1,−2) maps to A′(4,3). Find the vector and image of B(−2,0).
Give me a hint
Subtract to find the displacement before applying it to B.
Compare my reasoning
- The displacement is (4−1,3−(−2)) = (3,5).
- B′ = (−2+3,0+5) = (1,5).
- Both points move three right and five up.
Vector (3,5); B′(1,5).
Look for these in your work
- Used corresponding vertices.
- Applied one displacement consistently.
A shape was translated by (−6,2). Describe the translation taking the image back.
Give me a hint
Reverse both directed components.
Compare my reasoning
- The outward movement is six left and two up.
- Its reverse is six right and two down.
- Use vector (6,−2); the two displacements sum to zero.
(6,−2).
Look for these in your work
- Negated both components.
- Checked the return to the starting position.
Common mistakes
- Swapping horizontal and vertical components.
- Reporting image coordinates as the translation.
- Moving different vertices by different vectors.
What can you explain now?
Does translation change a polygon's area?
Compare with the explanation
No.
Every point moves together, preserving all lengths and angles and therefore area.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, translate a triangle and recover the vector from a different pair of corresponding vertices.
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.