Rotations and a stated centre
Turn vertices through a stated quarter- or half-turn and give angle, direction and centre.
Section C7.1 / E7.1 · Single Core and broader Extended transformations
Core and Extended. Core centres are the origin, a vertex or an edge midpoint; Extended allows any centre. 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 coordinates and compass directions.
- Distinguish clockwise from anticlockwise.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which information completes a rotation description?
What you will learn
- Rotate shapes through multiples of 90°.
- Describe the centre, angle and direction completely.
A centre stays fixed while the shape turns
Rotation preserves distances from the centre as well as lengths and angles within the shape. About the origin, 90° anticlockwise sends (x,y) to (−y,x); 90° clockwise sends it to (y,−x); 180° sends it to (−x,−y). A half-turn has the same result clockwise or anticlockwise.
For example, (3,1) becomes (−1,3) after 90° anticlockwise about the origin. Sketch the axes and turn the position direction to check the signs. Rotations preserve orientation, unlike reflections; a centre vertex remains fixed if it is part of the polygon.
Measure vectors from the actual centre
For a centre other than the origin, find each point's displacement from the centre, rotate that displacement and add the centre back. Rotating (4,1) about vertex (1,1) by 90° anticlockwise turns displacement (3,0) into (0,3), giving image (1,4).
A complete description states rotation, angle, direction and centre. Core uses the origin, vertices or edge midpoints for these multiples of ninety degrees. Extended also uses other centres. Do not use origin-only coordinate rules directly on a point when the specified centre is elsewhere.
Pause and explain
Rotate (3,1) 90° anticlockwise about the origin.
Worked example
Rotate (3,1) through 90° anticlockwise about the origin.
Show the worked solution
- The origin remains fixed.
- Use (x,y) → (−y,x), giving (−1,3).
- Both original and image are √10 from the centre.
Answer (−1,3).
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.
Rotate (2,−5) 180° about the origin.
Give me a hint
A half-turn negates both displacement components.
Compare my reasoning
- Original displacement is (2,−5).
- A half-turn gives (−2,5).
- The centre distance √29 is preserved.
(−2,5).
Look for these in your work
- Used a half-turn.
- Checked the distance from the centre.
Triangle vertices are O(0,0), A(4,0), B(4,2). Rotate 90° clockwise about vertex O.
Give me a hint
Use (x,y) → (y,−x).
Compare my reasoning
- O stays (0,0).
- A′ = (0,−4), B′ = (2,−4).
- Join the images with the same lengths as the original.
O′(0,0), A′(0,−4), B′(2,−4).
Look for these in your work
- Kept the centre fixed.
- Used the specified direction.
Rotate triangle A(1,1), B(4,1), C(1,3) 90° anticlockwise about vertex A.
Give me a hint
Subtract A before turning each displacement.
Compare my reasoning
- A stays (1,1).
- B displacement (3,0) becomes (0,3), giving B′(1,4).
- C displacement (0,2) becomes (−2,0), giving C′(−1,1).
A′(1,1), B′(1,4), C′(−1,1).
Look for these in your work
- Used the stated centre vertex.
- Added the centre back after rotating.
Common mistakes
- Omitting the centre.
- Confusing clockwise and anticlockwise.
- Using origin rules about another centre.
What can you explain now?
Does a 180° rotation need a direction to distinguish its image?
Compare with the explanation
No; both directions give the same half-turn.
A clockwise and anticlockwise half-turn end at the same point about a fixed centre.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, compare a quarter-turn and half-turn about one of a triangle's 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.