Rotate about the origin and specify the turn direction
Use quarter-turn and half-turn coordinate rules · Distinguish clockwise from counterclockwise · Keep each point's distance from the center
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
- Coordinates and degrees
- Circle radius
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
- Use quarter-turn and half-turn coordinate rules
- Distinguish clockwise from counterclockwise
- Keep each point's distance from the center
Quarter-turn rules
About the origin, 90° counterclockwise sends (x,y) to (−y,x); 90° clockwise sends it to (y,−x); 180° sends it to (−x,−y). A 270° counterclockwise turn is the same as 90° clockwise. Check a point on the positive x-axis to remember the direction.
Angles beyond memorised rules
For any stated rotation angle, join the center to a point, preserve that radius, and locate the new ray at the stated angle and direction. A compass arc and protractor can represent this geometrically; quarter-turn shortcuts are not a restriction on the concept.
Apply the same center, angle and direction to every vertex. Rotation preserves lengths, angles and orientation. A center, angle and direction give a complete description; a half-turn needs no direction choice because either way gives the same image.
Sources for this lesson
Pause and explain
Where does (1,0) move under 90° counterclockwise about O?
Worked example
Rotate (3,1) through 90° counterclockwise and 180° about the origin.
Show the worked solution
- 90° counterclockwise gives (−1,3).
- 180° negates both coordinates to (−3,−1).
Answer (−1,3); (−3,−1).
Try a topic test
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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,−3) 90° counterclockwise about the origin.
Give me a hint
Use (−y,x).
Compare my reasoning
- Negating y gives 3.
- The new y is the old x, 2.
(3,2).
Look for these in your work
- Use the stated direction
- Check distance from O
Rotate (−4,1) 90° clockwise and 180° about the origin.
Give me a hint
Use separate rules for the two turns.
Compare my reasoning
- Clockwise quarter-turn gives (1,4).
- Half-turn gives (4,−1).
(1,4); (4,−1).
Look for these in your work
- Keep both signs
- Distinguish the turns
A point is 5 units from O and is rotated by 60° clockwise. Is its image still 5 units from O? Describe how to locate it without using a quarter-turn rule.
Give me a hint
Keep the radius while changing its direction.
Compare my reasoning
- Rotation preserves the distance, so the image radius is 5.
- Draw the radius-5 arc and the ray turned 60° clockwise; their intersection locates the image.
Yes, radius 5; use the 60° clockwise ray and equal-radius arc.
Look for these in your work
- Use the full rotation concept
- Avoid an inapplicable shortcut
Common mistakes
- A 90° turn needs a direction.
- Rotate every vertex with the same rule.
What can you explain now?
Where does (1,0) move under 90° counterclockwise about O?
Compare with the explanation
(0,1).
The positive x-axis turns towards the positive y-axis.
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Make it stick
Where does (1,0) move under 90° counterclockwise about O?
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.