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Transformations

Transformations

Rotation

Rotation- On Grid Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Rotating a shape
  4. Describing a rotation
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 7.1: rotation
  • 7.1Draw a rotation of 90°, 180° or 270° about a centre.
  • 7.1Describe a rotation fully: angle, direction, centre.
  • 7.1Find the centre of a rotation.

Before you start: Reflection and translation. Next: Enlargement.

Formula bank

90° anticlockwise about OLearn it\((x,y)\to(-y,\ x)\)Use it when a quarter turn left.
90° clockwise about OLearn it\((x,y)\to(y,\ -x)\)same as 270° anticlockwise.Use it when a quarter turn right.
180° about OLearn it\((x,y)\to(-x,\ -y)\)direction does not matter.Use it when a half turn.
Any other centre CLearn itwork from C: (point − C) → turn → + CUse it when a centre that is not the origin.

Rotating a shape

Triangle A rotated 90° anticlockwise about the origin gives B-4-3-2-112345-112345xyAB
Every corner turns a quarter turn about the dot at O.

A rotation turns a shape about a fixed point, the centre. Every point stays the same distance from the centre. The image is the same size and shape, but turned.

General methodHow to draw a rotation (tracing paper)

  1. Trace the shape and the centre. Put your pencil point on the centre.
  2. Turn the tracing paper by the angle in the right direction (a quarter turn = 90°).
  3. Mark the new corners through the paper, then join them.

General methodHow to rotate a point without tracing paper

  1. Write the point relative to the centre: (point − centre).
  2. Turn it: 90° anticlockwise \((x,y)\to(-y,x)\); 90° clockwise \((x,y)\to(y,-x)\); 180° \((x,y)\to(-x,-y)\).
  3. Add the centre back on.

Example Rotate the point \((4,2)\) 90° clockwise about \((1,1)\).

Relative to the centre first.

  1. \((4,2)-(1,1)=(3,1)\)
  2. clockwise: \((3,1)\to(1,-3)\)
  3. \((1,-3)+(1,1)=(2,-2)\)

Answer \((2,-2)\)

Try it yourself

1
Rotate \((3,1)\) 90° anticlockwise about the origin.
Show answer

Answer (−1, 3)

2
Rotate \((3,2)\) 180° about \((2,0)\).
Show answer

Answer (1, −2)

Describing a rotation

General methodHow to describe a rotation fully

  1. Angle: is the image a quarter turn (90°) or a half turn (180°) from the object? Compare one side's direction.
  2. Direction: clockwise or anticlockwise (not needed for 180°).
  3. Centre: try the origin first; otherwise use tracing paper, or the point the same distance from each corner and its image (perpendicular bisectors of object–image joins meet there).
  4. Answer: 'Rotation, 90° clockwise, centre (a, b)'.

Example A has corners \((1,1),(4,1),(1,3)\); B has corners \((-1,1),(-1,4),(-3,1)\). Describe the single transformation from A to B.

The long side of A points right; in B it points up: a quarter turn anticlockwise. Test the origin with the rule.

  1. \((x,y)\to(-y,x)\): \((4,1)\to(-1,4)\) ✓
  2. \((1,3)\to(-3,1)\) ✓

Answer Rotation, 90° anticlockwise, centre (0, 0)

Try it yourself

1
\((4,3)\) maps to \((4,-1)\) by a rotation 90° clockwise. Is the centre \((2,1)\)? Check: rotate \((4,3)\) 90° clockwise about \((2,1)\).
Show answer

Answer (4, −1)

Ways they test it

Type 1 Draw a rotation

"Rotate triangle A 90° clockwise about (0, 1)." (2 marks)

Spot it angle, direction and centre given.

  1. Tracing paper, pencil on the centre.
  2. Or the point rule from the centre.

Type 2 Describe a rotation

"Describe fully the single transformation that maps A onto B." (3 marks)

Spot it the image is turned, same size.

  1. Rotation + angle + direction + centre.
  2. Three marks = three details.

Type 3 Find the centre

"Find the centre of rotation." (1-2 marks)

Spot it only the centre asked.

  1. Try likely grid points with tracing paper.
  2. Check two corners.

Mistakes to avoid

  • ✗ Rotating about the wrong point✓ pencil on the centre, not on a corner
  • ✗ Missing the direction✓ say clockwise or anticlockwise for 90°It costs a mark.
  • ✗ Writing 'turn' instead of 'rotation'✓ use the word rotationOnly the correct words score.
  • ✗ Mixing up the rules✓ anticlockwise \((-y,x)\), clockwise \((y,-x)\)Check with (1, 0): anticlockwise goes to (0, 1).

Summary

  • Rotation = centre + angle + direction.
  • About O: anticlockwise \((-y,x)\), clockwise \((y,-x)\), 180° \((-x,-y)\).
  • Other centres: subtract the centre, turn, add it back.

If you see …

  • same size, turnedrotation.
  • a 180° turnno direction needed.

Practice

Practice: easy, medium, hard

Easy

1
Rotate \((-3,2)\) 180° about the origin.
Show answer

Answer (3, −2)

2
Rotate \((2,5)\) 90° clockwise about the origin.
Show answer

Answer (5, −2)

3
Rotate \((4,-1)\) 90° anticlockwise about the origin.
Show answer

Answer (1, 4)

Medium

1
Rotate \((4,2)\) 90° clockwise about \((1,1)\).
Guide me

What is it asking? a point

What do we know? relative (3, 1)

Plan \((1,-3)\), add (1, 1)

Show working
  1. (2, −2)

Answer (2, −2)

2
Rotate \((5,1)\) 90° anticlockwise about \((2,1)\).
Guide me

What is it asking? a point

What do we know? relative (3, 0)

Plan \((0,3)\), add (2, 1)

Show working
  1. (2, 4)

Answer (2, 4)

3
Rotate \((3,2)\) 180° about \((2,0)\).
Guide me

What is it asking? a point

What do we know? relative (1, 2)

Plan \((-1,-2)\), add (2, 0)

Show working
  1. (1, −2)

Answer (1, −2)

Hard

1
Triangle \((1,1),(4,1),(1,3)\) is rotated 180° about \((0,-1)\). Image of \((4,1)\)?
Guide me

What is it asking? a point

What do we know? relative (4, 2)

Plan \((-4,-2)\), add (0, −1)

Show working
  1. (−4, −3)

Answer (−4, −3)

PartAnswer / workingMarksGuidance
(−4, −3)B1
Total: 1 mark
2
Same rotation: image of \((1,3)\)?
Guide me

What is it asking? a point

What do we know? relative (1, 4)

Plan \((-1,-4)\), add (0, −1)

Show working
  1. (−1, −5)

Answer (−1, −5)

PartAnswer / workingMarksGuidance
(−1, −5)B1
Total: 1 mark
3
\((1,1)\to(-1,1)\) and \((4,1)\to(-1,4)\) by a rotation about the origin. Angle in degrees (anticlockwise)?
Guide me

What is it asking? an angle

What do we know? the side pointing right now points up

Plan a quarter turn

Show working
  1. 90° anticlockwise

Answer 90

PartAnswer / workingMarksGuidance
90° anticlockwiseB1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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