Stage 8 - Q1 - 3 marks
Rotate P 180 degrees about K\((1,2)\).
Hint
A half turn places K at the midpoint of P and its image.
Stage 8 - Q2 - 3 marks
Rotate P 90 degrees anticlockwise about K\((2,1)\).
Hint
Subtract K, rotate the displacement, then add K back.
Stage 8 - Q3 - 4 marks
Rotate ABC 90 degrees clockwise about K\((1,1)\).
Hint
Rotate each displacement from K one quarter turn clockwise.
Stage 8 - Q4 - 2 marks
A half turn maps \((2,5)\) to \((-4,1)\). Find its centre.
Hint
A 180-degree rotation makes the centre lie halfway between corresponding points.
Stage 8 - Q5 - 4 marks
A point is rotated 90 degrees clockwise about \((3,-1)\) and its image is \((5,2)\). Find the original point.
Hint
Undo the rotation relative to the same centre.
Stage 8 - Q6 - 4 marks
A pupil says the midpoint of P and P′ always gives the centre of a rotation. Use a 90-degree rotation to disprove the claim.
Hint
Choose an easy point on an axis and make a quarter turn.
Stage 8 - Q7 - 4 marks
Two corresponding pairs under a proposed half turn are A\((0,2)\) to A′\((6,4)\) and B\((2,5)\) to B′\((4,1)\). Can one half turn do both? Give its centre.
Hint
Check the centre independently from each pair.
Stage 8 - Q8 - 4 marks
P\((4,1)\) rotates 90 degrees anticlockwise about K\((1,1)\), then 180 degrees about the same K. Find its final position and one equivalent rotation.
Hint
Rotation angles combine directly when the centre is the same.
Stage 8 - Q9 - 5 marks
A 90-degree anticlockwise rotation maps P\((3,1)\) to P′\((2,4)\). Find its centre K\((h,k)\) and check the result.
Hint
Write the rotation using displacement from the unknown centre.
Stage 8 - Q10 - 6 marks
Rotate a point first 180 degrees about A\((1,2)\), then 180 degrees about B\((4,-1)\). Prove that the result is the same translation for every point and find its vector.
Hint
Use a general point P to separate the fixed displacement from the starting coordinates.