Rotate the displacement from the centre and then add the centre back.
Core & Extended · Syllabus C7.1, E7.1
Notes and key methods
A complete rotation description states its centre, angle and direction. For 90° anticlockwise about the origin, (x,y) maps to (−y,x); clockwise gives (y,−x). A half-turn maps (x,y) to (−x,−y).
For a different centre, first subtract its coordinates, rotate the resulting displacement, and then add the centre back. The distance from the centre is preserved. The centre itself stays fixed.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Rotations about a stated centre
The centre is \(\displaystyle C=(0,1)\) and \(\displaystyle P=(1,5)\). Separate rotations are applied to P about C.
Find the x-coordinate after 90° anticlockwise.
Find the y-coordinate after 90° anticlockwise.
Find the x-coordinate after 90° clockwise.
Find the y-coordinate after a 180° rotation.
Show the reasoning
(a)
Subtract the centre, rotate the displacement and add the centre back.
Relative displacement (u,v) becomes (−v,u), so \(\displaystyle x'=-4\).
Answer: \(-4\).
(b)
Use the anticlockwise coordinate order for the relative vector.
The rotated vertical displacement is u, giving \(\displaystyle y'=2\).
Answer: \(2\).
(c)
Use the clockwise rotation rule about the stated centre.
The relative vector becomes (v,−u), so \(\displaystyle x'=4\).
Answer: \(4\).
(d)
Negate the displacement from C rather than the absolute coordinate.
A half-turn negates the relative vertical displacement, giving \(\displaystyle y'=-3\).
Answer: \(-3\).
Example 2 · Rotations about a stated centre
The centre is \(\displaystyle C=(1,2)\) and \(\displaystyle P=(3,7)\). Separate rotations are applied to P about C.
Find the x-coordinate after 90° anticlockwise.
Find the y-coordinate after 90° anticlockwise.
Find the x-coordinate after 90° clockwise.
Find the y-coordinate after a 180° rotation.
Show the reasoning
(a)
Subtract the centre, rotate the displacement and add the centre back.
Relative displacement (u,v) becomes (−v,u), so \(\displaystyle x'=-4\).
Answer: \(-4\).
(b)
Use the anticlockwise coordinate order for the relative vector.
The rotated vertical displacement is u, giving \(\displaystyle y'=4\).
Answer: \(4\).
(c)
Use the clockwise rotation rule about the stated centre.
The relative vector becomes (v,−u), so \(\displaystyle x'=6\).
Answer: \(6\).
(d)
Negate the displacement from C rather than the absolute coordinate.
A half-turn negates the relative vertical displacement, giving \(\displaystyle y'=-3\).
Answer: \(-3\).
Example 3 · Rotations about a stated centre
The centre is \(\displaystyle C=(2,3)\) and \(\displaystyle P=(5,9)\). Separate rotations are applied to P about C.
Find the x-coordinate after 90° anticlockwise.
Find the y-coordinate after 90° anticlockwise.
Find the x-coordinate after 90° clockwise.
Find the y-coordinate after a 180° rotation.
Show the reasoning
(a)
Subtract the centre, rotate the displacement and add the centre back.
Relative displacement (u,v) becomes (−v,u), so \(\displaystyle x'=-4\).
Answer: \(-4\).
(b)
Use the anticlockwise coordinate order for the relative vector.
The rotated vertical displacement is u, giving \(\displaystyle y'=6\).
Answer: \(6\).
(c)
Use the clockwise rotation rule about the stated centre.
The relative vector becomes (v,−u), so \(\displaystyle x'=8\).
Answer: \(8\).
(d)
Negate the displacement from C rather than the absolute coordinate.
A half-turn negates the relative vertical displacement, giving \(\displaystyle y'=-3\).
Answer: \(-3\).
Common mistakes
Keep intermediate values exact until the final step.