MathematicsExam code: 0580

Transformations

Vectors and Transformations

PractiseFlashcards

Doing one

  • A rotation turns the shape about a fixed point, the centre of rotation, through a multiple of 90°
  • Tracing paper may be used in the examination (ask for it, since you cannot bring your own), and it is by far the most reliable method
    • Trace the object, press a pencil point onto the centre, turn the paper through the given angle, then copy the image across
    • Drawing an arrow on the tracing paper first shows at a glance how far it has turned
Before rotating: the centre (0, 2) is marked with a red cross beside a curved arrow showing an anticlockwise turn, and the vertex (3, 1) of triangle A is marked, with (1, 5), where a quarter turn anticlockwise takes it
Source: Rotations by Save My Exams
  • The size and shape are unchanged, only the position and the orientation
  • If the centre lies on the shape, that point is invariant
Shape A rotated clockwise about a centre of rotation marked at (−4, 0), red lines join the centre to the vertex (−4, 4) and to its image (0, 0) at right angles, and a curved arrow shows the turn to the image B on the other side of the y-axis
Source: Rotations by Save My Exams

Describing one

  • Say rotation, then give the angle with its direction, then the centre as coordinates
Two clock faces with a red curved arrow on each: the first turns the way the hands move and is labelled clockwise, the second turns against them and is labelled anticlockwise
Source: Rotations by Save My Exams
  • Half turns need no direction, since 180° clockwise and anticlockwise give the same image
  • Mark schemes accept equivalents such as 270° clockwise for 90° anticlockwise, and abbreviations like acw

Worked example

Describing a rotation

Triangle P has vertices (1, 1), (1, 3) and (2, 1). Triangle Q has vertices (−1, 1), (−3, 1) and (−1, 2). Describe fully the single transformation that maps P onto Q.

A coordinate grid with x from −4 to 3 and y from 0 to 4. Triangle P is drawn with vertices at (1, 1), (1, 3) and (2, 1), and triangle Q with vertices at (−1, 1), (−3, 1) and (−1, 2). No centre of rotation is marked.

Solution:

  • P and Q are the same size and shape, and Q is P turned, not flipped: from the right angle, P's longer side points up and its shorter side right, while Q's longer side points left and its shorter side up, so this is a rotation
  • Tracing P and turning it shows a quarter turn fits, so the angle is 90°
  • Turning anticlockwise takes the vertex (1, 3) round to (−3, 1), which matches Q
  • Testing the origin as the centre: it is the same distance from (1, 1) as from (−1, 1), and likewise for the other pairs
  • The full description is a rotation, 90° anticlockwise, centre (0, 0)
  • Check: a 90° anticlockwise turn about the origin sends (x, y) to (−y, x), and (1, 1), (1, 3), (2, 1) do map to (−1, 1), (−3, 1), (−1, 2)

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