Mathematics (Core)Exam code: 0580

Transformations

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
  • The size and shape are unchanged, only the position and the orientation
  • If the centre lies on the shape, that point is invariant

Worked example

Rotating about a vertex

Triangle R has vertices (2, 1), (5, 1) and (2, 3). Rotate R through 90° clockwise about the centre (5, 1), and give the vertices of its image.

Solution:

  • The centre (5, 1) is a vertex of R, so that vertex stays where it is
  • Count every other vertex from the centre, not from the origin
  • A quarter turn clockwise turns right into down, down into left, left into up and up into right
  • (2, 1) is 3 squares left of the centre, so its image is 3 squares above the centre: (5, 4)
  • (2, 3) is 3 squares left of and 2 squares above the centre, so its image is 3 squares above and 2 squares right of it: (7, 4)
  • Join the three points with a ruler: the image has vertices (5, 1), (5, 4) and (7, 4)
  • Check with tracing paper: trace R, hold the pencil point on (5, 1), turn the paper a quarter turn clockwise, and the tracing sits on the image
  • Turning about the origin by mistake would give a triangle of the right shape in the wrong place
A grid from x = 0 to 8 and y = −1 to 5 with triangle R, vertices (2, 1), (5, 1) and (2, 3), the centre (5, 1) marked with a cross, a curved arrow showing a clockwise quarter turn, and the image R′, vertices (5, 1), (5, 4) and (7, 4)

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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