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Exam questions · Maths · Transformations and Similarity

Rotations

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Write down [2 marks]

    Write down the coordinates of the image of the point \((5, -3)\) after a rotation of \(180^\circ\) about the origin. [2 marks]

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    Model answer

    A \(180^\circ\) rotation about the origin changes the sign of both coordinates, so the image is \((-5, 3)\).

    Mark scheme

    • Both coordinates change sign — M1
    • \((-5, 3)\) — A1
  2. 2 Rotate [3 marks]

    Rotate triangle \(T\) through \(90^\circ\) anticlockwise about the origin \(O\). [3 marks]

    Triangle T on a grid with the origin O marked.
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    Model answer

    A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\). The vertices \((1, 2)\), \((4, 2)\) and \((1, 4)\) go to \((-2, 1)\), \((-2, 4)\) and \((-4, 1)\).

    Mark scheme

    • Rotates at least two vertices by \(90^\circ\) about the origin — M1
    • At least two vertices correct — A1
    • Triangle with vertices \((-2, 1)\), \((-2, 4)\) and \((-4, 1)\) — A1
  3. 3 Describe [3 marks]

    Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\). [3 marks]

    Triangle A and its image triangle B on a grid.
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    Model answer

    The point \((3, 2)\) goes to \((1, 0)\) and \((5, 2)\) goes to \((-1, 0)\). The midpoint of \((3, 2)\) and \((1, 0)\) is \((2, 1)\), and the same is true for the other pair, so it is a rotation of \(180^\circ\) about \((2, 1)\).

    Mark scheme

    • Rotation — B1
    • \(180^\circ\) — B1
    • About the point \((2, 1)\) — B1
  4. 4 Calculate [3 marks]

    The point \((-1, 4)\) is rotated through \(90^\circ\) clockwise about the origin. (a) Calculate the coordinates of the image. [2 marks] (b) The original point is instead rotated through \(180^\circ\) about the origin. Write down the coordinates of the image. [1 mark]

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    Model answer

    (a) A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\), so \((-1, 4)\) goes to \((4, 1)\). (b) \((1, -4)\).

    Mark scheme

    • (a) Uses \((x, y) \to (y, -x)\) — M1
    • (a) \((4, 1)\) — A1
    • (b) \((1, -4)\) — B1
  5. 5 Find [2 marks]

    A rotation of \(180^\circ\) maps the point \((6, -1)\) onto the point \((0, 5)\). Find the coordinates of the centre of the rotation. [2 marks]

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    Model answer

    The centre is the midpoint: \(\left(\dfrac{6 + 0}{2}, \dfrac{-1 + 5}{2}\right) = (3, 2)\).

    Mark scheme

    • \(\dfrac{6 + 0}{2}\) or \(\dfrac{-1 + 5}{2}\) — M1
    • \((3, 2)\) — A1
  6. 6 Calculate [3 marks]

    The point \(P\) is \((2, 3)\). \(P\) is rotated through \(90^\circ\) anticlockwise about the point \((1, 1)\) to give \(Q\). Calculate the coordinates of \(Q\). [3 marks]

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    Model answer

    \(P\) is 1 right and 2 up from the centre. A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\), so \(Q\) is 2 left and 1 up from the centre. \(Q = (1 - 2, 1 + 1) = (-1, 2)\).

    Mark scheme

    • Position relative to the centre, \((1, 2)\) — M1
    • Rotates it to \((-2, 1)\) relative to the centre — M1
    • \((-1, 2)\) — A1

Quick check

  1. 1

    What three details describe a rotation?

    1. AThe mirror line and the angle
    2. BThe scale factor and the centre
    3. CThe centre, the angle and the direction
    4. DThe column vector and the angle
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    C: The centre, the angle and the direction

    A rotation is described by its centre, its angle and its direction.

  2. 2

    What is the image of \((3, -2)\) in a \(180^\circ\) rotation about the origin?

    1. A\((3, 2)\)
    2. B\((-3, 2)\)
    3. C\((2, -3)\)
    4. D\((-3, -2)\)
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    B: \((-3, 2)\)

    A \(180^\circ\) turn about the origin changes the sign of both coordinates.

  3. 3

    What is the image of \((2, 5)\) in a \(90^\circ\) anticlockwise rotation about the origin?

    1. A\((-5, 2)\)
    2. B\((5, -2)\)
    3. C\((-2, -5)\)
    4. D\((5, 2)\)
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    A: \((-5, 2)\)

    A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\).

  4. 4

    What is the image of \((3, 1)\) in a \(90^\circ\) clockwise rotation about the origin?

    1. A\((-1, 3)\)
    2. B\((-3, -1)\)
    3. C\((-1, -3)\)
    4. D\((1, -3)\)
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    D: \((1, -3)\)

    A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\).

  5. 5

    Why is no direction needed to describe a \(180^\circ\) rotation?

    1. AThe shape does not move
    2. BThe direction is always clockwise
    3. CA half turn is the same in both directions
    4. DThe centre decides the direction
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    C: A half turn is the same in both directions

    A half turn clockwise ends in the same place as a half turn anticlockwise.

  6. 6

    Is the image of a rotation congruent to the object?

    1. ANo, it is always larger
    2. BYes, it has the same size and shape
    3. CNo, it is always smaller
    4. DOnly for a half turn
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    B: Yes, it has the same size and shape

    A rotation does not change lengths or angles.

  7. 7

    What is the image of \((-2, 4)\) in a \(90^\circ\) clockwise rotation about the origin?

    1. A\((4, 2)\)
    2. B\((-4, -2)\)
    3. C\((2, 4)\)
    4. D\((-4, 2)\)
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    A: \((4, 2)\)

    \((x, y)\) goes to \((y, -x)\), so \((-2, 4)\) goes to \((4, 2)\).

  8. 8

    The point \((4, 3)\) is rotated through \(180^\circ\) about the point \((1, 1)\). What is the image?

    1. A\((-4, -3)\)
    2. B\((-3, -2)\)
    3. C\((2, 1)\)
    4. D\((-2, -1)\)
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    D: \((-2, -1)\)

    The point is 3 right and 2 up from the centre, so the image is 3 left and 2 down: \((1 - 3, 1 - 2) = (-2, -1)\).

  9. 9

    A \(180^\circ\) rotation takes \((1, 5)\) to \((5, 1)\). What is the centre of rotation?

    1. A\((0, 0)\)
    2. B\((4, 4)\)
    3. C\((3, 3)\)
    4. D\((6, 6)\)
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    C: \((3, 3)\)

    The centre is the midpoint of a point and its image: \(\left(\dfrac{1 + 5}{2}, \dfrac{5 + 1}{2}\right) = (3, 3)\).