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Exam questions · Maths · Transformations and Similarity

Combined Transformations

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Describe [4 marks]

    Triangle \(A\) is reflected in the line \(y = x\) to give triangle \(B\). Triangle \(B\) is reflected in the \(x\)-axis to give triangle \(C\). (a) Draw triangles \(B\) and \(C\). [2 marks] (b) Describe fully the single transformation that maps \(A\) onto \(C\). [2 marks]

    Triangle A on a grid with the line y equals x.
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    Model answer

    (a) \(B\) has vertices \((1, 2)\), \((1, 4)\) and \((3, 2)\), and \(C\) has vertices \((1, -2)\), \((1, -4)\) and \((3, -2)\). (b) \((x, y)\) goes to \((y, x)\) and then to \((y, -x)\), which is a rotation of \(90^\circ\) clockwise about the origin.

    Mark scheme

    • (a) \(B\) correct — B1
    • (a) \(C\) correct — B1
    • (b) Rotation, \(90^\circ\) clockwise — B1
    • (b) About the origin — B1
  2. 2 Describe [3 marks]

    The point \((1, 4)\) is reflected in the line \(y = 2\), and the image is then reflected in the line \(y = 6\). Describe the single transformation that has the same effect. [3 marks]

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    Model answer

    The first reflection gives \((1, 0)\) and the second gives \((1, 12)\). The point has moved 8 up, which is twice the distance between the lines. It is a translation by \(\begin{pmatrix} 0 \\ 8 \end{pmatrix}\).

    Mark scheme

    • \((1, 0)\) seen — M1
    • \((1, 12)\) seen — M1
    • Translation by \(\begin{pmatrix} 0 \\ 8 \end{pmatrix}\) — A1
  3. 3 Write down [2 marks]

    (a) A shape is rotated through \(180^\circ\) about the origin. Write down the coordinates of the point that does not move. [1 mark] (b) A shape is translated. How many points stay in the same place? [1 mark]

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    Model answer

    (a) The centre of rotation, \((0, 0)\). (b) None, because every point moves by the same vector.

    Mark scheme

    • (a) \((0, 0)\) — B1
    • (b) None — B1
  4. 4 Work out [3 marks]

    The point \((2, 1)\) is translated by the vector \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and the image is then rotated through \(180^\circ\) about the origin. Write down the coordinates of the final image. [3 marks]

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    Model answer

    The translation gives \((3, 3)\), and the rotation changes the sign of both coordinates, giving \((-3, -3)\).

    Mark scheme

    • \((3, 3)\) — B1
    • Changes the sign of both coordinates of their image — M1
    • \((-3, -3)\) — A1
  5. 5 Describe [4 marks]

    The point \((3, 1)\) is reflected in the \(x\)-axis, and the image is then reflected in the line \(y = x\). (a) Write down the coordinates of the final image. [2 marks] (b) Describe the single transformation that maps \((3, 1)\) onto the final image, and explain how you know it is a rotation. [2 marks]

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    Model answer

    (a) The first reflection gives \((3, -1)\), and the second swaps the coordinates, giving \((-1, 3)\). (b) \((x, y)\) goes to \((-y, x)\), which is a rotation of \(90^\circ\) anticlockwise about the origin.

    Mark scheme

    • (a) \((3, -1)\) — B1
    • (a) \((-1, 3)\) — B1
    • (b) Rotation, \(90^\circ\) anticlockwise — B1
    • (b) About the origin — B1
  6. 6 Work out [3 marks]

    A shape is rotated through \(90^\circ\) clockwise about the origin and then through \(90^\circ\) clockwise about the origin again. Describe the single transformation that has the same effect, and use the point \((2, 3)\) to check. [3 marks]

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    Model answer

    The first rotation sends \((2, 3)\) to \((3, -2)\) and the second sends it to \((-2, -3)\). This is the same as a rotation of \(180^\circ\) about the origin.

    Mark scheme

    • \((3, -2)\) seen — M1
    • \((-2, -3)\) seen — M1
    • Rotation of \(180^\circ\) about the origin — A1

Quick check

  1. 1

    What single transformation is a reflection in the \(x\)-axis followed by a reflection in the \(y\)-axis?

    1. AA rotation of \(180^\circ\) about the origin
    2. BA translation
    3. CA reflection in the line \(y = x\)
    4. DAn enlargement with scale factor \(-1\)
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    A: A rotation of \(180^\circ\) about the origin

    \((x, y)\) goes to \((x, -y)\) and then to \((-x, -y)\), which is a half turn.

  2. 2

    What do two reflections in parallel lines give?

    1. AA rotation
    2. BA reflection
    3. CAn enlargement
    4. DA translation
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    D: A translation

    The shape is moved in a straight line, at right angles to the mirror lines.

  3. 3

    The point \((1, 1)\) is reflected in \(x = 2\) and then in \(x = 5\). Which translation has the same effect?

    1. A\(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 0 \\ 6 \end{pmatrix}\)
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    C: \(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\)

    The lines are 3 apart, and the translation is twice that distance, 6, at right angles to them.

  4. 4

    Which points are invariant in a reflection?

    1. AEvery point
    2. BThe points on the mirror line
    3. COnly the origin
    4. DThere are none
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    B: The points on the mirror line

    Points on the mirror line do not move.

  5. 5

    Which point is invariant in a rotation?

    1. AThe centre of rotation
    2. BEvery point on the shape
    3. CThe corner of the shape
    4. DThere are none
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    A: The centre of rotation

    The centre stays fixed while everything else turns around it.

  6. 6

    The point \((2, 1)\) is reflected in the line \(y = x\) and then in the \(x\)-axis. What is the final image?

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

    Swapping gives \((1, 2)\), and changing the sign of \(y\) gives \((1, -2)\).

  7. 7

    Two mirror lines cross at a point, with an angle of \(30^\circ\) between them. What single transformation is a reflection in one followed by the other?

    1. AA rotation of \(30^\circ\) about the crossing point
    2. BA translation of 30 units
    3. CA rotation of \(60^\circ\) about the crossing point
    4. DA reflection in the crossing point
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    C: A rotation of \(60^\circ\) about the crossing point

    The rotation is through twice the angle between the lines.

  8. 8

    Does a translation have an invariant point?

    1. AYes, the origin
    2. BNo, every point moves
    3. CYes, the centre
    4. DOnly for small shapes
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    B: No, every point moves

    A translation moves every point by the same vector.

  9. 9

    Two rotations of \(90^\circ\) clockwise about the same centre are carried out one after the other. What single transformation is this?

    1. AA rotation of \(180^\circ\) about that centre
    2. BA rotation of \(90^\circ\) about that centre
    3. CA reflection
    4. DA translation
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    A: A rotation of \(180^\circ\) about that centre

    \(90^\circ + 90^\circ = 180^\circ\).