OpenRevise

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 \(y\)-axis 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 in the first quadrant.
    Show answerHide answer

    Model answer

    (a) \(B\) has vertices \((-1, 2)\), \((-3, 2)\) and \((-1, 5)\), and \(C\) has vertices \((-1, -2)\), \((-3, -2)\) and \((-1, -5)\). (b) \((x, y)\) goes to \((-x, -y)\), which is a rotation of \(180^\circ\) about the origin.

    Mark scheme

    • (a) \(B\) correct — B1
    • (a) \(C\) correct — B1
    • (b) Rotation, \(180^\circ\) — B1
    • (b) About the origin — B1
  2. 2 Describe [3 marks]

    The point \((2, 3)\) is reflected in the line \(x = 1\), and the image is then reflected in the line \(x = 5\). Describe the single transformation that has the same effect.

    Show answerHide answer

    Model answer

    The first reflection gives \((0, 3)\) and the second gives \((10, 3)\). The point has moved 8 to the right, which is twice the distance between the lines, \(2 \times 4 = 8\). So it is a translation by \(\begin{pmatrix} 8 \\ 0 \end{pmatrix}\).

    Mark scheme

    • \((0, 3)\) seen — M1
    • \((10, 3)\) seen — M1
    • Translation by \(\begin{pmatrix} 8 \\ 0 \end{pmatrix}\) — A1
  3. 3 Write down [2 marks]

    (a) A shape is rotated through \(90^\circ\) about the point \((2, 1)\). Write down the coordinates of the point that does not move. (1 mark) (b) A shape is reflected in the line \(y = 3\). Describe the points that do not move. (1 mark)

    Show answerHide answer

    Model answer

    (a) The centre of rotation, \((2, 1)\), is invariant. (b) Every point on the line \(y = 3\) stays where it is.

    Mark scheme

    • (a) \((2, 1)\) — B1
    • (b) The points on the line \(y = 3\) — B1
  4. 4 Work out [3 marks]

    The point \(P\) is \((3, 1)\). \(P\) is translated by the vector \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) and the image is then reflected in the \(x\)-axis. Write down the coordinates of the final image.

    Show answerHide answer

    Model answer

    The translation gives \((1, 4)\), and the reflection in the \(x\)-axis gives \((1, -4)\).

    Mark scheme

    • \((1, 4)\) — B1
    • Changes the sign of the \(y\)-coordinate of their image — M1
    • \((1, -4)\) — A1
  5. 5 Show that [3 marks]

    Show that a reflection in the line \(y = x\) followed by a reflection in the line \(y = -x\) is the same as a rotation of \(180^\circ\) about the origin. Use the point \((2, 5)\).

    Show answerHide answer

    Model answer

    Reflecting \((2, 5)\) in \(y = x\) swaps the coordinates, giving \((5, 2)\). Reflecting in \(y = -x\) swaps and changes both signs, giving \((-2, -5)\). A \(180^\circ\) rotation about the origin sends \((2, 5)\) to \((-2, -5)\), the same point.

    Mark scheme

    • \((5, 2)\) seen — M1
    • \((-2, -5)\) seen — M1
    • Compares with the \(180^\circ\) rotation, \((-2, -5)\) — C1
  6. 6 Describe [4 marks]

    The point \((4, -1)\) is rotated through \(90^\circ\) clockwise about the origin, and the image is then reflected in the \(y\)-axis. Describe fully the single transformation that has the same effect.

    Show answerHide answer

    Model answer

    The rotation gives \((-1, -4)\), and the reflection gives \((1, -4)\). In general \((x, y)\) goes to \((y, -x)\) and then to \((-y, -x)\). That is a reflection in the line \(y = -x\).

    Mark scheme

    • \((-1, -4)\) seen — M1
    • \((1, -4)\) seen — M1
    • Reflection — A1
    • In the line \(y = -x\) — 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 \(-1\)
    Show answerHide answer

    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
    Show answerHide answer

    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\(\begin{pmatrix} 0 \\ 6 \end{pmatrix}\)
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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\((1, -2)\)
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    A: A rotation of \(180^\circ\) about that centre

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