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Exam questions · Maths · Transformations and Similarity

Reflections and Translations

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

    Write down the coordinates of the image of the point \((-2, 6)\) after a reflection in the \(x\)-axis. [2 marks]

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    Model answer

    A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate, so the image is \((-2, -6)\).

    Mark scheme

    • The \(y\)-coordinate changes sign — M1
    • \((-2, -6)\) — A1
  2. 2 Reflect [4 marks]

    Triangle \(S\) is drawn on the grid. (a) Reflect triangle \(S\) in the line \(y = -1\). [2 marks] (b) Translate triangle \(S\) by the vector \(\begin{pmatrix} -5 \\ 2 \end{pmatrix}\). [2 marks]

    Triangle S on a grid with the mirror line y equals minus 1.
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    Model answer

    (a) The vertices \((1, 1)\), \((4, 1)\) and \((1, 3)\) are 2, 2 and 4 above \(y = -1\), so the image has vertices \((1, -3)\), \((4, -3)\) and \((1, -5)\). (b) Subtracting 5 from each \(x\) and adding 2 to each \(y\) gives \((-4, 3)\), \((-1, 3)\) and \((-4, 5)\).

    Mark scheme

    • (a) Reflects at least two vertices correctly — M1
    • (a) Triangle with vertices \((1, -3)\), \((4, -3)\) and \((1, -5)\) — A1
    • (b) Translates at least two vertices correctly — M1
    • (b) Triangle with vertices \((-4, 3)\), \((-1, 3)\) and \((-4, 5)\) — A1
  3. 3 Describe [3 marks]

    Triangle \(B\) is the image of triangle \(A\) after a single transformation. (a) Describe fully the single transformation. [2 marks] (b) Write down the coordinates of the image of the vertex \((3, 3)\) of triangle \(A\). [1 mark]

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

    (a) The coordinates swap and change sign, so \((x, y)\) goes to \((-y, -x)\). It is a reflection in the line \(y = -x\). (b) \((3, 3)\) goes to \((-3, -3)\).

    Mark scheme

    • (a) Reflection — B1
    • (a) In the line \(y = -x\) — B1
    • (b) \((-3, -3)\) — B1
  4. 4 Calculate [3 marks]

    (a) Calculate the column vector of the translation that maps \((3, -2)\) onto \((-1, 4)\). [2 marks] (b) The same translation maps \((5, 5)\) onto the point \(Q\). Write down the coordinates of \(Q\). [1 mark]

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    Model answer

    (a) The change in \(x\) is \(-1 - 3 = -4\) and the change in \(y\) is \(4 - (-2) = 6\), so the vector is \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\). (b) \(Q = (5 - 4, 5 + 6) = (1, 11)\).

    Mark scheme

    • (a) \(-1 - 3\) or \(4 - (-2)\) — M1
    • (a) \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\) — A1
    • (b) \((1, 11)\) — B1 (follow through from (a))
  5. 5 Find [2 marks]

    The point \(P\) is \((4, -3)\). \(P\) is reflected in the line \(x = 1\) to give \(P'\). Find the coordinates of \(P'\). [2 marks]

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    Model answer

    \(P\) is 3 to the right of the line, so \(P'\) is 3 to the left, at \((1 - 3, -3) = (-2, -3)\).

    Mark scheme

    • Distance 3 from the line — M1
    • \((-2, -3)\) — A1
  6. 6 Calculate [3 marks]

    The point \(A\) is \((1, 4)\). \(A\) is reflected in the line \(y = -x\) to give \(B\). \(B\) is translated by the vector \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\) to give \(C\). Calculate the coordinates of \(C\). [3 marks]

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    Model answer

    Reflecting in \(y = -x\) sends \((x, y)\) to \((-y, -x)\), so \(B = (-4, -1)\). Then \(C = (-4 + 2, -1 + 1) = (-2, 0)\).

    Mark scheme

    • \(B = (-4, -1)\) — B1
    • Adds the vector to their \(B\) — M1
    • \((-2, 0)\) — A1

Quick check

  1. 1

    What is the image of the point \((3, 5)\) in the \(x\)-axis?

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

    A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate.

  2. 2

    What is the image of the point \((2, 3)\) in the \(y\)-axis?

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

    A reflection in the \(y\)-axis changes the sign of the \(x\)-coordinate.

  3. 3

    What is the image of \((4, 1)\) in the line \(y = x\)?

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

    In the line \(y = x\) the coordinates swap.

  4. 4

    The point \((5, 2)\) is reflected in the line \(x = 3\). What is the image?

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

    \((5, 2)\) is 2 right of the line, so the image is 2 left of it, at \((1, 2)\).

  5. 5

    What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

    1. A2 right and 5 up
    2. B2 left and 5 up
    3. C5 left and 2 up
    4. D2 left and 5 down
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    B: 2 left and 5 up

    The top number is left or right, and the bottom number is up or down.

  6. 6

    The point \((4, 1)\) is translated by \(\begin{pmatrix} -3 \\ 2 \end{pmatrix}\). What is the image?

    1. A\((1, 3)\)
    2. B\((7, 3)\)
    3. C\((1, -1)\)
    4. D\((7, -1)\)
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    A: \((1, 3)\)

    \((4 - 3, 1 + 2) = (1, 3)\).

  7. 7

    A translation takes \((2, 5)\) to \((7, 3)\). What is the column vector?

    1. A\(\begin{pmatrix} -5 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 9 \\ 8 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)
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    D: \(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)

    The change in \(x\) is \(7 - 2 = 5\) and the change in \(y\) is \(3 - 5 = -2\).

  8. 8

    What must you give to describe a reflection fully?

    1. AThe scale factor
    2. BThe centre and the angle
    3. CThe equation of the mirror line
    4. DA column vector
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    C: The equation of the mirror line

    A reflection is described by its mirror line, such as \(x = 1\).

  9. 9

    What is the image of \((4, 1)\) in the line \(y = -x\)?

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

    In the line \(y = -x\) the coordinates swap and both change sign.