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Exam questions · Maths · Transformations and Similarity

Reflections and Translations

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

    The point \((4, -1)\) is translated by the vector \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\). Write down the coordinates of the image. [2 marks]

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    Model answer

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

    Mark scheme

    • \(4 - 2\) or \(-1 + 3\) — M1
    • \((2, 2)\) — A1
  2. 2 Reflect [3 marks]

    Triangle \(T\) is drawn on the grid. (a) Reflect triangle \(T\) in the line \(y = x\). [2 marks] (b) Write down the coordinates of the image of the point \((6, 2)\) in the line \(y = x\). [1 mark]

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

    (a) The coordinates swap, so \((3, 1)\), \((5, 1)\) and \((5, 4)\) go to \((1, 3)\), \((1, 5)\) and \((4, 5)\). (b) \((2, 6)\).

    Mark scheme

    • (a) Reflects at least two vertices correctly — M1
    • (a) Triangle with vertices \((1, 3)\), \((1, 5)\) and \((4, 5)\) — A1
    • (b) \((2, 6)\) — B1
  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) Triangle \(A\) has an area of 3 square units. Write down the area of triangle \(B\). [1 mark]

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

    (a) The point \((1, 2)\) goes to \((4, -1)\), which is 3 right and 3 down. It is a translation by \(\begin{pmatrix} 3 \\ -3 \end{pmatrix}\). (b) A translation does not change the size, so the area is 3 square units.

    Mark scheme

    • (a) Translation — B1
    • (a) \(\begin{pmatrix} 3 \\ -3 \end{pmatrix}\) — B1
    • (b) 3 — B1
  4. 4 Find [2 marks]

    A reflection maps the point \((5, 1)\) onto the point \((5, 7)\). Find the equation of the mirror line. [2 marks]

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    Model answer

    The mirror line is halfway between the two points, and the points differ only in \(y\). The midpoint of the \(y\)-values is \(\dfrac{1 + 7}{2} = 4\), so the mirror line is \(y = 4\).

    Mark scheme

    • \(\dfrac{1 + 7}{2}\) or 4 seen — M1
    • \(y = 4\) — A1
  5. 5 Work out [3 marks]

    Triangle \(T\) has vertices \((1, 1)\), \((4, 1)\) and \((1, 3)\). \(T\) is reflected in the line \(x = -1\). Write down the coordinates of the image of (a) the vertex \((4, 1)\), (b) the vertex \((1, 3)\). [3 marks]

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    Model answer

    (a) \((4, 1)\) is 5 to the right of \(x = -1\), so the image is 5 to the left, at \((-6, 1)\). (b) \((1, 3)\) is 2 to the right, so the image is at \((-3, 3)\).

    Mark scheme

    • (a) Distance 5 from the line, or 5 seen — M1
    • (a) \((-6, 1)\) — A1
    • (b) \((-3, 3)\) — B1
  6. 6 Work out [3 marks]

    The point \(P\) is \((2, 5)\). \(P\) is reflected in the line \(y = x\) to give \(Q\). \(Q\) is translated by the vector \(\begin{pmatrix} -3 \\ -2 \end{pmatrix}\) to give \(R\). Write down the coordinates of \(R\). [3 marks]

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    Model answer

    Reflecting in \(y = x\) gives \(Q = (5, 2)\). Translating gives \(R = (5 - 3, 2 - 2) = (2, 0)\).

    Mark scheme

    • \(Q = (5, 2)\) — B1
    • Adds the vector to their \(Q\) — 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.