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Exam questions · Maths · Transformations and Similarity

Enlargements

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

    The point \((4, 6)\) is enlarged by a scale factor of \(\dfrac{1}{2}\) with the centre \((0, 0)\). Write down the coordinates of the image. [2 marks]

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    Model answer

    Multiply each coordinate by \(\dfrac{1}{2}\): \((2, 3)\).

    Mark scheme

    • \(4 \times \dfrac{1}{2}\) or \(6 \times \dfrac{1}{2}\) — M1
    • \((2, 3)\) — A1
  2. 2 Enlarge [3 marks]

    Enlarge triangle \(T\) by a scale factor of 2 with the centre \(C\). [3 marks]

    Triangle T on a grid with the centre of enlargement C at (1, 1).
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    Model answer

    The centre is \((1, 1)\). \((2, 2)\) is 1 right and 1 up, so the image is 2 right and 2 up, at \((3, 3)\). \((4, 2)\) goes to \((1 + 6, 1 + 2) = (7, 3)\) and \((2, 3)\) goes to \((1 + 2, 1 + 4) = (3, 5)\).

    Mark scheme

    • Enlarges at least two vertices from the centre \((1, 1)\) with a scale factor of 2 — M1
    • At least two vertices correct — A1
    • Triangle with vertices \((3, 3)\), \((7, 3)\) and \((3, 5)\) — A1
  3. 3 Describe [3 marks]

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

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

    The bottom of \(A\) is 1 long and the bottom of \(B\) is 3 long, so the scale factor is 3. Matching corners \((1, 1)\) and \((3, 3)\), and \((2, 1)\) and \((6, 3)\), lie on lines through the origin. So it is an enlargement, scale factor 3, centre \((0, 0)\).

    Mark scheme

    • Enlargement — B1
    • Scale factor 3 — B1
    • Centre \((0, 0)\) — B1
  4. 4 Work out [3 marks]

    Triangle \(ABC\) is enlarged by a scale factor of 2.5 to give triangle \(DEF\). \(AB = 6\) cm and the angle at \(A\) is \(40^\circ\). (a) Work out the length of \(DE\). [2 marks] (b) Write down the size of the angle at \(D\). [1 mark]

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    Model answer

    (a) \(DE = 6 \times 2.5 = 15\) cm. (b) Angles do not change in an enlargement, so the angle at \(D\) is \(40^\circ\).

    Mark scheme

    • (a) \(6 \times 2.5\) — M1
    • (a) 15 cm — A1
    • (b) \(40^\circ\) — B1
  5. 5 Work out [3 marks]

    The point \(A\) is \((3, 2)\). \(A\) is enlarged by a scale factor of 3 with the centre \((2, 1)\). Work out the coordinates of the image of \(A\). [3 marks]

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    Model answer

    \(A\) is 1 right and 1 up from the centre. The image is 3 right and 3 up from the centre, at \((2 + 3, 1 + 3) = (5, 4)\).

    Mark scheme

    • Position relative to the centre, \((1, 1)\) — M1
    • \((3, 3)\) relative to the centre — M1
    • \((5, 4)\) — A1
  6. 6 Work out [3 marks]

    The point \(P\) is \((2, -1)\). \(P\) is enlarged by a scale factor of \(-3\) with the centre \((0, 0)\). Write down the coordinates of the image of \(P\). [3 marks]

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    Model answer

    Multiply each coordinate by \(-3\): \((2 \times (-3), -1 \times (-3)) = (-6, 3)\).

    Mark scheme

    • \(2 \times (-3)\) or \(-1 \times (-3)\) — M1
    • \((-6, \ldots)\) or \((\ldots, 3)\) — A1
    • \((-6, 3)\) — A1

Quick check

  1. 1

    A side of 3 cm is enlarged to 12 cm. What is the scale factor?

    1. A\(9\)
    2. B\(\dfrac{1}{4}\)
    3. C\(36\)
    4. D\(4\)
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    D: \(4\)

    \(\dfrac{12}{3} = 4\).

  2. 2

    The point \((2, 3)\) is enlarged by scale factor 3 with centre \((0, 0)\). What is the image?

    1. A\((5, 6)\)
    2. B\((6, 3)\)
    3. C\((6, 9)\)
    4. D\((9, 6)\)
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    C: \((6, 9)\)

    Multiply both coordinates by 3.

  3. 3

    What happens to the angles in an enlargement?

    1. AThey are multiplied by the scale factor
    2. BThey stay the same
    3. CThey are halved
    4. DThey are added to the scale factor
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    B: They stay the same

    An enlargement keeps the angles, so the shapes are similar.

  4. 4

    What does an enlargement with scale factor \(\dfrac{1}{2}\) do to a shape?

    1. AIt halves every length
    2. BIt doubles every length
    3. CIt halves every angle
    4. DIt leaves the shape unchanged
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    A: It halves every length

    A fractional scale factor less than 1 makes the shape smaller.

  5. 5

    The centre of enlargement is \((1, 2)\) and the scale factor is 3. What is the image of \((3, 3)\)?

    1. A\((9, 9)\)
    2. B\((5, 4)\)
    3. C\((6, 5)\)
    4. D\((7, 5)\)
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    D: \((7, 5)\)

    \((3, 3)\) is 2 right and 1 up from the centre, so the image is 6 right and 3 up: \((7, 5)\).

  6. 6

    What must you give to describe an enlargement fully?

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

    An enlargement is described by its scale factor and its centre.

  7. 7

    A triangle with sides 4, 6 and 8 is enlarged to give a triangle with sides 10, 15 and 20. What is the scale factor?

    1. A\(2\)
    2. B\(2.5\)
    3. C\(6\)
    4. D\(3\)
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    B: \(2.5\)

    \(\dfrac{10}{4} = 2.5\).

  8. 8

    Which transformation is the same as an enlargement with scale factor \(-1\)?

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

    Every point goes to the opposite side of the centre at the same distance.

  9. 9

    The point \((1, 3)\) is enlarged by scale factor \(-2\) with centre \((0, 0)\). What is the image?

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

    Multiply both coordinates by \(-2\).