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Exam questions · Maths · Transformations and Similarity

Congruence and Similarity

  • 6 exam questions
  • 17 marks
  • 9 quick checks
  1. 1 Give a reason [2 marks]

    Triangle \(ABC\) has sides of 3 cm, 4 cm and 5 cm. Triangle \(PQR\) has sides of 3 cm, 4 cm and 5 cm. Are the triangles congruent? Give a reason for your answer.

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    Model answer

    Yes. All three sides of one triangle are equal to the three sides of the other, so they are congruent by SSS.

    Mark scheme

    • Yes — B1
    • All three pairs of sides are equal, or SSS — C1
  2. 2 Prove [4 marks]

    \(ABCD\) is a kite with \(AB = AD\) and \(BC = DC\). (a) Prove that triangles \(ABC\) and \(ADC\) are congruent. (3 marks) (b) Hence write down the size of angle \(ABC\) compared with angle \(ADC\). (1 mark)

    A kite ABCD with AB equal to AD, BC equal to DC and the diagonal AC drawn.
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    Model answer

    (a) \(AB = AD\) (given), \(BC = DC\) (given) and \(AC\) is a side of both triangles. All three sides are equal, so the triangles are congruent by SSS. (b) Corresponding angles in congruent triangles are equal, so angle \(ABC\) equals angle \(ADC\).

    Mark scheme

    • \(AB = AD\) and \(BC = DC\) given — M1
    • \(AC\) is a common side — M1
    • SSS, so the triangles are congruent — C1
    • (b) Angle \(ABC\) = angle \(ADC\) — B1
  3. 3 Work out [3 marks]

    Triangles \(ABC\) and \(DEF\) are similar. (a) Work out the scale factor of the enlargement from \(ABC\) to \(DEF\). (1 mark) (b) Work out the length of \(EF\). (1 mark) (c) Work out the length of \(FD\). (1 mark)

    Two similar triangles ABC and DEF, with the sides of ABC 5, 6 and 7 centimetres and DE equal to 10 centimetres.
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    Model answer

    (a) \(DE\) matches \(AB\), so the scale factor is \(\dfrac{10}{5} = 2\). (b) \(EF = 6 \times 2 = 12\) cm. (c) \(FD = 7 \times 2 = 14\) cm.

    Mark scheme

    • (a) 2 — B1
    • (b) 12 — B1
    • (c) 14 — B1
  4. 4 Work out [2 marks]

    Triangles \(ABC\) and \(DEF\) are similar. \(AB = 4\) cm, \(BC = 6\) cm and \(DE = 10\) cm, where \(DE\) corresponds to \(AB\) and \(EF\) corresponds to \(BC\). Work out the length of \(EF\).

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    Model answer

    The scale factor is \(\dfrac{10}{4} = 2.5\), so \(EF = 6 \times 2.5 = 15\) cm.

    Mark scheme

    • \(\dfrac{10}{4}\) or 2.5 — M1
    • 15 cm — A1
  5. 5 Work out [3 marks]

    Two similar shapes have lengths in the ratio \(2 : 3\). The area of the smaller shape is 20 cm\(^2\). Work out the area of the larger shape.

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    Model answer

    The area scale factor is \(\left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\). The larger area is \(20 \times \dfrac{9}{4} = 45\) cm\(^2\).

    Mark scheme

    • \(\left(\dfrac{3}{2}\right)^2\) or \(\dfrac{9}{4}\) — M1
    • \(20 \times \dfrac{9}{4}\) — M1
    • 45 cm\(^2\) — A1
  6. 6 Work out [3 marks]

    Two similar solids have volumes in the ratio \(8 : 27\). The surface area of the smaller solid is 36 cm\(^2\). Work out the surface area of the larger solid.

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    Model answer

    The volumes are in the ratio \(8 : 27\), so the lengths are in the ratio \(\sqrt[3]{8} : \sqrt[3]{27} = 2 : 3\). The area ratio is \(4 : 9\), so the larger surface area is \(36 \times \dfrac{9}{4} = 81\) cm\(^2\).

    Mark scheme

    • Length ratio \(2 : 3\) — M1
    • Area ratio \(4 : 9\) or the scale factor \(\dfrac{9}{4}\) — M1
    • 81 cm\(^2\) — A1

Quick check

  1. 1

    Which of these is not enough to show that two triangles are congruent?

    1. AThree equal sides
    2. BThree equal angles
    3. CTwo sides and the angle between them
    4. DA right angle, the hypotenuse and another side
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    B: Three equal angles

    Three equal angles give similar triangles, which may be different sizes.

  2. 2

    Two triangles have two equal sides and the equal angle between them. Which condition is this?

    1. ASAS
    2. BSSS
    3. CASA
    4. DRHS
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    A: SAS

    Side, angle, side with the angle between the sides is SAS.

  3. 3

    Two similar triangles have matching sides of 6 cm and 15 cm. What is the scale factor from the smaller to the larger?

    1. A\(9\)
    2. B\(0.4\)
    3. C\(90\)
    4. D\(2.5\)
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    D: \(2.5\)

    \(\dfrac{15}{6} = 2.5\).

  4. 4

    A triangle with sides 3, 4 and 5 cm is similar to a triangle with sides 9 cm, \(x\) cm and 15 cm. What is \(x\)?

    1. A\(8\)
    2. B\(10\)
    3. C\(12\)
    4. D\(7\)
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    C: \(12\)

    The scale factor is \(\dfrac{9}{3} = 3\), so \(x = 4 \times 3 = 12\).

  5. 5

    Triangles \(ABC\) and \(DEF\) are similar, with \(AB = 4\), \(DE = 10\) and \(BC = 6\). What is \(EF\)?

    1. A4 cm
    2. B15 cm
    3. C9.6 cm
    4. D7.5 cm
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    B: 15 cm

    The scale factor is \(\dfrac{10}{4} = 2.5\), so \(EF = 6 \times 2.5 = 15\) cm.

  6. 6

    Two triangles have all three angles equal. What can you say?

    1. AThey are similar
    2. BThey are congruent
    3. CThey have the same area
    4. DThey are enlargements with scale factor 1
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    A: They are similar

    Equal angles make the shapes similar, but not necessarily the same size.

  7. 7

    The lengths of a shape are doubled. By what factor is the area multiplied?

    1. A2
    2. B8
    3. C16
    4. D4
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    D: 4

    The area scale factor is \(2^2 = 4\).

  8. 8

    The lengths of a solid are multiplied by 3. By what factor is the volume multiplied?

    1. A9
    2. B3
    3. C27
    4. D6
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    C: 27

    The volume scale factor is \(3^3 = 27\).

  9. 9

    Two similar shapes have areas in the ratio \(9 : 25\). What is the ratio of their lengths?

    1. A\(9 : 25\)
    2. B\(3 : 5\)
    3. C\(81 : 625\)
    4. D\(4.5 : 12.5\)
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    B: \(3 : 5\)

    Take square roots: \(\sqrt{9} : \sqrt{25} = 3 : 5\).