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Exam questions · Maths · Transformations and Similarity

Congruence and Similarity

  • 6 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Explain [2 marks]

    Two triangles each have angles of \(40^\circ\), \(60^\circ\) and \(80^\circ\). Are the triangles congruent? Explain your answer. [2 marks]

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    Model answer

    Not necessarily. Equal angles show the triangles are similar, but they could be different sizes, so they are not necessarily congruent.

    Mark scheme

    • Not necessarily — B1
    • They could be different sizes, or the angles only show similarity — B1
  2. 2 Prove [4 marks]

    Triangle \(ABC\) is isosceles with \(AB = AC\). \(D\) is a point on \(BC\) such that \(AD\) is perpendicular to \(BC\). (a) Prove that triangles \(ABD\) and \(ACD\) are congruent. [3 marks] (b) Hence show that \(BD = DC\). [1 mark]

    An isosceles triangle ABC with AB equal to AC and the line AD drawn perpendicular to BC.
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    Model answer

    (a) Both triangles have a right angle at \(D\), the hypotenuses \(AB\) and \(AC\) are equal (given), and \(AD\) is a common side. The triangles are congruent by RHS. (b) Corresponding sides of congruent triangles are equal, so \(BD = DC\).

    Mark scheme

    • Right angle at D in both triangles — M1
    • \(AB = AC\) and \(AD\) common — M1
    • RHS, so the triangles are congruent — A1
    • (b) \(BD = DC\) from the congruent triangles — B1
  3. 3 Calculate [3 marks]

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

    Two similar triangles ABC and DEF, with the sides of ABC 10, 8 and 6 centimetres and DE equal to 15 centimetres.
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    Model answer

    (a) \(DE\) matches \(AB\), so the scale factor is \(\dfrac{15}{10} = 1.5\). (b) \(EF = 8 \times 1.5 = 12\) cm. (c) \(FD = 6 \times 1.5 = 9\) cm.

    Mark scheme

    • (a) 1.5 — B1
    • (b) 12 — B1
    • (c) 9 — B1
  4. 4 Calculate [3 marks]

    A photograph is 10 cm by 15 cm. It is enlarged so that the shorter side is 25 cm. (a) Calculate the length of the longer side of the enlarged photograph. [2 marks] (b) Calculate the perimeter of the enlarged photograph. [1 mark]

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    Model answer

    (a) The scale factor is \(\dfrac{25}{10} = 2.5\), so the longer side is \(15 \times 2.5 = 37.5\) cm. (b) \(2 \times (25 + 37.5) = 125\) cm.

    Mark scheme

    • (a) \(\dfrac{25}{10}\) or 2.5 — M1
    • (a) 37.5 cm — A1
    • (b) 125 cm — B1 (follow through from (a))
  5. 5 Calculate [3 marks]

    Two similar shapes have corresponding lengths of 6 cm and 15 cm. The area of the smaller shape is 8 cm\(^2\). Calculate the area of the larger shape. [3 marks]

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    Model answer

    The length scale factor is \(\dfrac{15}{6} = 2.5\), so the area scale factor is \(2.5^2 = 6.25\). The larger area is \(8 \times 6.25 = 50\) cm\(^2\).

    Mark scheme

    • \(\dfrac{15}{6}\) or 2.5 — M1
    • \(8 \times 2.5^2\) or \(8 \times 6.25\) — M1
    • 50 cm\(^2\) — A1
  6. 6 Calculate [3 marks]

    Two similar bottles have heights of 15 cm and 20 cm. The smaller bottle holds 270 ml. Calculate the volume of the larger bottle. [3 marks]

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    Model answer

    The length scale factor is \(\dfrac{20}{15} = \dfrac{4}{3}\), so the volume scale factor is \(\left(\dfrac{4}{3}\right)^3 = \dfrac{64}{27}\). The larger bottle holds \(270 \times \dfrac{64}{27} = 640\) ml.

    Mark scheme

    • \(\dfrac{20}{15}\) or \(\dfrac{4}{3}\) — M1
    • \(270 \times \left(\dfrac{4}{3}\right)^3\) or \(270 \times \dfrac{64}{27}\) — M1
    • 640 ml — 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: \(\sqrt{9} : \sqrt{25} = 3 : 5\).