OpenRevise

Exam questions · Maths · Transformations and Similarity

Congruence and Similarity

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

    Triangle \(ABC\) has \(AB = 6\) cm, \(BC = 8\) cm and angle \(B = 50^\circ\). Triangle \(PQR\) has \(PQ = 6\) cm, \(QR = 8\) cm and angle \(Q = 50^\circ\). Show that the triangles are congruent. [2 marks]

    Show answerHide answer

    Model answer

    Two sides and the angle between them are equal: \(AB = PQ\), \(BC = QR\) and angle \(B\) equals angle \(Q\). The triangles are congruent by SAS.

    Mark scheme

    • Two pairs of equal sides and the equal angle between them — M1
    • SAS — C1
  2. 2 Prove [4 marks]

    \(ABCD\) is a parallelogram. The diagonal \(AC\) is drawn. (a) Prove that triangles \(ABC\) and \(CDA\) are congruent. [3 marks] (b) Hence write down the size of angle \(ABC\) compared with angle \(CDA\). [1 mark]

    A parallelogram ABCD with opposite sides marked equal and the diagonal AC drawn.
    Show answerHide answer

    Model answer

    (a) \(AB = CD\) and \(BC = DA\), because opposite sides of a parallelogram are equal. \(AC\) is a side of both triangles. All three sides are equal, so the triangles are congruent by SSS. (b) Angle \(ABC\) equals angle \(CDA\).

    Mark scheme

    • Opposite sides equal, with the reason — M1
    • \(AC\) is a common side — M1
    • SSS, so the triangles are congruent — Q1
    • (b) Angle \(ABC\) = angle \(CDA\) — B1
  3. 3 Work out [3 marks]

    Triangles \(ABC\) and \(DEF\) are similar. (a) Work out the scale factor 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 2, 3 and 4 centimetres and DE equal to 6 centimetres.
    Show answerHide answer

    Model answer

    (a) \(DE\) matches \(AB\), so the scale factor is \(\dfrac{6}{2} = 3\). (b) \(EF = 3 \times 3 = 9\) cm. (c) \(FD = 4 \times 3 = 12\) cm.

    Mark scheme

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

    Rectangles \(A\) and \(B\) are similar. Rectangle \(A\) is 4 cm long and 6 cm wide. The length of rectangle \(B\) is 10 cm. (a) Work out the width of rectangle \(B\). [2 marks] (b) Work out the perimeter of rectangle \(B\). [1 mark]

    Show answerHide answer

    Model answer

    (a) The scale factor is \(\dfrac{10}{4} = 2.5\), so the width is \(6 \times 2.5 = 15\) cm. (b) \(2 \times (10 + 15) = 50\) cm.

    Mark scheme

    • (a) \(\dfrac{10}{4}\) or 2.5 — M1
    • (a) 15 cm — A1
    • (b) 50 cm — B1 (follow through from (a))
  5. 5 Work out [3 marks]

    Two similar prisms have lengths in the ratio \(3 : 5\). The surface area of the smaller prism is 36 cm\(^2\). Work out the surface area of the larger prism. [3 marks]

    Show answerHide answer

    Model answer

    The area scale factor is \(\left(\dfrac{5}{3}\right)^2 = \dfrac{25}{9}\). The larger surface area is \(36 \times \dfrac{25}{9} = 100\) cm\(^2\).

    Mark scheme

    • \(\left(\dfrac{5}{3}\right)^2\) or \(\dfrac{25}{9}\) — M1
    • \(36 \times \dfrac{25}{9}\) — M1
    • 100 cm\(^2\) — A1
  6. 6 Work out [3 marks]

    Two similar cones have volumes of 40 cm\(^3\) and 135 cm\(^3\). The radius of the smaller cone is 6 cm. Work out the radius of the larger cone. [3 marks]

    Show answerHide answer

    Model answer

    The volume ratio is \(40 : 135 = 8 : 27\), so the length ratio is \(2 : 3\). The radius of the larger cone is \(6 \times \dfrac{3}{2} = 9\) cm.

    Mark scheme

    • \(40 : 135 = 8 : 27\) — M1
    • Length scale factor \(\dfrac{3}{2}\) — M1
    • 9 cm — 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
    Show answerHide answer

    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
    Show answerHide answer

    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\(2.5\)
    Show answerHide answer

    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\(7\)
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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\(4.5 : 12.5\)
    Show answerHide answer

    B: \(3 : 5\)

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