[discia] GCSE Maths › Geometry & Measures › The Cosine Rule

The Cosine Rule

discia.org 10 questions · 43 marks
Name Date
  1. Calculator 3 marks

    \(ABC\) is a triangle.

    10 cmx cm15 cmABC68°Diagram NOT accurately drawn

    Work out the length of \(BC\).

    Give your answer to 3 significant figures.

    Show answer
    \(14.6\ \text{cm}\)
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  2. Calculator 4 marks

    In triangle \(ABC\), \(AB = 9\) cm, \(BC = 16\) cm and angle \(ABC = 127^\circ\).

    (a)

    Write down which of the sine rule and the cosine rule you would use to work out the length of \(AC\).

    (b)

    Use that rule to find the length of \(AC\).

    Give your answer to 1 decimal place.

    Show answer
    (a) the cosine rule
    (b) \(22.6\ \text{cm}\)
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  3. Calculator 3 marks

    \(ABC\) is a triangle.

    9 cm6 cm7 cmABCxDiagram NOT accurately drawn

    Work out the size of the angle marked \(x\).

    Give your answer to 1 decimal place.

    Show answer
    \(41.8^\circ\)
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  4. Calculator 4 marks

    Two straight paths leave a youth hostel. The angle between them is \(101^\circ\).

    Zainab walks \(11\) km along one path. Oscar walks \(13\) km along the other.

    Work out the distance between Zainab and Oscar.

    Give your answer correct to 1 decimal place.

    Show answer
    \(18.6\ \text{km}\)
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  5. Calculator 4 marks

    A triangular picnic area at a campsite has sides

    \(AB = 30\) m, \(BC = 55\) m and \(AC = 45\) m.

    Work out the size of the largest angle of the picnic area.

    Give your answer correct to 1 decimal place.

    Show answer
    \(92.1^\circ\)
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  6. Calculator 4 marks

    \(ABCD\) is a parallelogram.

    17 cm13 cmABCDDiagram NOT accurately drawn

    \(AB = 17\) cm, \(BC = 13\) cm and angle \(ABC = 124^\circ\).

    Calculate the length of the diagonal \(BD\).

    Give your answer correct to 1 decimal place.

    Show answer
    \(14.5\ \text{cm}\)
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  7. Calculator 5 marks

    \(B\) is \(47\) miles from \(A\) on a bearing of \(217^\circ\).

    N217°47 milesAB

    A fishing boat sails from \(A\) on a bearing of \(284^\circ\).

    It travels at a constant speed of \(11\) mph for \(2\) hours \(30\) minutes.

    Is the fishing boat now closer to \(B\) than it was when it left \(A\)?

    You must show your working.

    Show answer
    Yes. The fishing boat is now \(44.2\) miles from \(B\), which is less than the \(47\) miles it was from \(B\) at the start.
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  8. Calculator 6 marks

    \(ABC\) is a triangle.

    12 cm15 cmABC88°Diagram NOT accurately drawn
    (a)

    Find the length of \(AC\).

    Give your answer correct to 1 decimal place.

    (b)

    Find the size of angle \(BAC\).

    Give your answer correct to the nearest degree.

    Show answer
    (a) \(18.9\ \text{cm}\)
    (b) \(53^\circ\)
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  9. Calculator 5 marks

    A helicopter flies from airfield \(A\) to hospital \(B\) on a bearing of \(127^\circ\).

    It then flies from hospital \(B\) to point \(C\) on a bearing of \(186^\circ\).

    \(AB = 22\) km and \(BC = 25\) km.

    Work out the distance \(AC\).

    Give your answer correct to 1 decimal place.

    Show answer
    \(40.9\ \text{km}\)
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  10. Calculator 5 marks

    \(ABCDEFGHI\) is a regular nonagon. Each of its sides is \(11\) cm long.

    Find the length of \(AC\).

    Give your answer correct to 1 decimal place.

    Show answer
    \(20.7\ \text{cm}\)
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discia.org The Cosine Rule · sheet 1
Answers The Cosine Rule · sheet 1
  1. \(14.6\ \text{cm}\)
  2. (a) the cosine rule
    (b) \(22.6\ \text{cm}\)
  3. \(41.8^\circ\)
  4. \(18.6\ \text{km}\)
  5. \(92.1^\circ\)
  6. \(14.5\ \text{cm}\)
  7. Yes. The fishing boat is now \(44.2\) miles from \(B\), which is less than the \(47\) miles it was from \(B\) at the start.
  8. (a) \(18.9\ \text{cm}\)
    (b) \(53^\circ\)
  9. \(40.9\ \text{km}\)
  10. \(20.7\ \text{cm}\)
discia.org The Cosine Rule · sheet 1