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Cosine rule with an obtuse included angle, a side expressed in terms of x, and the exact area of the triangle

Cosine rule with an obtuse included angle, a side expressed in terms of x, and the exact area of the triangle

14 marks

In triangle \(DEF\), \(DF=8\), \(EF=7\) and \(D\hat{F}E=120^\circ\).

  1. Find the length of \(DE\).[3]

In another triangle \(ABC\), \(AC=x+1\), \(BC=3x-1\) and \(A\hat{C}B=120^\circ\), where \(x>\frac13\).

  1. Show that \(AB=\sqrt{13x^2-2x+1}\).[5]
  2. It is now given that \(AB=7\).
    1. Find the value of \(x\).
    2. Find the exact area of triangle \(ABC\).[6]

Solution:

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Answer:(a) \(DE=13\) (c)(i) \(x=2\) (c)(ii) \(\frac{15\sqrt3}{4}\)

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