1. 1
   In triangle $PQR$, $PQ=8$ cm, $PR=5$ cm and $Q\hat{P}R=60^\circ$.

   (a)

   Find the length of $QR$.

   [1 mark]

   • A$49$ cm
   • B$\sqrt{129}$ cm
   • C$\sqrt{89}$ cm
   • D$7$ cm

   (b)

   Find the area of triangle $PQR$.

   [1 mark]

   • A$20\sqrt3$ cm$^2$
   • B$10$ cm$^2$
   • C$10\sqrt3$ cm$^2$
   • D$20$ cm$^2$

   (c)

   Find the exact shortest distance from $R$ to the line $PQ$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Triangle $ABC$ is right-angled at $B$, with $AB=5$ cm and $BC=12$ cm.

   (a)

   Find the value of $\tan B\hat{A}C$.

   [1 mark]

   • A$\frac{5}{12}$
   • B$\frac{12}{5}$
   • C$\frac{12}{13}$
   • D$\frac{5}{13}$

   (b)

   Find the value of $\sin A\hat{C}B$.

   [1 mark]

   • A$\frac{5}{13}$
   • B$\frac{12}{13}$
   • C$\frac{5}{12}$
   • D$\frac{13}{5}$

   (c)

   Find the exact shortest distance from $B$ to the line $AC$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In triangle $XYZ$, $XY=10$ cm, $Y\hat{X}Z=45^\circ$ and $X\hat{Y}Z=75^\circ$.

   (a)

   Find the exact length of $YZ$.

   [3 marks]

   (b)

   Using a calculator, find the length of $XZ$ and hence find the area of triangle $XYZ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A farmer's field is in the shape of a triangle $ABC$, where $AB=80$ m, $AC=70$ m and $BC=110$ m. A calculator may be used.

   (a)

   (i) Show that $\cos B\hat{A}C=-\frac{1}{14}$.
   (ii) Hence explain why $B\hat{A}C$ is obtuse, and find its size.
   (iii) Find the area of the field.

   [6 marks]

   (b)

   The farmer builds a straight fence from $A$ to a point $D$ on $[BC]$ that divides the field into two parts of equal area. Justify why $D$ is the midpoint of $[BC]$, and find the length of the fence.

   [6 marks]

   Total for question 4: 12 marks

End of questions