Sine and cosine rules and area of a triangleAQA A-Level Maths: Subtopic test

1. 1
   In triangle $ABC$, $AB=9$ cm, $BC=14$ cm and $A\hat{B}C=40^\circ$.

   (a)

   Find the length of $AC$, to 3 significant figures.

   [1 mark]

   • A$21.7$ cm
   • B$9.16$ cm
   • C$16.6$ cm
   • D$84.0$ cm

   (b)

   Find the area of triangle $ABC$, to 3 significant figures.

   [1 mark]

   • A$81.0$ cm$^2$
   • B$48.3$ cm$^2$
   • C$40.5$ cm$^2$
   • D$63.0$ cm$^2$

   (c)

   Find the size of angle $B\hat{A}C$, to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In triangle $LMN$, $LM=6$ m, $LN=10$ m and $MN=14$ m.

   (a)

   Find the value of $\cos M\hat{L}N$.

   [1 mark]

   • A$\frac{1}{2}$
   • B$-\frac{1}{4}$
   • C$-\frac{3}{5}$
   • D$-\frac{1}{2}$

   (b)

   Find the area of triangle $LMN$.

   [1 mark]

   • A$15\sqrt{3}$ m$^2$
   • B$30\sqrt{3}$ m$^2$
   • C$30$ m$^2$
   • D$15$ m$^2$

   (c)

   Find the size of the smallest angle of the triangle, to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Points $A$ and $B$ are $120$ m apart on a straight road on level ground. A tower $T$ stands in a field, with $T\hat{A}B=52^\circ$ and $T\hat{B}A=71^\circ$. A calculator may be used.

   (a)

   Find the distance $AT$, to 3 significant figures.

   [3 marks]

   (b)

   Find the shortest distance from $T$ to the road, and hence the area of triangle $ABT$. Give both answers to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A yacht leaves a harbour $H$ and sails $8$ km on a bearing of $040^\circ$ to a buoy $B$. It then sails $11$ km on a bearing of $125^\circ$ to a lighthouse $L$. A calculator may be used.

   (a)

   (i) Show that $H\hat{B}L=95^\circ$.
   (ii) Find the distance $HL$, to 3 significant figures.
   (iii) Find the area of triangle $HBL$.

   [6 marks]

   (b)

   The yacht returns directly from $L$ to $H$. Find the bearing on which it must sail, to the nearest degree.

   [6 marks]

   Total for question 4: 12 marks

End of questions