Sine, Cosine Rule & Area of Triangles (Cambridge (CIE) O Level Maths): Exam Questions

The diagram shows a field, ABCD, on horizontal ground.
BC = 192 m, CD = 287.9 m and BD = 168 m.

Calculate angle CBD and show that it rounds to 106.0°, correct to 1 decimal place.

Calculate the value of $x$.

$x=$................................................

Calculate the area of the triangle.

 .......................................... cm²

AD  = 10 cm, BC  = 8 cm, BX  = 5 cm, CX  = 7 cm.   

The diagonals of the cyclic quadrilateral ABCD intersect at X.  

Calculate angle BXC.  

 Angle BXC = ................................................

The diagram shows a quadrilateral $PQRS$ formed from two triangles, $PQS$ and $QRS$.

Calculate

i) $QR$,

$QR=$ ............................................ cm [3]

ii) $PS$,

$PS=$ ............................................ cm [3]

iii) the area of quadrilateral $PQRS$.

........................................... cm² [4]

The diagram shows a trapezium $ABCD$.
$AB$ is parallel to $DC$.
$AB=55\mathrm{m}$, $BD=70\mathrm{m}$, angle $ABD=40°$ and angle $BCD=32°$.

Calculate $AD$.  

$AD=$ ........................................... m

Calculate $BC$.

$BC=$ ........................................... m

Calculate the area of $ABCD$.

 .......................................... m²

Calculate the obtuse angle $x$ in this triangle.

$x$ = ...................................................

The diagram shows two triangles $ABD$ and $BCD$.
$AD=16.5\text{cm}$ and $BD=12.4\text{cm}$.
Angle $ADB=64°$, angle $BDC=53°$ and angle $DBC=95°$.

i) Find $AB$.   

$AB$ = ........................................ cm [4]

ii) Find $BC$.   

$BC$ = ........................................ cm [4]

In the diagram, $BC$ is a vertical wall standing on horizontal ground $AB$.
$D$ is the point on $AB$ where $AD$ = 58 m.
The angle of elevation of $C$ from $A$ is 26°.
The angle of elevation of $C$ from $D$ is 72°.

i) Show that $AC$ = 76.7 m, correct to 1 decimal place.  

[5]

ii) Calculate $BD$.  

$BD$ = .......................................... m [3]

Calculate SR.  

SR = ............................................... cm

Calculate RQ.  

RQ = ............................................... cm

In the diagram, AB = 18 cm, BC = 21.3 cm and BD = 11.6 cm. Angle BDC = 123.5° and angle ABC is a right angle.

i) Calculate angle BCD.

Angle BCD = ............................................... [3]

ii) Calculate AD.

AD = ......................................... cm [5]

The diagram shows the positions of three cities, Geneva (G), Budapest (B) and Hamburg (H).

Use the cosine rule to calculate the distance from Geneva to Budapest.  

.......................................... km

The bearing of Budapest from Hamburg is 133°.

Calculate the bearing of Budapest from Geneva.

Calculate angle ACB.  

Angle ACB = .................................................

Calculate angle ACD.  

Angle ACD = ..................................................

Calculate the area of the quadrilateral ABCD.  

........................................... cm²

In the pentagon $ABCDE$, angle $ACB$ = angle $AED$ = 90°. Triangle $ACD$ is equilateral with side length 12 cm.
$DE$ = $BC$ = 6 cm.

Calculate angle $BAE$.

Angle $BAE$ = .............................................

Calculate.

i) $AB$.

$AB$ = ......................................... cm [2]

ii) $AE$.

$AE$ = ......................................... cm [3]

Calculate the area of the pentagon.

......................................... cm²

$ABC$is a triangle.

$$\begin{gathered} AB=8.7\mathrm{cm}. \\ \mathrm{Angle}ABC=49°. \\ \mathrm{Angle}ACB=64°. \end{gathered}$$

Calculate the area of triangle $ABC$.
Give your answer correct to $3$ significant figures.

The diagram shows triangle $LMN$.

Calculate the length of $LN$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.
$D$ is a point on $AC$.
Angle $BAD$= 45°
Angle $ADB$= 80°
$AB$ = 7.4cm
$DC$ = 5.8 cm

Work out the length of $BC$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.

$AC$= 8.4m
Angle $ACB$= 40°

The area of the triangle = 100 m².

Work out the length of $AB$.
Give your answer correct to 3 significant figures.
You must show all your working.

Work out the length of  $AD$.
Give your answer correct to 3 significant figures.

$ABCD$is a quadrilateral with $AB=500\mathrm{m}$, $BC=650\mathrm{m}$ and $DA=350\mathrm{m}$. Angle $DAB=135°$ and angle $BDC=52°$.

Calculate $DB$.

Calculate the obtuse angle $BCD$.

$ABCD$ is a quadrilateral with $AB=700\mathrm{m}$, $BC=850\mathrm{m}$ and $CD=450\mathrm{m}$.

Angle $ABC=40°$ and angle $CAD=54°$.

Calculate $AC$.

$AC=$ ........................................... $\mathrm{m}$

Calculate the obtuse angle $ADC$.

Find the area of a regular hexagon with side length 7.4 cm.

.......................................... cm²

The diagram shows a cuboid.
AB = 8 cm, AD = 6 cm and DH = 6 cm.  

Calculate angle HAF.  

Angle HAF = .................................................

The diagram shows triangle ABC with point G inside.
CB = 11 cm, CG = 5.3 cm and BG = 6.9 cm. 
Angle CAB = 42° and angle ACG = 54°.

Calculate the value of $x$.  

$x=$................................................ 

Calculate AC.  

AC = ........................................... cm

The area of triangle DEF is 70 m².

i) Show that  $4x^2+11x-300=0$.

[4]

ii) Use the quadratic formula to solve $4x^2+11x-300=0$.
Show all your working and give your answers correct to 2 decimal places.

$x$ = .......................... or $x$ = .......................... [4]

iii) Find the length of DE.

DE = ............................................... m [1]

The diagram shows a field ABCDE.

Calculate the perimeter of the field ABCDE.  

................................................ m

Calculate angle ABD.  

Angle ABD =  ..........................................................

i) Calculate angle CBD.  

Angle CBD = .................................................... [2]

ii) The point C is due north of the point B.  

Find the bearing of D from B.  

[2]

Calculate the area of the field ABCDE.
Give your answer in hectares.
[1 hectare = 10 000m²]  

...................................... hectares

Triangle $EFG$ has an area of 70 m².
$EF:FG=1:2$ and angle $EFG$=40°.

Calculate $EF$.  

$EF$ = .......................................... m

A different triangle $PQR$ also has an area of 70 m².
$PQ:QR=1:2$ and $PQ=EF$  

Find angle $PQR$.  

Angle $PQR$ = ..............................................

The diagram shows a design made from a triangle AOC joined to a sector OCB.
AC = 8cm, OB = OC = 7 cm and angle ACO = 78°.

Use the cosine rule to show that OA = 9.47 cm, correct to 2 decimal places.

Calculate angle OAC.

Angle OAC = ................................................

The perimeter of the design is 29.5 cm.

Show that angle COB = 41.2°, correct to 1 decimal place.

Calculate the total area of the design.

......................................... cm²

The diagram shows two ports, L and P, and a buoy, M.
The bearing of L from P is 201° and LP = 248 km.
The bearing of M from P is 127°.
Angle PML = 42°.

Use the sine rule to calculate LM.  

LM = ......................................... km

A ship sails directly from L to P.

Calculate the shortest distance from M to LP.  

 .......................................... km

The area of triangle $ABC$ is $6\sqrt{2}$m².

Calculate the value of $x$.
Give your answer correct to $3$significant figures.

$ABC$and $ADC$are triangles.

The area of triangle $ADC$is 56 m²

Work out the length of $AB.$
Give your answer correct to 1 decimal place.

$ABCD$is a parallelogram.

$AC$ = 9 cm
$DC$ = 11 cm
Angle $DAC$ = 100°

Calculate the area of the parallelogram.
Give your answer correct to 3 significant figures.

$ABC$is an acute-angled triangle.
$BA$= 7cm
$BC$  = 8cm

The area of triangle $ABC$ is 18 cm².

Work out the size of angle $BAC$.
Give your answer correct to 3 significant figures.
You must show all your working.

In triangle $ABC$, $A\hat{B}C$= 45° and $B\hat{A}C$= 65°.
$AC$is 5 cm shorter than $BC$.

i) Show that $BC=\frac{5\sin 65}{\sin 65-\sin 45}$

[3]

ii) Find the length of $BC$.

$BC$= ..................cm[1]

In triangle $PQR$, $PQ$= 13 cm, $QR$= 6 cm and $RP$= 10 cm.  $QR$is produced to $S$.  

i) Find the value of cos $P\hat{R}Q$, giving your answer as a fraction in its simplest form.

[3]

ii) Hence write down the value of cos $P\hat{R}S$

[1]