IGCSEMaths: ExtendedCIETopic Questions3. Geometry3.10 Sine, Cosine Rule & Area of Triangles

Syllabus Edition

First teaching 2021

Last exams 2024

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Sine, Cosine Rule & Area of Triangles (CIE IGCSE Maths: Extended)

Topic Questions

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1
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cie-igcse-2020-oct-nov-p4-tz1-q6

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.

  

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2
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cie-igcse-2020-oct-nov-p4-tz2-q8c

 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 = ................................................

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3a
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cie-igcse-2020-specimen-p4-q9a

The diagram shows a trapezium A B C D.
A B is parallel to D C.
A B space equals space 55 space straight m, B D space equals space 70 space straight m, angle A B D space equals space 40 degree and angle B C D space equals space 32 degree.

Calculate A D.
 

A D space equals ........................................... m

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3b
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Calculate B C.


B C space equals ........................................... m

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3c
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Calculate the area of A B C D.


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

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4
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cie-igcse-2019-may-jun-p4-tz2-q8a

The diagram shows two triangles A B D and B C D.
A D equals 16.5 space text cm end text and B D equals 12.4 space text cm end text.
Angle A D B equals 64 degree, angle B D C equals 53 degree and angle D B C equals 95 degree.

i)
Find A B. 
 
A B = ........................................ cm [4]
ii)
Find B C. 
 
B C = ........................................ cm [4]

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5
Sme Calculator8 marks

cie-igcse-2019-may-jun-p4-tz3-q9a

In the diagram, B C is a vertical wall standing on horizontal ground A B.
D is the point on A B where A D = 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 A C = 76.7 m, correct to 1 decimal place.
 
[5]
ii)
Calculate B D.
 
B D = .......................................... m [3]

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6
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cie-igcse-2018-oct-nov-p4-tz1-q7b
 

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]

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7a
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The diagram shows the positions of three cities, Geneva (G), Budapest (B) and Hamburg (H).

cie-igcse-2018-oct-nov-p4-tz2-q8

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

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

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7b
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The bearing of Budapest from Hamburg is 133°.

Calculate the bearing of Budapest from Geneva.

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8a
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cie-igcse-2018-oct-nov-p4-tz3-q6
 

Calculate angle ACB.
 

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

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8b
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Calculate angle ACD.
 

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

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8c
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Calculate the area of the quadrilateral ABCD.
 

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

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9a
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cie-igcse-2018-may-jun-3-p4-q6a
In the pentagon A B C D E, angle A C B = angle A E D = 90°.
Triangle A C D is equilateral with side length 12 cm.
D E = B C = 6 cm.

Calculate angle B A E.

Angle B A E = .............................................

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9b
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Calculate the area of the pentagon.

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

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9c
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Calculate.

i)
A B.

A B = ......................................... cm [2]
ii)
A E.

A E = ......................................... cm [3]

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10
Sme Calculator10 marks

cie-igcse-2020-march-p4-tz2-q8a

The diagram shows a quadrilateral P Q R S formed from two triangles, P Q S and Q R S.

Calculate

i)
Q R,


Q R equals ............................................ cm [3]

ii)
P S,


P S equals ............................................ cm [3]

iii)
the area of quadrilateral P Q R S.


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

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1a
Sme Calculator3 marks

cie-igcse-2020-may-june-p4-tz1-q7

The diagram shows the positions of three points A, B and C in a field.

Show that B C is 118.1 m, correct to 1 decimal place.

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1b
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Calculate angle A B C.

Angle A B C equals ................................................

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2a
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cie-igcse-2020-may-jun-p4-tz2-q4

The diagram shows two triangles.

Calculate QR.
 

QR = ............................................ m

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2b
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Calculate RS.
 

RS = ............................................ m

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2c
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Calculate the total area of the two triangles.
 

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

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3
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cie-igcse-2019-oct-nov-p4-tz1-q5a

Use the cosine rule to find angle BAC.

Angle BAC = ...................................................

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4
Sme Calculator7 marks

cie-igcse-2019-oct-nov-p4-tz2-q6a


The diagram shows triangle ABC.

i)
Use the cosine rule to calculate BC.

 

BC = ............................................... m [4]

ii)
Use the sine rule to calculate angle ACB.

 

Angle ACB = ................................................... [3]

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5a
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cie-igcse-2019-oct-nov-p4-tz3-q4

The diagram shows a field, ABCD, on horizontal ground.

Use the cosine rule to find angle BAC.


Angle BAC = ...................................................

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5b
Sme Calculator3 marks

Use the sine rule to find angle CAD.

Angle CAD = ...................................................

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5c
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Calculate the area of the field.


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

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6
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cie-igcse-2018-oct-nov-p4-tz1-q7a
 Calculate the area of triangle PQR.
 

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

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7a
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cie-igcse-2020-oct-nov-p4-tz1-q6

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

Angle CBD  rounds to 106.0°, correct to 1 decimal place.

Calculate the area of triangle BCD.


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

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7b
Sme Calculator2 marks

Tomas buys the triangular part of the field, BCD.
The cost is $35 750 per hectare.

Calculate the amount he pays.
Give your answer correct to the nearest $100.
[1 hectare = 10000m2]

   
$ ................................................

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1a
Sme Calculator4 marks

cie-igcse-2020-may-jun-p4-tz3-q6a

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 equals................................................ 

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1b
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Calculate AC.
 

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

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2
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cie-igcse-2019-oct-nov-p4-tz2-q6b

The area of triangle DEF is 70 m2.

i)
Show that  4 x squared plus 11 x minus 300 equals 0.

 

[4]

ii)
Use the quadratic formula to solve 4 x squared plus 11 x minus 300 equals 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]

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3a
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cie-igcse-2020-may-jun-p4-tz1-q3

The diagram shows a field ABCDE.

Calculate the perimeter of the field ABCDE.
 

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

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3b
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Calculate angle ABD.
 

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

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3c
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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]

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3d
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Calculate the area of the field ABCDE.
Give your answer in hectares.
[1 hectare = 10 000m2]
 

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

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4a
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Triangle E F G has an area of 70 m2.
E F colon F G equals 1 colon 2 and angle E F G=40°.

Calculate E F.
 

E F = .......................................... m

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4b
Sme Calculator1 mark

A different triangle P Q R also has an area of 70 m2.
P Q colon Q R equals 1 colon 2 and P Q equals E F
 
Find angle P Q R.
 

Angle P Q R = ..............................................

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5a
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cie-igcse-2018-may-jun-2-p4-q5

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.

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5b
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Calculate angle OAC.

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

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5c
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The perimeter of the design is 29.5 cm.

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

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5d
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Calculate the total area of the design.

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

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6a
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cie-igcse-2018-march-p4-q8

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

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6b
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A ship sails directly from L to P.

Calculate the shortest distance from M to LP.
 

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

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