IGCSEMaths: ExtendedCIEExam Questions3. Geometry3.4 Circle Theorems

Syllabus Edition

First teaching 2021

Last exams 2024

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Circle Theorems (CIE IGCSE Maths: Extended)

Exam Questions

2 hours36 questions

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3.4 Circle Theorems

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3.4 Circle Theorems

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1a
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cie-igcse-2018-oct-nov-p4-tz2-q7

In the diagram, A, B, C and D lie on the circle, centre O.
EA is a tangent to the circle at A.
Angle EAB = 61° and angle BAC = 55°.

Find angle BAO.
 

Angle BAO = ................................................

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1b
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Find angle AOC.
 

Angle AOC = ................................................

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1c
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Find angle ABC.
 

Angle ABC = ................................................

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1d
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Find angle CDA.
 

Angle CDA = ................................................

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2
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cie-igcse-2018-may-jun-1-8b

A, B, C and D are points on the circumference of the circle, centre O.
DOB is a straight line and angle DAC = 58°.

Find angle CDB.

Angle CDB = ...............................................

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3
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cie-igcse-2018-may-jun-1-8c

P, Q and R are points on the circumference of the circle, centre O.
PO is parallel to QR and angle POQ = 48°.

Find angle OPR.

Angle OPR = ..............................................

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4a
Sme Calculator5 marks

cie-igcse-2018-may-jun-2-p4-q9a
A, B, C, D and E lie on the circle, centre O.
Angle AEB = 35°, angle ODE = 28° and angle ACD = 109°.

Work out the following angles, giving reasons for your answers.

i)
Angle EBD = ............................. because .............................

[3]

ii)
Angle EAD = ............................. because ............................. 

[2]

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4b
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Work out angle BEO.

Angle BEO = ................................................

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5
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cie-igcse-2020-may-jun-p4-tz3-q8bi

K, L and M are points on the circle.
KS is a tangent to the circle at K.
KM is a diameter and triangle KLM is isosceles.
 
Find the value of z.
 

z equals...............................................

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6
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cie-igcse-2020-may-jun-p4-tz3-q8biii

F, G, H and J are points on the circle.
EFG is a straight line parallel to JH.
 
Find the value of y.
 

y equals................................................

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1
Sme Calculator5 marks

cie-igcse-2020-oct-nov-p4-tz2-q8a

The points A, B, C, D and E lie on the circle.
 PAQ is a tangent to the circle at A and EC = EB.
 Angle ECB = 80° and angle ABE = 40°.

Find the values of v comma space w comma space x comma space y and z.

 
v space equals ...................  space w space equals space...................  space x space equals space...................  space y space equals space...................  z space equals ..................

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2
Sme Calculator3 marks

cie-igcse-2020-oct-nov-p4-tz2-q8b

In the diagram, K, L and M lie on the circle, centre O.
Angle KML = space 2 x degree spaceand reflex angle KOL = 11 x degree.

Find the value of x.
  

x space equals ................................................

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3
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cie-igcse-2020-oct-nov-p4-tz3-q5a

A, B, C and D are points on the circle, centre O.
EF is a tangent to the circle at D.
Angle ADE = 42° and angle COD = 162°.

Find the following angles, giving reasons for each of your answers.

i)
Angle x
   
x space equals ............................ because ............................ [2]
ii)
Angle y
   
y space equals ............................ because ........................... [2]
iii)
Angle z
   
z space equals ............................ because ........................... [3]

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4
Sme Calculator2 marks

cie-igcse-2020-may-jun-p4-tz3-q8bii

AT is a tangent to the circle at A.
 
Find the value of x.
 

x equals................................................

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5
Sme Calculator6 marks

cie-igcse-2019-oct-nov-1-q1d

A, B, C and D lie on the circle, centre O, with diameter AC.
PQ is a tangent to the circle at A.
Angle PAD = 60° and angle BAC = 20°.

Find the values of u, v, w, x and y.

u = ..................... , v = ....................., w = ..................... ,   

x = ..................... , y = .....................

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6a
Sme Calculator3 marks

cie-igcse-2019-oct-nov-p4-tz3-q6a

In the diagram, A, B, C and D lie on the circle, centre O.
Angle ADC = 128°, angle ACD = 28° and angle BCO = 30°.

Show that obtuse angle AOC = 104°.

Give a reason for each step of your working.

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6b
Sme Calculator3 marks
i)

Find angle BAO.
 

Angle BAO = ................................................... [2]

ii)

Find angle ABD.

Angle ABD = ................................................... [1]

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7
Sme Calculator5 marks

cie-igcse-2019-may-jun-p4-tz2-q2b

The diagram shows a circle with diameter P Q.
S P T is a tangent to the circle at P.

Find the value of y. 
 

y = ..............................................

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8
Sme Calculator6 marks

cie-igcse-2018-oct-nov-p4-tz3-q8b

P, Q, R, S and T lie on the circle, centre O.
Angle PST = 75° and angle QTS = 85°.
 
Find the values of v, w, x and y.
 

v = .................................................     
w = .................................................     
x = .................................................     
y = .................................................     

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1
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cie-igcse-2020-oct-nov-p4-tz1-q4c

A, B, C and D are points on the circle, centre O.
Angle ABD = 21° and CD = 12 cm.

Calculate the area of the circle.


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

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2
Sme Calculator3 marks

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

 The diagonals of the cyclic quadrilateral ABCD intersect at X.  

Explain why triangle ADX is similar to triangle BCX.
Give a reason for each statement you make.

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3
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cie-igcse-2019-oct-nov-1-q1e

A, B and C lie on the circle, centre O.
Angle AOC = left parenthesis 3 x plus 22 right parenthesis degree and angle ABC = 5 x degree.

Find the value of x.

x = ...................................................

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4a
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q9-4-4-hard-edexcel-gcse-maths

A comma space B comma space C and D are points on the circumference of a circle, centre O.
F D E is a tangent to the circle.

Show that y space minus space x space equals space 90
You must give a reason for each stage of your working.

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4b
Sme Calculator1 mark

Dylan was asked to give some possible values for x and y.

He said, "y could be 200 and x could be 110, because 200 − 110 = 90"

Is Dylan correct?
You must give a reason for your answer.

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5
Sme Calculator5 marks

q5-4-4-very-hard-edexcel-gcse-maths

P Q R S T is a regular pentagon.
R comma space U and T are points on a circle, centre O.
Q R and P T are tangents to the circle.
R S U is a straight line.

Prove that S T space equals space U T.

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