Circle Theorems (Cambridge (CIE) O Level Maths): Exam Questions

3 hours41 questions
1a
Sme Calculator
1 mark
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 = ................................................

1b
Sme Calculator
2 marks

Find angle AOC.  

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

1c
Sme Calculator
1 mark

Find angle ABC.  

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

1d
Sme Calculator
1 mark

Find angle CDA.  

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

Did this page help you?

2
Sme Calculator
3 marks
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 = ...............................................

Did this page help you?

3
Sme Calculator
2 marks
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 = ..............................................

Did this page help you?

4a
Sme Calculator
3 marks

Work out angle BEO.

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

4b
Sme Calculator
5 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]

Did this page help you?

5
Sme Calculator
2 marks
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...............................................

Did this page help you?

6
Sme Calculator
6 marks
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................................................

Did this page help you?

7
Sme Calculator
6 marks

The diagram shows two circles with equal radii.
A comma space E and C are points on the circle with centre
B. B comma space E comma space D and F are points on the circle with centre C.
A B C D spaceis a straight line.

q8a-june-2018-4024-2-cambridge-o-level-maths-d

Explain why space E B spaceis parallel to D F

[2]

Did this page help you?

1
Sme Calculator
5 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 ..................

Did this page help you?

2
Sme Calculator
7 marks
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]

Did this page help you?

3
Sme Calculator
2 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................................................

Did this page help you?

4
Sme Calculator
6 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 = .....................

Did this page help you?

5a
Sme Calculator
3 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.

5b
Sme Calculator
3 marks

i) Find angle BAO.  

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

ii) Find angle ABD.

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

Did this page help you?

6
Sme Calculator
5 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 = ..............................................

Did this page help you?

7
Sme Calculator
6 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 = ................................................. 

Did this page help you?

8
Sme Calculator
4 marks
q8a-june-2021-4024-21-cambridge-o-level-maths-d

P comma space Q and R are points on the circumference of a circle, centre O.  Angle P O Q space equals space 8 x to the power of degree, angle R P O space equals space x degree and angle O Q R space equals space 24 to the power of degree.  

Calculate angle P Q O.

Angle P Q O = ................................................ 

Did this page help you?

1
Sme Calculator
5 marks
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

Did this page help you?

2
Sme Calculator
3 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.

Did this page help you?

3
Sme Calculator
4 marks
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 = ...................................................

Did this page help you?

4a
Sme Calculator
3 marks
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.

4b
Sme Calculator
1 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.

Did this page help you?

5
Sme Calculator
5 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.

Did this page help you?