Circle Theorems
----------------------------------------Question 1
The diagram shows a cross section of a tunnel.
It shows a segment of a circle of centre O and radius 5 m.
AB is a chord of the circle.
AB = 8 m.
The length of the tunnel is 40 m.
Calculate the curved surface area of the tunnel.
Give your answer correct to 3 significant figures.
.................... m2
(6 marks)
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Question 2
The diagram shows a circle centre O.
PQ and QR are tangents to the circle at P and Q respectively.
S is a point on the circle.
Angle PSR = 70º.
PS = SR.
(a) i) Calculate the size of angle PQR.
ii) State the reason for your answer.
(b) i) Calculate the size of angle SPO.
ii) Explain why PQRS cannot be a cyclic quadrilateral.
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Question 3
A, B, C and T are points on the circumference of a circle.
Angle BAC = 25º.
The line PTS is the tangent at T to the circle.
AT = AP.
AB is parallel to TC.
(a) Calculate the size of angle APT. Give reasons for your answer.
(b) Calculate the size of angle BTS. Give reasons for your answer.
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Question 4
A, B, C and D are points on the circumference of a circle.
TA and TC are tangents to the circle.
The centre of the circle is at O.
ODT is a straight line.
Angle OTC = 42.
(a) Calculate the size of angle COT. Give reasons for your answer.
(2 marks)
(b) Calculate the size of angle DCT. Give reasons for your answer.
(2 marks)
(c) Calculate the size of the angle ABC. Give reasons for your answer.
(2 marks)
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Question 5
A, B, C and D are four points on the circumference of a circle.
TA is the tangent to the circle at A.
Angle DAT = 30.
Angle ADC = 132.
(a) (i) Calculate the size of angle ABC.
(ii) Explain your method.
(b) (i) Calculate the size of angle CBD.
(ii) Explain your method.
(c) Explain why AC cannot be a diameter of the circle.
.................. 
(2 marks)
.................. 
(3 marks)
(1 mark)
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Question 6
Points A, B and C lie on the circumference of a circle with centre O.
DA is the tangent to the circle at A.
BCD is a straight line.
OC and AB intersect at E.
Angle BOC = 80°
Angle CAD = 38°.
(a) Calculate the size of angle BAC.
.................. 
(1 mark)
(b) Calculate the size of angle OBA.
.................. 
(3 marks)
(c) Give a reason why it is not possible to draw a circle with
diameter ED through the point A.
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Question 7
The diagram shows the shape PQRST.
(1 mark)
RST is a circular arc with centre P and radius 18 cm.
Angle RPT = 40.
(a) Calculate the length of the circular arc RST.
Give your answer correct to 3 significant figures.
.................. cm
(2 marks)
PQR is a semicircle with centre O.
(b) Calculate the total area of the shape PQRST.
Give your answer correct to 3 significant figures.
.................. cm2
(2 marks)
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Question 8
A, B, C and D are points on the circumference of a circle centre O.
A tangent is drawn from E to touch the circle at C.
Angle AEC = 36°
EAO is a straight line.
(a) Calculate the size of angle ABC.
Give reasons for your answer.
.................. 
(4 marks)
(b) Calculate the size of angle ADC.
Give a reason for your answer.
.................. 
(2 marks)
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Question 9
The diagram shows a circle, centre O.
A, B, C and D are points on the circumference of the circle.
Angle ACD = 72°.
ABD is an isosceles triangle with BA = BD.
(a) Calculate the size of angle ADB.
............................ °
(2 marks)
AOC is a diameter.
(b) Calculate the size of angle BAC.
............................ °
(2 marks)
Tangents are drawn to the circle at A and B.
These tangents meet at P.
(c) Calculate the size of the angle APB.
............................ °
(2 marks)
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Question 10
R, S and T are points on the circumference of a circle, centre O.
PS and PT are tangents to the circle.
PSN and TORN are straight lines.
PQ is parallel to SR.
SR = NR.
Angle OPT = angle OPS.
(a) Work out the size of angle PNT.
............................ 
(3 marks)
(b) Show that PS = SN.
(3 marks)
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Question 11
A, B, C and D are points on a circle, centre O.
Angle BOD = 116
(a) Calculate the size of angle BAD.
............................ 
(1 mark)
BC = CD.
(b) Calculate the size of angle DBC.
............................ 
(2 marks)
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Question 12
In the diagram, A, B and C are points on the circle, centre O.
Angle BCE = 63.
FE is a tangent to the circle at point C.
(i) Calculate the size of angle ACB.
Give reasons for your answer.
............................ 
(ii) Calculate the size of angle BAC.
Give reasons for your answer.
............................ 
(4 marks)
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Question 13
B and C are points on the circle, centre O.
AB and AC are tangents to the circle.
Angle BAC = 48
(a) Find the size of angle ABO.
Give a reason for your answer.
............................ 
(2 marks)
(b) Find the size of angle BOC.
Give reasons for your answer.
............................ 
(2 marks)
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Question 14
A and B are points on the circumference of the circle, centre O.
PA and PB are tangents to the circle.
Angle APB = 56o.
Calculate the size of angle AOP.
Give a reason for each stage in your working.
............................ 
(3 marks)
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Question 15
P, Q and R are
points on a circle.
O is the centre of
the circle.
RT is the tangent to
the circle at R.
Angle QRT = 56.
(a) Find
(i) the size of angle RPQ,
............................ 
(ii) the size of angle ROQ.
............................ 
(2 marks)
A, B and D are
points on a circle.
AC is a diameter of
the circle.
Angle CAD = 25
and angle BCD =
132.
(b) Calculate
(i) the size of angle BAC,
............................ 
(ii) the size of angle ABD.
............................ 
(3 marks)