Circle theorems

PTR and QTS are chords of a circle.

PT = (3y – 5) cm QT = (y + 3) cm RT = 4xcm ST = (x + 2) cm

Find an expression for y in terms of x

A, B, C and D are points on a circle, centre O

Angle BCD = 128°

Work out the size of angle OBD Give a reason for each stage of your working.

angle = OBD_........

A, B and C are points on a circle, centre O

Angle OAC = 52°

Find the size of angle ABC Give reasons for your working.

P, Q, R and S are points on a circle, centre O ROP is a diameter of the circle. Angle PRQ = 22°

Find the size of angle RQP

Give a reason for your answer.

Find the size of angle PSQ

Give a reason for your answer.

A, B and C are points on a circle, centre O

The radius of the circle is 8.5 cm Angle ABC = 132°

Work out the perimeter of the shaded sector AOC Give your answer correct to 3 significant figures.

A, B, C and D are points on a circle, centre O

EBF is the tangent to the circle at B

Work out the size of angle DCB

Give a reason for your answer to (a)(i)

Work out the size of angle ADO

AEC and BED are chords of a circle.

AE = (x + 5) cm BE = xcm CE = (5x – 12) cm DE = (x + 12) cm

Angle DAE = 48°

Work out the size of angle ADE Give your answer correct to one decimal place.

P, Q, R and S are points on a circle with centre O

PS is a diameter of the circle.

Angle PQR = 136

Work out the size of angle RPS

D, E, F and G are points on a circle, centre O

EOG is a diameter of the circle.

Angle EGD = 42°

Calculate the size of angle DFG Give a reason for each stage of your working.

PTQ is a diameter of a circle. RTS is a chord of the circle.

TQ = 3 cm ST = 4 cm TR = 12 cm

Calculate the radius of the circle.

B, D, E and F are points on a circle.

ABC is the tangent at B to the circle. Angle ABD = 39 Angle EFD = 18

Work out the size of angle BDE. Give reasons for your working.

A, B and C are points on a circle with centre O.

Angle ABC = 75

The area of the shaded segment is 200 cm²

Calculate the radius of the circle. Give your answer correct to 3 significant figures.

B, D, E and F are points on a circle. ABC is the tangent to the circle at B.

Angle EDF = 40 Angle FBC = 70

Prove that the tangent ABC is parallel to EF. Give a reason for each stage of your working.

L, M, N and P are points on a circle, centre O

Angle MNP = 58°

Find the size of angle MLP

Give a reason for your answer.

Find the size of the reflex angle MOP

A, B and C are points on a circle, centre O. Angle ABC = 38°

Work out the size of angle OAC. Give a reason for each stage of your working.

APB and CPD are chords of a circle.

AP = 9 cm PB = 6 cm CP = 8 cm

Calculate the length of PD.

A, B, C and D are points on a circle. TDV is the tangent to the circle at D.

AB = AD Angle ADT = 71°

Work out the size of angle BCD. Give a reason for each stage of your working.

B, C, D and F are points on a circle. ABC, AFD, BFE and CDE are straight lines.

Work out the size of angle x. Show your working clearly.

A, B, C and D are points on a circle, centre O.

AOC is a diameter of the circle.

Angle AOD = 98°

Work out the size of angle DBC. Give a reason for each stage in your working.

L, M and N are points on a circle, centre O. QMT is the tangent to the circle at M.

Find the size of angle NOM.

Give a reason for your answer.

Find the size of angle NMQ.

Give a reason for your answer.

The points B, C, Y and X lie on a circle.

AXY and ABC are straight lines.

AX = 12 cm XY = x cm AB = x cm BC = 4 cm

Show that x²x

Find the length of AC. Show your working clearly. Give your answer correct to 3 significant figures.

A, C and D are points on a circle, centre O. AB and CB are tangents to the circle.

Angle ABC = 74°

Work out the size of angle ADC. Show your working clearly.

A, D, B and E are points on a circle, centre O. AFBC, OEC and OFD are straight lines.

AF = 7 cm, FB = 4 cm, BC = 5 cm, FD = 2 cm and CE = x cm.

Work out the value of x. Show your working clearly.