Circle theorems

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

Find the size of angle OBC. Write down any circle theorems that you use.

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

CD is a tangent to the circle. Angle BCD = 40° Angle OAB = 3angle OAC Work out the size of angle ACD. Write down any circle theorems that you use.

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

MAP and NBP are tangents to the circle. Prove that AP = BP

A, B, C and D are points on the circumference of a circle, centre O. AC is a diameter of the circle. ADE and BCE are straight lines. Work out the size of angle BDC. Write down any circle theorems that you use.

A, B and D are points on a circle with centre O. CDE is the tangent to the circle at D.

Work out the size of angle ADC. Write down any circle theorems you use.

A and B are points on a circle, centre O. DBC is the tangent to the circle at B. Angle AOB = x° Show that angle ABC = x° You must give a reason for each stage of your working.

ABDE is a cyclic quadrilateral. ABC and EDC are straight lines. Angle DBC = 60° Given that size of angle EAB: size of angle BCD = 2: 1 work out the size of angle BCD. You must show all your working.

A, B, C and D are points on the circumference of a circle, centre O. ADE and BCE are straight lines.

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

P, Q, R and S are four points on a circle.

PXR and SXQ are straight lines. Prove that triangle PQX and triangle SRX are similar.

A, B, C and D are four points on a circle. SBT is a tangent to the circle. Angle ABD = 20 the size of angle BAD: the size of angle BCD = 3: 1 Find the size of angle SBA. Give a reason for each stage of your working.

C is a circle with centre the origin. A tangent to C passes through the points (–20, 0) and (0, 10) Work out an equation of C. You must show all your working.

Points B, D, E and F lie on a circle. ABC is the tangent to the circle at B. Find the size of angle ABD. You must give a reason for each stage of your working.

A and B are points on a circle with centre O. CAD is the tangent to the circle at A. BOD is a straight line.

Angle ODA = 32°

Work out the size of angle CAB. You must show all your working.

The points A, B, C and D lie on a circle. CDE is a straight line.

Angle ABD = 40° Work out the size of angle ADE. You must give a reason for each stage of your working.

The diagram shows a circle, centre O.

AB is the tangent to the circle at the point A. Angle OBA 30° Point B has coordinates (16, 0) Point P has coordinates (3 p, p) Find the value of p. Give your answer correct to 1 decimal place. You must show all your working.

A, B and C are points on the circumference of a circle, centre O. DAE is the tangent to the circle at A.

Angle BAE = 56° Angle CBO = 35°

Work out the size of angle CAO. You must show all your working.

A, B, R and P are four points on a circle with centre O. A, O, R and C are four points on a different circle. The two circles intersect at the points A and R.

CPA, CRB and AOB are straight lines.

Prove that angle CAB = angle ABC.

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

BC is a tangent to the circle. AOC is a straight line. Angle ABO = x °.

Find the size of angle ACB, in terms of x. Give your answer in its simplest form. Give reasons for each stage of your working.

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

Show that y – x = 90 You must give a reason for each stage of your working.

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.

A, B, C and D are four points on the circumference of a circle.

AEC and BED are straight lines.

Prove that triangle ABE and triangle DCE are similar. You must give reasons for each stage of your working.

A, B and C are points on a circle of radius 5 cm, centre O. DA and DC are tangents to the circle. DO = 9 cm

Work out the length of arc ABC. Give your answer correct to 3 significant figures.