Circle theorems

A, C and D are points on a circle, diameter AC. ABC is an isosceles triangle with AC = BC BCD is a straight line.

Work out the size of angle x.

Line DAE is a tangent at A to the circle with centre O.

Work out the size of angle x.

Here is a cyclic quadrilateral.

f: g = 2: 3

Work out f: h Give your answer in its simplest form.

A, B and C are points on a circle, centre O. AP and BP are tangents to the circle.

Work out the size of angle x.

Here is a cyclic quadrilateral. a: b: c = 9: 5: 3

Work out the size of angle d.

d = °

PQ is a diameter of a circle. QR is a tangent to the circle. PQ = QR PR = 10 cm

Work out the radius of the circle. Give your answer as a decimal.

P, Q and R are points on a circle.

Assume that S is the centre of the circle. Work out the size of angle x.

x = °

In fact, the centre of the circle is on PS but not at S. What does this mean about the size of angle x ? Tick one box.

For a different circle, AB is a tangent at A C and D are on the circumference of the circle

Here is Simon’s method to work out the size of angle y. Angle ADC = 70 ° (alternate segment theorem) Therefore y = 70 ° (angles in an isosceles triangle)

Is he correct? Give a reason for your answer.

P, Q and R are points on a circle. SP is a tangent to the circle.

Prove that y = 90 ° – x

A, B and C are points on a circle, centre O. BD is a tangent to the circle. OCD is a straight line.

Work out the size of angle x.

A, B and C are points on a circle, centre O. DC is a tangent to the circle.

Show that angle ABO: angle ACO = 3: 1

A, B and C are three points on a circle. The radii from A, B and C are shown.

Is AC a diameter of the circle? You must show your working.

A, B and C are three points on the circumference of a circle, centre O. BD and CD are tangents to the circle. ABDC is a kite.

Prove that angle ABO is 45° –

ABCD is a quadrilateral.

Prove that ABCD is not a cyclic quadrilateral.

A and B are points on a circle. PA and PB are tangents.

Work out the size of angle APB.

C, D and E are points on a different circle.

Is X the centre of the circle? Tick a box.

Show working to support your answer.

C, D and E are points on a circle. CE = DE The tangent at D is shown. ACD and BCE are straight lines.

Prove that y = 3 x

A, B and C are points on a circle. CD is a tangent.

Assume that triangle ABC is isosceles with AC = BC Prove that AB is parallel to DC.

In fact, triangle ABC is equilateral. Tick the two boxes for the statements that must be correct.

P, Q, R and S are points on a circle. PXR and QXS are straight lines.

Prove that QS is not a diameter of the circle.

A, B and C are points on a circle. DCB is a straight line. PAQ is a tangent to the circle.

Sam is trying to work out the size of angle m. Here is his working.

angle ACB = 56° angles in the same segment are equal m = 180° – 56° angles at a point on a straight line add up to 180° m = 124°

Make a criticism of his working.

Here is a cyclic quadrilateral.

x: y = 5: 7 Work out the size of angle w.

A, B and C are points on a circle. CD is a tangent to the circle.

Write down the size of angle x. Give a reason for your answer.

A, B, C, D and E are points on a circle. BFD and AFC are straight lines.

Work out the size of angle x. You must show your working which may be on the diagram.