Robinson Tuition
Grade 9 · Higher · free GCSE maths worksheet

Multi-step circle theorem reasoning

Chains of reasons in geometry problems.

17 original questions, 70 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

AB is a diameter of a circle with centre O.

C is a point on the circle, not at A or B.

Prove that angle ACB = 90°.

OABCDiagram NOT accurately drawn
3 marks
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Using the isosceles triangles OAC and OBC, angle ACB = x + y where 2x + 2y = 180, so angle ACB = 90 degrees
2

AB is a chord of a circle with centre O.

M is the point on AB such that OM is perpendicular to AB.

Prove that M is the midpoint of AB.

OABMDiagram NOT accurately drawn
4 marks
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Triangles OMA and OMB are congruent (RHS), so AM = BM and M is the midpoint of AB
3

P is a point outside a circle with centre O.

The two tangents from P touch the circle at A and at B.

Prove that PA = PB.

OABPDiagram NOT accurately drawn
4 marks
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Triangles OAP and OBP are congruent (RHS), so PA = PB
4

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

O lies inside triangle ABP.

Prove that angle AOB = 2 × angle APB.

OABPDiagram NOT accurately drawn
4 marks
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Extending PO to D and using the exterior angles of the isosceles triangles OAP and OBP gives angle AOB = 2x + 2y = 2 angle APB
5

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

P and Q are on the same side of the chord AB, and both are on the major arc AB.

You may use the theorem that the angle at the centre of a circle is twice the angle at the circumference standing on the same arc.

Prove that angle APB = angle AQB.

OABPQDiagram NOT accurately drawn
4 marks
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Both angles equal half of angle AOB by the angle at the centre theorem, so they are equal
6

ABCD is a cyclic quadrilateral in a circle with centre O.

You may use the theorem that the angle at the centre of a circle is twice the angle at the circumference standing on the same arc.

Prove that angle ABC + angle ADC = 180°.

OABCDDiagram NOT accurately drawn
4 marks
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The two angles at the centre standing on arcs ADC and ABC sum to 360 degrees and equal 2 angle ABC and 2 angle ADC, so angle ABC + angle ADC = 180 degrees
7

The line ST is a tangent to a circle with centre O, touching the circle at A.

AB is a chord of the circle.

C is a point on the major arc AB, so that C is in the alternate segment to angle TAB.

Prove that angle TAB = angle ACB.

OABCSTDiagram NOT accurately drawn
4 marks
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Drawing the diameter AD: angle TAB = 90 - angle DAB = angle ADB (angles of right-angled triangle ABD), and angle ADB = angle ACB (same segment), so angle TAB = angle ACB
8

A, B and C are points on a circle.

DAE is a tangent to the circle at A.

B is on the opposite side of the chord AC from E.

Angle CAE = 63° and angle BAC = 44°.

Work out the size of

  1. angle ABC,
  2. angle ACB.

Give a reason for each stage of your working.

63°44°OACBDEDiagram NOT accurately drawn
4 marks
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a) 63° b) 73°
9

AB and CD are chords of a circle with centre O.

AB = CD

M is the point on AB such that OM is perpendicular to AB.
N is the point on CD such that ON is perpendicular to CD.

Prove that OM = ON.

OABCDMNDiagram NOT accurately drawn
5 marks
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The perpendicular from the centre bisects each chord, so AM = CN; then Pythagoras (or RHS congruence) in triangles OMA and ONC gives OM = ON
10

A, B and C are points on a circle with centre O. Angle ACB is acute. TA is a tangent to the circle at A, with T and C on opposite sides of the line AB.

Prove that angle TAB = angle ACB.

(You may use the fact that the angle at the centre is twice the angle at the circumference.)

4 marks
Show answer
Let OAB = y. Then TAB = 90 - y (tangent perpendicular to radius), AOB = 180 - 2y (isosceles triangle OAB), ACB = 90 - y (angle at centre is twice angle at circumference). So TAB = ACB.
11

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

PA and PB are tangents to the circle from the point P.

C lies on the major arc AB.

Angle APB = 42°.

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

42°OABPCDiagram NOT accurately drawn
4 marks
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69°
12

A, B, C and D are points on a circle, in that order, so ABCD is a cyclic quadrilateral.

TA is a tangent to the circle at A. T and D are on opposite sides of the line AB.

Angle TAB = 48°.

Angle BDC = 35°.

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

4 marks
Show answer
97
13

A, B, C and D are points on a circle, in that order.

The chords AC and BD intersect at the point E inside the circle.

Prove that triangle ABE is similar to triangle DCE.

ABCDEDiagram NOT accurately drawn
4 marks
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Proof using angles in the same segment (twice) and vertically opposite angles, so the triangles are similar (AA)
14

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

PA and PB are tangents to the circle from the point P.

C lies on the major arc AB.

Angle APB = x°.

Find the size of angle ACB in terms of x. Give your answer in its simplest form and give a reason for each stage of your working.

x°OABPCDiagram NOT accurately drawn
5 marks
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(90 - x2)°
15

A, B, C and D are points on the circumference of a circle, in that order, so ABCD is a cyclic quadrilateral.

Angle DAB = (3x + 10)°

Angle BCD = (5x − 30)°

Angle ABC = (2x + 25)°

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

4 marks
Show answer
105
16

ABCD is a cyclic quadrilateral.

angle BAD : angle BCD = 4 : 5

angle ABC = angle BAD + 30°

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

OABCDDiagram NOT accurately drawn
5 marks
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70°
17

A, B and C are points on a circle. AB = AC.

The line TAS is a tangent to the circle at A. T and C are on opposite sides of the line AB.

Prove that TS is parallel to BC.

4 marks
Show answer
Angle TAB = angle ACB (alternate segment theorem). Angle ABC = angle ACB (isosceles triangle ABC). So TAB = ABC, which are equal alternate angles, so TS is parallel to BC.

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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