Revision Library

revisionlibrary.co.uk

HIGHER

6.10 Circle Theorems

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

This question tests circle vocabulary.

(a)Write down the mathematical name for a straight line from the centre of a circle to a point on the circumference.(1)

(b)Write down the mathematical name for a straight line joining two points on the circumference of a circle.(1)

(c)Write down the mathematical name for a straight line that touches a circle at exactly one point.(1)

(d)ABCD is a cyclic quadrilateral, meaning all four vertices lie on the circumference of a circle. Write down the rule connecting the sum of each pair of opposite angles.(1)

(Total for Question 1 is 4 marks)

A, B and C are points on the circumference of a circle with centre O. Angle AOB = 130 degrees, where angle AOB is the angle at the centre standing on the same arc AB as angle ACB.
Diagram: circle, centre O marked, points A, B and C on the circumference, angle AOB = 130 degrees marked at the centre, angle ACB marked at the circumference on the major arc.

Diagram NOT accurately drawn

(a)Work out the size of angle ACB.(2)

(b)Give a reason for your answer to part (a).(1)

(Total for Question 2 is 3 marks)

AB is a diameter of a circle with centre O. C is a point on the circumference. Angle CAB = 34 degrees.
Diagram: circle, centre O, AB drawn as a diameter (a straight line through O), C marked on the circumference above AB, triangle ABC drawn with angle CAB = 34 degrees marked at A.

Diagram NOT accurately drawn

(a)Write down the size of angle ACB. Give a reason for your answer.(2)

(b)Work out the size of angle ABC.(2)

(Total for Question 3 is 4 marks)

A, B, C and D are points on the circumference of a circle. C and D are on the same side of the chord AB, so angle ACB and angle ADB are angles in the same segment. Angle ADB = 42 degrees.
Diagram: circle with chord AB drawn, points C and D on the circumference on the same side of AB, angle ADB = 42 degrees marked at D, angle ACB marked at C.

Diagram NOT accurately drawn

(a)Work out the size of angle ACB.(2)

(b)State the circle theorem used in part (a).(1)

(Total for Question 4 is 3 marks)

ABCD is a cyclic quadrilateral. Side AB is extended to a point E, so that angle CBE is the exterior angle of the quadrilateral at B. Angle CBE = 115 degrees.
Diagram: circle with cyclic quadrilateral ABCD drawn on the circumference in order A, B, C, D, side AB extended beyond B to point E, angle CBE = 115 degrees marked at B.

Diagram NOT accurately drawn

(a)Work out the size of angle ADC. Give a reason for your answer.(2)

(b)Work out the size of angle ABC.(2)

(Total for Question 5 is 4 marks)

ABCD is a cyclic quadrilateral. Angle A = (3x + 10) degrees and angle C = (2x + 30) degrees.
Diagram: circle with cyclic quadrilateral ABCD on the circumference, angle A marked (3x + 10) degrees, angle C marked (2x + 30) degrees.

Diagram NOT accurately drawn

(a)Explain why angle A + angle C = 180 degrees.(1)

(b)Work out the value of x.(2)

(c)Work out the size of angle A and the size of angle C.(2)

(Total for Question 6 is 5 marks)

TP is a tangent to a circle with centre O, touching the circle at T. OT = 5 cm and OP = 13 cm. Since OT is a radius and TP is a tangent at T, angle OTP = 90 degrees.
Diagram: circle, centre O, radius OT drawn to the point of contact T, tangent TP drawn perpendicular to OT, OP drawn from O to the external point P, right angle marked at T, OT = 5 cm, OP = 13 cm.

Diagram NOT accurately drawn

(a)Work out the length TP.(2)

(b)Work out the size of angle TPO. Give your answer correct to 1 decimal place.(2)

(Total for Question 7 is 4 marks)

PA and PB are tangents to a circle with centre O, touching the circle at A and B respectively. Angle APB = 50 degrees.
Diagram: circle, centre O, external point P, tangents PA and PB drawn touching the circle at A and B, radii OA and OB drawn, right angles marked at A and B, angle APB = 50 degrees marked at P.

Diagram NOT accurately drawn

(a)Work out the size of angle AOB.(3)

(b)Work out the size of angle OAB.(2)

(Total for Question 8 is 5 marks)

TA is a tangent to a circle at the point A. AB is a chord of the circle, and C is a point on the circle in the alternate segment. The angle between the tangent TA and the chord AB is 58 degrees.
Diagram: circle, tangent TA drawn touching at A, chord AB drawn, angle between TA and AB marked 58 degrees at A, point C marked on the circle in the alternate segment, angle ACB marked at C.
Write down the size of angle ACB, giving a reason for your answer.

Diagram NOT accurately drawn

(Total for Question 9 is 2 marks)

A, B and C are points on the circumference of a circle with centre O. B lies on the minor arc AC. The reflex angle AOC = 250 degrees.
Diagram: circle, centre O, points A, B and C on the circumference, B on the minor arc AC, the reflex angle AOC = 250 degrees marked at the centre going the long way round, angle ABC marked at B.
Work out the size of angle ABC.

Diagram NOT accurately drawn

(Total for Question 10 is 3 marks)

A chord AB of a circle with centre O has length 16 cm. The radius of the circle is 10 cm. M is the midpoint of AB, and OM is drawn perpendicular to AB.
Diagram: circle, centre O, chord AB = 16 cm, M marked as the midpoint of AB, OM drawn perpendicular to AB from the centre, radius OA = 10 cm marked.

Diagram NOT accurately drawn

(a)Explain why AM = 8 cm.(1)

(b)Work out the length OM.(2)

(Total for Question 11 is 3 marks)

O is the centre of a circle of radius 6 cm. P is a point outside the circle with OP = 10 cm. PT is a tangent to the circle, touching it at T.
Diagram: circle, centre O, radius OT = 6 cm drawn to point of contact T, tangent PT drawn, OP = 10 cm drawn from centre to external point P, right angle marked at T.
Show that PT = 8 cm.

Diagram NOT accurately drawn

(Total for Question 12 is 3 marks)

PA and PB are tangents to a circle with centre O, touching the circle at A and B. Angle APB = (3x + 10) degrees and angle AOB = (2x + 20) degrees.
Diagram: circle, centre O, external point P, tangents PA and PB touching at A and B, radii OA and OB, right angles marked at A and B, angle APB = (3x + 10) degrees at P, angle AOB = (2x + 20) degrees at O.

Diagram NOT accurately drawn

(a)Show that angle APB + angle AOB = 180 degrees.(2)

(b)Work out the value of x.(2)

(c)Work out the size of angle APB.(1)

(Total for Question 13 is 5 marks)

TA is a tangent to a circle at the point A. AB and AC are chords of the circle. The angle between the tangent and the chord AB is (2x + 5) degrees. The angle in the alternate segment, angle ACB, is (3x - 15) degrees.
Diagram: circle, tangent TA at A, chords AB and AC drawn, tangent-chord angle (2x + 5) degrees marked at A, angle ACB = (3x - 15) degrees marked at C in the alternate segment.

Diagram NOT accurately drawn

(a)Use the alternate segment theorem to form an equation in x, and solve it to find the value of x.(2)

(b)Work out the size of angle ACB.(1)

(Total for Question 14 is 3 marks)

A, B, C and D are points on a circle, with AC a diameter of the circle. Angle ACB = 45 degrees and angle CAD = 32 degrees.
Diagram: circle with AC drawn as a diameter through the centre, points B and D on the circumference on opposite sides of AC, triangle ABC drawn with angle ACB = 45 degrees at C, triangle ACD drawn with angle CAD = 32 degrees at A.
Work out the size of angle BAD and the size of angle BCD, giving reasons at each stage of your working.

Diagram NOT accurately drawn

(Total for Question 15 is 6 marks)

AB is a diameter of a circle, centre O. C is a point on the circle. The tangent to the circle at C meets the line AB extended beyond B at the point T. Angle CAB = 35 degrees.
Diagram: circle, centre O, AB a diameter extended beyond B to meet point T, C on the circumference, tangent to the circle at C drawn meeting line AB extended at T, angle CAB = 35 degrees marked at A.
Work out the size of angle CTB, giving reasons at each stage of your working.

Diagram NOT accurately drawn

(Total for Question 16 is 6 marks)

PA and PB are tangents to a circle with centre O, touching the circle at A and B. C is a point on the major arc AB. Angle ACB = y degrees.
Diagram: circle, centre O, external point P, tangents PA and PB touching at A and B, radii OA and OB drawn, right angles marked at A and B, point C on the major arc AB with angle ACB = y degrees marked, angle AOB (the angle standing on the minor arc AB) and angle APB shown.
Prove that angle APB = 180 - 2y.

Diagram NOT accurately drawn

(Total for Question 17 is 5 marks)

Mark scheme · 6.10 Circle Theorems

Question 1

(a) B1 radius cao
(a) Answer: Radius
(b) B1 chord cao
(b) Answer: Chord
(c) B1 tangent cao
(c) Answer: Tangent
(d) B1 opposite angles sum to 180 degrees oe
(d) Answer: Each pair of opposite angles sums to 180 degrees.

Question 2

(a) M1 130 / 2 oe
(a) A1 65 degrees cao
(a) Answer: 65 degrees
(b) B1 the angle at the centre is twice the angle at the circumference, subtended by the same arc oe
(b) Answer: The angle at the centre is twice the angle at the circumference standing on the same arc.

Question 3

(a) B1 90 degrees cao
(a) B1 reason: the angle in a semicircle is 90 degrees oe
(a) Answer: 90 degrees, because the angle in a semicircle is a right angle.
(b) M1 180 - 90 - 34 oe
(b) A1 56 degrees cao
(b) Answer: 56 degrees

Question 4

(a) M1 identify angle ACB = angle ADB (same segment)
(a) A1 42 degrees cao
(a) Answer: 42 degrees
(b) B1 angles in the same segment (subtended by the same arc) are equal oe
(b) Answer: Angles in the same segment are equal.

Question 5

(a) M1 use the exterior angle of a cyclic quadrilateral equals the interior opposite angle
(a) A1 115 degrees cao, with reason stated
(a) Answer: 115 degrees
(b) M1 180 - 115 oe, using angles on a straight line ABE or opposite angles in a cyclic quadrilateral
(b) A1 65 degrees cao
(b) Answer: 65 degrees

Question 6

(a) B1 opposite angles in a cyclic quadrilateral sum to 180 degrees oe
(a) Answer: Opposite angles in a cyclic quadrilateral sum to 180 degrees, and A and C are opposite angles.
(b) M1 3x + 10 + 2x + 30 = 180 oe
(b) A1 x = 28 cao
(b) Answer: x = 28
(c) M1 substitute their x into 3x + 10 and 2x + 30 (ft)
(c) A1 angle A = 94 degrees and angle C = 86 degrees, both cao
(c) Answer: Angle A = 94 degrees, angle C = 86 degrees

Question 7

(a) M1 √13² - 5² oe
(a) A1 12 cm cao
(a) Answer: 12 cm
(b) M1 sin(TPO) = 5/13 oe, or use tan = 5/12 with their TP
(b) A1 awrt 22.6 degrees
(b) Answer: 22.6 degrees (1 d.p.)

Question 8

(a) M1 angle OAP = angle OBP = 90 degrees (tangent perpendicular to radius)
(a) M1 360 - 90 - 90 - 50 oe (angle sum of quadrilateral OAPB)
(a) A1 130 degrees cao
(a) Answer: 130 degrees
(b) M1 (180 - their AOB) / 2 oe, using triangle OAB isosceles since OA = OB (radii)
(b) A1 25 degrees cao
(b) Answer: 25 degrees

Question 9

B1 58 degrees cao
B1 reason: alternate segment theorem, the angle between a tangent and a chord equals the angle in the alternate segment oe
Answer: 58 degrees, by the alternate segment theorem.

Question 10

M1 identify that the angle at the centre theorem applies to the reflex angle AOC
A1 125 degrees cao
B1 reason: the angle at the centre is twice the angle at the circumference, subtended by the same (major) arc AC oe
Answer: 125 degrees

Question 11

(a) B1 the perpendicular from the centre to a chord bisects the chord, so AM = 16 / 2 = 8 cm oe
(a) Answer: The perpendicular from the centre to a chord bisects the chord, so AM = 16 / 2 = 8 cm.
(b) M1 √10² - 8² oe
(b) A1 6 cm cao
(b) Answer: 6 cm

Question 12

B1 angle OTP = 90 degrees, since the tangent is perpendicular to the radius at the point of contact
M1 PT² = 10² - 6² oe
A1 PT = 8 cm cso
Answer: PT = 8 cm (shown)

Question 13

(a) M1 angle OAP = angle OBP = 90 degrees (tangent perpendicular to radius)
(a) A1 angle sum of quadrilateral OAPB = 360, so angle APB + angle AOB = 360 - 90 - 90 = 180 degrees cso
(a) Answer: Angle APB + angle AOB = 180 degrees (shown)
(b) M1 (3x + 10) + (2x + 20) = 180 oe
(b) A1 x = 30 cao
(b) Answer: x = 30
(c) B1 100 degrees cao (ft their x)
(c) Answer: 100 degrees

Question 14

(a) M1 2x + 5 = 3x - 15 oe (equating tangent-chord angle to angle in alternate segment)
(a) A1 x = 20 cao
(a) Answer: x = 20
(b) B1 45 degrees cao (ft their x)
(b) Answer: 45 degrees

Question 15

B1 angle ABC = 90 degrees, since AC is a diameter (angle in a semicircle)
M1 angle BAC = 180 - 90 - 45 = 45 degrees (angle sum of triangle ABC)
B1 angle ADC = 90 degrees, since AC is a diameter (angle in a semicircle)
M1 angle ACD = 180 - 90 - 32 = 58 degrees (angle sum of triangle ACD)
A1 angle BAD = angle BAC + angle CAD = 45 + 32 = 77 degrees cao
A1 angle BCD = angle BCA + angle ACD = 45 + 58 = 103 degrees cao (ft)
Answer: Angle BAD = 77 degrees, angle BCD = 103 degrees

Question 16

B1 angle ACB = 90 degrees, since AB is a diameter (angle in a semicircle)
M1 angle ABC = 180 - 90 - 35 = 55 degrees (angle sum of triangle ABC)
M1 angle CBT = 180 - 55 = 125 degrees (angles on the straight line ABT)
B1 angle BCT = 35 degrees (alternate segment theorem: tangent-chord angle at C between CT and chord CB equals angle CAB in the alternate segment)
M1 angle CTB = 180 - 125 - 35 (angle sum of triangle BCT)
A1 20 degrees cao
Answer: 20 degrees

Question 17

B1 angle OAP = 90 degrees (tangent perpendicular to radius at A)
B1 angle OBP = 90 degrees (tangent perpendicular to radius at B)
M1 angle AOB = 2y, using the angle at the centre is twice the angle at the circumference (angle AOB and angle ACB subtend the same arc AB)
M1 angle sum of quadrilateral OAPB: angle APB + angle AOB + 90 + 90 = 360
A1 angle APB = 360 - 90 - 90 - 2y = 180 - 2y cso
Answer: Angle APB = 180 - 2y (proved)

Circle Theorems - Worksheets, Questions and Revision

6.10 Circle Theorems

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17