AC is a diameter of a circle. B is a point on the circumference. Angle BAC = 34 degrees. Diagram: circle, AC drawn as a diameter (a straight line through the centre O), B marked on the circumference above AC, triangle ABC drawn with angle BAC = 34 degrees at A.
Diagram NOT accurately drawn
(a)State the size of angle ABC, giving a reason for your answer.(2)
(b)Calculate the size of angle BCA.(2)
(Total for Question 1 is 4 marks)
2
PA and PB are tangents drawn from an external point P to a circle with centre O, touching the circle at A and B. Angle APB = 50 degrees. Diagram: circle, centre O, external point P outside the circle, tangents PA and PB touching the circle at A and B, radii OA and OB drawn, angle APB = 50 degrees marked at P.
Diagram NOT accurately drawn
(a)State the size of angle OAP, giving a reason for your answer.(2)
(b)Calculate the size of angle AOB.(2)
(Total for Question 2 is 4 marks)
3
O is the centre of a circle. A, B and C are points on the circumference, with C on the major arc AB. Angle AOB = 148 degrees. Diagram: circle, centre O, A and B on the circumference with angle AOB = 148 degrees marked at the centre, C a separate point on the major arc AB, joined to A and B. Calculate the size of angle ACB, giving a reason for your answer.
Diagram NOT accurately drawn
(Total for Question 3 is 3 marks)
4
ABCD is a cyclic quadrilateral (all four vertices lie on a circle). Angle A = (2x + 10) degrees and angle C = (3x - 20) degrees, where A and C are opposite angles of the quadrilateral. Diagram: circle with four points A, B, C, D on the circumference in order, joined to form cyclic quadrilateral ABCD, angle A marked (2x + 10) degrees, angle C marked (3x - 20) degrees.
Diagram NOT accurately drawn
(a)Find the value of x.(3)
(b)Hence find the size of angle A.(1)
(Total for Question 4 is 4 marks)
5
A circle has centre O and radius 8.4 cm. PT is a tangent to the circle at point T, and OP = 15.6 cm. Diagram: circle centre O, radius OT = 8.4 cm drawn to point T on the circumference, tangent PT drawn at T meeting external point P, OP = 15.6 cm drawn from centre O to P, forming triangle OTP. Calculate the length of PT, correct to 3 significant figures.
Diagram NOT accurately drawn
(Total for Question 5 is 4 marks)
6
XAY is a tangent to a circle at point A. B and C are points on the circle such that AB = BC. Angle BAY = 58 degrees (the angle between the tangent AY and the chord AB). Diagram: circle with tangent line XAY touching at A, chord AB drawn with angle BAY = 58 degrees marked between tangent and chord, point C on the major arc such that AB = BC, with BC and AC drawn to form triangle ABC.
Diagram NOT accurately drawn
(a)State the size of angle ACB, giving a reason for your answer.(2)
(b)Given that AB = BC, calculate the size of angle ABC.(3)
(Total for Question 6 is 5 marks)
7
O is the centre of a circle. A, B, C and D are points on the circumference such that ABCD is a cyclic quadrilateral (in that order), with B and D on opposite arcs formed by chord AC. The reflex angle AOC = 250 degrees. Diagram: circle centre O, chord AC drawn, the reflex angle AOC = 250 degrees marked at O on the side of D, B on the opposite arc from D, forming cyclic quadrilateral ABCD.
Diagram NOT accurately drawn
(a)Calculate the size of angle ABC, giving a reason for your answer.(3)
(b)Hence calculate the size of angle ADC, giving a reason for your answer.(2)
(Total for Question 7 is 5 marks)
8
A circle has centre O and radius 10 cm. AB is a chord of length 12 cm. M is the midpoint of AB, and OM is perpendicular to AB. Diagram: circle centre O, radius 10 cm, chord AB = 12 cm, M the midpoint of AB, line OM drawn from centre to M perpendicular to AB, right angle marked at M.
Diagram NOT accurately drawn
(a)Calculate the length of OM, giving a reason for your method.(3)
(b)Hence calculate the size of angle AOB, giving your answer correct to 1 decimal place.(3)
(Total for Question 8 is 6 marks)
9
PA and PB are tangents from an external point P to a circle with centre O, touching the circle at A and B. Diagram: circle centre O, external point P, tangents PA and PB touching circle at A and B, radii OA and OB drawn, forming quadrilateral OAPB.
Diagram NOT accurately drawn
(a)State the size of angle OAP, giving a reason for your answer.(2)
(b)Show that angle APB + angle AOB = 180 degrees.(2)
(c)Given that angle APB = (2x + 20) degrees and angle AOB = (3x + 40) degrees, find the value of x and hence the size of angle AOB.(3)
(Total for Question 9 is 7 marks)
10
A, B, C and D are points on the circumference of a circle. C and D lie on the same segment relative to chord AB, so angle ADB and angle ACB are angles in the same segment. Angle ADB = (4x - 8) degrees and angle ACB = (2x + 14) degrees. Diagram: circle with chord AB drawn, C and D on the same major arc, both joined to A and B, forming angle ACB and angle ADB.
Diagram NOT accurately drawn
(a)State the reason why angle ADB = angle ACB, and hence find the value of x.(3)
(b)Hence state the size of angle ACB.(1)
(Total for Question 10 is 4 marks)
11
ABCD is a cyclic quadrilateral. Side BC is extended beyond C to a point E, forming exterior angle DCE = 108 degrees. Diagram: circle with cyclic quadrilateral ABCD (vertices in order round the circle), side BC extended beyond C to point E outside the circle, exterior angle DCE = 108 degrees marked.
Diagram NOT accurately drawn
(a)State the size of angle DAB, giving a reason for your answer.(2)
(b)Given that angle ABC = (3x + 15) degrees and angle ADC = (x + 45) degrees, find the value of x and hence the size of angle ABC.(4)
(Total for Question 11 is 6 marks)
12
TA and TB are tangents from an external point T to a circle with centre O, touching the circle at A and B. C is a point on the major arc AB. Angle ATB = 40 degrees. Diagram: circle centre O, external point T with tangents TA, TB touching circle at A, B, radii OA, OB drawn forming quadrilateral OATB, C a separate point on the major arc AB, joined to A and B.
Diagram NOT accurately drawn
(a)Calculate the size of angle AOB.(2)
(b)Hence calculate the size of angle ACB.(2)
(Total for Question 12 is 4 marks)
13
O is the centre of a circle. A, B and C are points on the circumference, with C on the major arc AB. OA = OB (radii), and angle OAB = 32 degrees. Diagram: circle centre O, A and B on circumference with radii OA, OB drawn (triangle OAB isosceles), angle OAB = 32 degrees marked, C on major arc AB joined to A and B.
Diagram NOT accurately drawn
(a)Calculate the size of angle AOB.(2)
(b)Hence calculate the size of angle ACB.(2)
(Total for Question 13 is 4 marks)
14
O is the centre of a circle. A, B and C are points on the circumference. The reflex angle AOC = (3x + 40) degrees, and the angle at the circumference ABC (standing on the same arc AC) = (x + 50) degrees. Diagram: circle centre O, A and C on circumference, reflex angle AOC marked at the centre on the far side from B, B on the minor arc joined to A and C, forming angle ABC.
Diagram NOT accurately drawn
(a)Using the circle theorem connecting the angle at the centre and the angle at the circumference, find the value of x.(3)
(b)Hence state the size of angle ABC.(1)
(Total for Question 14 is 4 marks)
15
PAQ is a straight tangent line touching a circle at A. B and C are points on the circle, with chords AB and AC drawn, and BC also drawn to complete triangle ABC. Angle QAB = 47 degrees and angle PAC = 65 degrees (the angles between the tangent and each chord). Diagram: circle with straight tangent line PAQ touching at A, chords AB and AC drawn to points B and C on either side, chord BC also drawn, angle QAB = 47 degrees and angle PAC = 65 degrees marked between tangent and each chord.
Diagram NOT accurately drawn
(a)State the size of angle ACB, giving a reason for your answer.(2)
(b)State the size of angle ABC, giving a reason for your answer.(2)
(c)Calculate the size of angle BAC.(2)
(Total for Question 15 is 6 marks)
16
O is the centre of a circle. A, B and C are points on the circumference, with angle ABC = (3x + 4) degrees at the circumference. The angle AOC at the centre (standing on the same arc as angle ABC, on the side of the circle not containing B) = (8x - 14) degrees. Diagram: circle centre O, chord AC drawn, angle AOC = (8x - 14) degrees marked at the centre on the arc side away from B, B on the other (major) arc forming angle ABC = (3x + 4) degrees, D a fourth point on the same arc as the marked centre angle so that ABCD is a cyclic quadrilateral.
Diagram NOT accurately drawn
(a)Show that x = 11.(3)
(b)Hence find the size of angle ABC.(1)
(c)D is a fourth point on the circle such that ABCD is a cyclic quadrilateral. Find the size of angle ADC.(2)
(Total for Question 16 is 6 marks)
17
Prove that the angle at the centre of a circle is twice the angle at the circumference, when both angles are subtended by the same arc. Use a clearly labelled diagram and full geometric reasoning in your answer. Diagram: circle centre O, A, B and C points on the circumference with C on the major arc AB, line CO drawn from C through the centre O and extended to meet the circle again at D (so CD is a straight line through the centre), radii OA and OB drawn.; this is a general proof, not tied to specific angle values.
Diagram NOT accurately drawn
(Total for Question 17 is 5 marks)
18
AB is a diameter of a circle with centre O. C is a point on the circumference (not equal to A or B). Prove that angle ACB = 90 degrees. Diagram: circle centre O, AB a diameter (a straight line through O), C a point on the circumference, joined to A, to B, and to O (radius OC drawn, splitting angle ACB into two parts).; this is a general proof, not tied to specific angle values.
Diagram NOT accurately drawn
(Total for Question 18 is 5 marks)
19
ABCD is a cyclic quadrilateral, with vertices in order around a circle with centre O. Prove that angle ABC + angle ADC = 180 degrees. Diagram: circle centre O, points A, B, C, D on the circumference in order, forming cyclic quadrilateral ABCD, O joined to A and to C by radii OA and OC.; this is a general proof, not tied to specific angle values.
Diagram NOT accurately drawn
(Total for Question 19 is 5 marks)
20
TAS is a tangent to a circle at point A. AB is a chord, and C is a point on the major arc (the alternate segment relative to the tangent-chord angle at A). Diagram: circle with tangent line TAS at A, chord AB drawn, C a point on the major arc AB (alternate segment), a diameter drawn from A through the centre O to a point D on the circle, and BD joined, forming triangle ABD.; this is a general proof, not tied to specific angle values.
Diagram NOT accurately drawn
(a)Prove that angle BAT (the angle between the tangent and the chord) equals angle ACB (the angle in the alternate segment). This is known as the alternate segment theorem.(5)
(b)Hence, in a different circle, the angle between a tangent and a chord is 63 degrees. Write down the size of the angle in the alternate segment.(1)
(Total for Question 20 is 6 marks)
Mark scheme · 8.8 Proof of the Circle Theorems
Question 1
(a) B1 angle ABC = 90 degrees
(a) B1 reason: the angle in a semicircle is 90 degrees oe
(a) Answer: 90 degrees
(b) M1 180 - 90 - 34 oe
(b) A1 56 degrees cao
(b) Answer: 56 degrees
Question 2
(a) B1 angle OAP = 90 degrees
(a) B1 reason: the tangent is perpendicular to the radius at the point of contact
(a) Answer: 90 degrees
(b) M1 360 - 90 - 90 - 50 oe
(b) A1 130 degrees cao
(b) Answer: 130 degrees
Question 3
M1 148 / 2 oe, using angle at centre = 2 x angle at circumference
A1 74 degrees cao
B1 reason: the angle at the centre is twice the angle at the circumference, standing on the same arc AB
Answer: 74 degrees
Question 4
(a) M1 (2x + 10) + (3x - 20) = 180 oe
(a) M1 5x - 10 = 180 (oe simplifying)
(a) A1 x = 38 cao
(a) Answer: x = 38
(b) A1 86 degrees, ft their x
(b) Answer: 86 degrees
Question 5
B1 angle OTP = 90 degrees (tangent perpendicular to radius)
M1 15.62 - 8.42 oe
M1√172.8
A1 awrt 13.1 cm
Answer: PT = 13.1 cm (3 s.f.)
Question 6
(a) B1 angle ACB = 58 degrees
(a) B1 reason: alternate segment theorem, the angle between a tangent and a chord equals the angle in the alternate segment
(a) Answer: 58 degrees
(b) B1 angle BAC = 58 degrees, since AB = BC means triangle ABC is isosceles with equal base angles at A and C
(b) M1 180 - 58 - 58 oe
(b) A1 64 degrees cao
(b) Answer: 64 degrees
Question 7
(a) M1 250 / 2 oe
(a) A1 125 degrees cao
(a) B1 reason: the angle at the centre is twice the angle at the circumference, standing on the same arc AC
(a) Answer: 125 degrees
(b) B1 reason: opposite angles in a cyclic quadrilateral sum to 180 degrees
(b) A1 55 degrees, ft (180 - their 125)
(b) Answer: 55 degrees
Question 8
(a) B1 AM = 6 cm, reason: the perpendicular from the centre to a chord bisects the chord
(a) M1√102 - 62 oe
(a) A1 8 cm cao
(a) Answer: OM = 8 cm
(b) M1 sin(angle AOM) = 6/10 oe (or cos/tan ratio using triangle OMA)
(b) M1 angle AOM = 36.869...(degrees), then doubling
(b) A1 73.7 degrees, awrt, ft
(b) Answer: angle AOB = 73.7 degrees (1 d.p.)
Question 9
(a) B1 90 degrees
(a) B1 reason: tangent is perpendicular to the radius at the point of contact
(a) Answer: 90 degrees
(b) M1 360 - 90 - 90 oe, using angle sum of quadrilateral OAPB = 360
(b) A1 angle APB + angle AOB = 180 cso
(b) Answer: 180 degrees (shown)
(c) M1 (2x + 20) + (3x + 40) = 180 oe
(c) A1 x = 24
(c) A1 angle AOB = 112 degrees, ft
(c) Answer: x = 24, angle AOB = 112 degrees
Question 10
(a) B1 reason: angles in the same segment, subtended by the same arc AB, are equal
(a) M1 4x - 8 = 2x + 14 oe
(a) A1 x = 11
(a) Answer: x = 11
(b) A1 36 degrees, ft
(b) Answer: 36 degrees
Question 11
(a) B1 108 degrees
(a) B1 reason: the exterior angle of a cyclic quadrilateral equals the interior opposite angle
(a) Answer: 108 degrees
(b) M1 (3x + 15) + (x + 45) = 180 oe
(b) M1 4x + 60 = 180 (oe simplifying)
(b) A1 x = 30
(b) A1 angle ABC = 105 degrees, ft
(b) Answer: x = 30, angle ABC = 105 degrees
Question 12
(a) M1 360 - 90 - 90 - 40 oe
(a) A1 140 degrees cao
(a) Answer: 140 degrees
(b) M1 140 / 2 oe
(b) A1 70 degrees cao, ft
(b) Answer: 70 degrees
Question 13
(a) M1 180 - 32 - 32 oe, using OA = OB so triangle OAB is isosceles with angle OBA = 32
(a) A1 116 degrees cao
(a) Answer: 116 degrees
(b) M1 116 / 2 oe
(b) A1 58 degrees cao, ft
(b) Answer: 58 degrees
Question 14
(a) M1 3x + 40 = 2(x + 50) oe
(a) M1 3x + 40 = 2x + 100 (oe rearranging)
(a) A1 x = 60
(a) Answer: x = 60
(b) A1 110 degrees, ft
(b) Answer: 110 degrees
Question 15
(a) B1 47 degrees
(a) B1 reason: alternate segment theorem
(a) Answer: 47 degrees
(b) B1 65 degrees
(b) B1 reason: alternate segment theorem
(b) Answer: 65 degrees
(c) M1 180 - 47 - 65 oe
(c) A1 68 degrees cao
(c) Answer: 68 degrees
Question 16
(a) M1 8x - 14 = 2(3x + 4) oe, using angle at centre = 2 x angle at circumference
(c) B1 reason: opposite angles of a cyclic quadrilateral sum to 180 degrees
(c) A1 143 degrees, ft (180 - their 37)
(c) Answer: 143 degrees
Question 17
B1 states OA = OB = OC (all radii of the circle), so triangles OAC and OBC are isosceles
B1 lets angle OAC = angle OCA and angle OBC = angle OCB (base angles of isosceles triangles)
M1 uses the exterior angle of a triangle = sum of the two interior opposite angles to write angle AOD = angle OAC + angle OCA = 2 x angle OCA, and similarly angle BOD = 2 x angle OCB
M1 combines angle AOB = angle AOD + angle BOD (using the correct configuration with D on the far side of O from A, B, C)
C1 concludes angle AOB = 2 x angle OCA + 2 x angle OCB = 2(angle OCA + angle OCB) = 2 x angle ACB, as required; fully correct and complete chain of reasoning shown
Answer: Proved: angle AOB = 2 x angle ACB
Question 18
B1 states OA = OC = OB (all radii), so triangles OAC and OBC are isosceles
B1 lets angle OAC = angle OCA = a and angle OBC = angle OCB = b
M1 uses the angle sum of triangle ABC: angle CAB + angle ABC + angle ACB = 180, i.e. a + b + (a + b) = 180
A1 simplifies to 2(a + b) = 180, so a + b = 90
C1 concludes angle ACB = angle OCA + angle OCB = a + b = 90 degrees, as required
Answer: Proved: angle ACB = 90 degrees
Question 19
B1 states the reflex angle AOC = 2 x angle ABC (angle at the centre is twice the angle at the circumference, both standing on arc ADC)
B1 states the non-reflex angle AOC = 2 x angle ADC (angle at the centre is twice the angle at the circumference, both standing on arc ABC)
M1 uses angles around a point: reflex angle AOC + non-reflex angle AOC = 360
M1 substitutes to get 2 x angle ABC + 2 x angle ADC = 360
C1 divides by 2 to conclude angle ABC + angle ADC = 180 degrees, as required