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6.10D Circle Theorems: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 80 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 fraction of the circumference of a circle.(1)

(b)Write down the mathematical name for the region enclosed by two radii and an arc.(1)

(c)Write down the mathematical name for the region enclosed by a chord and an arc.(1)

(d)A straight line touches a circle at exactly one point. Write down the mathematical name for this point.(1)

(Total for Question 1 is 4 marks)

AB is a diameter of a circle with centre O. C is a point on the circumference. Angle ABC = 26 degrees.
Diagram: circle, centre O, AB drawn as a diameter, C on the circumference above AB, triangle ABC drawn with angle ABC = 26 degrees marked at B.

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 BAC.(2)

(Total for Question 2 is 4 marks)

X, Y and Z are points on the circumference of a circle with centre O. Angle XOY = 96 degrees, where angle XOY is the angle at the centre standing on the same arc XY as angle XZY.
Diagram: circle, centre O, points X, Y and Z on the circumference, angle XOY = 96 degrees marked at the centre, angle XZY marked at the circumference on the major arc.
Work out the size of angle XZY, giving a reason for your answer.

Diagram NOT accurately drawn

(Total for Question 3 is 3 marks)

O is the centre of a circle. P and Q are points on the circumference, so OP and OQ are radii. Angle POQ = 68 degrees.
Diagram: circle, centre O, radii OP and OQ drawn to points P and Q on the circumference, chord PQ drawn, angle POQ = 68 degrees marked at O.

Diagram NOT accurately drawn

(a)Explain why triangle OPQ is isosceles.(1)

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

(Total for Question 4 is 3 marks)

WXYZ is a cyclic quadrilateral, with all four vertices on the circumference of a circle. Angle W = 78 degrees.
Diagram: circle with cyclic quadrilateral WXYZ drawn on the circumference in order W, X, Y, Z, angle W = 78 degrees marked.
Work out the size of angle Y, giving a reason for your answer.

Diagram NOT accurately drawn

(Total for Question 5 is 3 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 = 37 degrees.
Diagram: circle with chord AB drawn, points C and D on the circumference on the same side of AB, angle ADB = 37 degrees marked at D, 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 6 is 2 marks)

TP is a tangent to a circle with centre O, touching the circle at T. OT = 9 cm and OP = 41 cm.
Diagram: circle, centre O, radius OT drawn to the point of contact T, tangent TP drawn, OP drawn from O to the external point P, right angle marked at T, OT = 9 cm, OP = 41 cm.

Diagram NOT accurately drawn

(a)State the size of angle OTP. Give a reason for your answer.(1)

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

(Total for Question 7 is 3 marks)

PA and PB are tangents to a circle with centre O, touching the circle at A and B respectively. Angle APB = 44 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 = 44 degrees marked at P.

Diagram NOT accurately drawn

(a)Give a reason why PA = PB.(1)

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

(Total for Question 8 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 = 108 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 = 108 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 9 is 4 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 63 degrees.
Diagram: circle, tangent TA drawn touching at A, chord AB drawn, angle between TA and AB marked 63 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 10 is 2 marks)

A chord PQ of a circle with centre O has length 24 cm. The radius of the circle is 13 cm. M is the midpoint of PQ, and OM is drawn perpendicular to PQ.
Diagram: circle, centre O, chord PQ = 24 cm, M marked as the midpoint of PQ, OM drawn perpendicular to PQ from the centre, radius OP = 13 cm marked.

Diagram NOT accurately drawn

(a)Explain why PM = 12 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 7 cm. P is a point outside the circle with OP = 25 cm. PT is a tangent to the circle, touching it at T.
Diagram: circle, centre O, radius OT = 7 cm drawn to point of contact T, tangent PT drawn, OP = 25 cm drawn from centre to external point P, right angle marked at T.
Show that PT = 24 cm.

Diagram NOT accurately drawn

(Total for Question 12 is 3 marks)

ABCD is a cyclic quadrilateral. Angle A = (4x - 5) degrees and angle C = (2x + 35) degrees.
Diagram: circle with cyclic quadrilateral ABCD on the circumference, angle A marked (4x - 5) degrees, angle C marked (2x + 35) 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 13 is 5 marks)

AB is a diameter of a circle with centre O. C is a point on the circumference, and OC is drawn. Angle OCB = 33 degrees.
Diagram: circle, centre O, AB drawn as a diameter, C on the circumference, radius OC drawn, angle OCB = 33 degrees marked at C.
Work out the size of angle BAC, giving reasons at each stage of your working.

Diagram NOT accurately drawn

(Total for Question 14 is 4 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 = 264 degrees.
Diagram: circle, centre O, points A, B and C on the circumference, B on the minor arc AC, the reflex angle AOC = 264 degrees marked at the centre going the long way round, angle ABC marked at B.

Diagram NOT accurately drawn

(a)Work out the size of the non-reflex angle AOC.(1)

(b)Work out the size of angle ABC.(3)

(Total for Question 15 is 4 marks)

PA and PB are tangents to a circle with centre O, touching the circle at A and B. Angle AOB = 152 degrees.
Diagram: circle, centre O, external point P, tangents PA and PB touching the circle at A and B, radii OA and OB, right angles marked at A and B, angle AOB = 152 degrees marked at O.

Diagram NOT accurately drawn

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

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

(Total for Question 16 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 (3x + 8) degrees. The angle in the alternate segment, angle ACB, is (5x - 32) degrees.
Diagram: circle, tangent TA at A, chords AB and AC drawn, tangent-chord angle (3x + 8) degrees marked at A, angle ACB = (5x - 32) 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 17 is 3 marks)

A, B, C and D are points on a circle, with AC a diameter of the circle. Angle ACB = 52 degrees and angle CAD = 24 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 = 52 degrees at C, triangle ACD drawn with angle CAD = 24 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 18 is 6 marks)

AB is a diameter of a circle, centre O. D is a point on the circle. The tangent to the circle at D meets the line BA extended beyond A at the point T. Angle DBA = 41 degrees.
Diagram: circle, centre O, AB a diameter, D on the circumference, tangent at D meeting line BA extended beyond A at T, angle DBA = 41 degrees marked at B, angle DTA required.
Work out the size of angle DTA, giving reasons at each stage of your working.

Diagram NOT accurately drawn

(Total for Question 19 is 6 marks)

A, B and C are points on a circle, with BC a diameter. The tangent to the circle at A meets the line BC extended beyond C, at the point T. Angle ABC = a degrees.
Diagram: circle, BC a diameter, A on the circumference, tangent at A meeting line BC extended beyond C at T, angle ABC = a degrees marked at B, angle ATC to be found in terms of a.
Prove that angle ATC = (90 - 2a) degrees.

Diagram NOT accurately drawn

(Total for Question 20 is 6 marks)

Mark scheme · 6.10D Circle Theorems: Fluency and Exam Drill

Question 1

(a) B1 arc cao
(a) Answer: Arc
(b) B1 sector cao
(b) Answer: Sector
(c) B1 segment cao
(c) Answer: Segment
(d) B1 point of contact oe (point of tangency)
(d) Answer: The point of contact

Question 2

(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 - 26 oe
(b) A1 64 degrees cao
(b) Answer: 64 degrees

Question 3

M1 96 / 2 oe
A1 48 degrees cao
B1 reason: the angle at the centre is twice the angle at the circumference, subtended by the same arc oe
Answer: 48 degrees

Question 4

(a) B1 OP = OQ, since both are radii of the same circle oe
(a) Answer: OP and OQ are both radii of the circle, so OP = OQ, making triangle OPQ isosceles.
(b) M1 (180 - 68) / 2 oe
(b) A1 56 degrees cao
(b) Answer: 56 degrees

Question 5

B1 reason: opposite angles in a cyclic quadrilateral sum to 180 degrees oe
M1 180 - 78 oe
A1 102 degrees cao
Answer: 102 degrees

Question 6

B1 37 degrees cao
B1 reason: angles in the same segment (subtended by the same arc) are equal oe
Answer: 37 degrees, because angles in the same segment are equal.

Question 7

(a) B1 90 degrees, since the tangent is perpendicular to the radius at the point of contact oe
(a) Answer: 90 degrees, because a tangent is perpendicular to the radius at the point of contact.
(b) M1 √41² - 9² oe
(b) A1 40 cm cao
(b) Answer: 40 cm

Question 8

(a) B1 tangents drawn to a circle from the same external point are equal in length oe
(a) Answer: PA = PB because tangents drawn to a circle from the same external point are equal in length.
(b) M1 (180 - 44) / 2 oe, using triangle PAB isosceles since PA = PB
(b) A1 68 degrees cao
(b) Answer: 68 degrees

Question 9

(a) M1 use the exterior angle of a cyclic quadrilateral equals the interior opposite angle
(a) A1 108 degrees cao, with reason stated
(a) Answer: 108 degrees
(b) M1 180 - 108 oe, using angles on a straight line ABE or opposite angles in a cyclic quadrilateral
(b) A1 72 degrees cao
(b) Answer: 72 degrees

Question 10

B1 63 degrees cao
B1 reason: alternate segment theorem, the angle between a tangent and a chord equals the angle in the alternate segment oe
Answer: 63 degrees, by the alternate segment theorem.

Question 11

(a) B1 the perpendicular from the centre to a chord bisects the chord, so PM = 24 / 2 = 12 cm oe
(a) Answer: The perpendicular from the centre to a chord bisects the chord, so PM = 24 / 2 = 12 cm.
(b) M1 √13² - 12² oe
(b) A1 5 cm cao
(b) Answer: 5 cm

Question 12

B1 angle OTP = 90 degrees, since the tangent is perpendicular to the radius at the point of contact
M1 PT² = 25² - 7² oe
A1 PT = 24 cm cso
Answer: PT = 24 cm (shown)

Question 13

(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 4x - 5 + 2x + 35 = 180 oe
(b) A1 x = 25 cao
(b) Answer: x = 25
(c) M1 substitute their x into 4x - 5 and 2x + 35 (ft)
(c) A1 angle A = 95 degrees and angle C = 85 degrees, both cao
(c) Answer: Angle A = 95 degrees, angle C = 85 degrees

Question 14

B1 angle ACB = 90 degrees, since AB is a diameter (angle in a semicircle)
M1 angle OCA = 90 - 33 = 57 degrees
M1 OA = OC (radii), so triangle OAC is isosceles, angle OAC = angle OCA
A1 angle BAC = 57 degrees cao
Answer: 57 degrees

Question 15

(a) B1 96 degrees cao
(a) Answer: 96 degrees
(b) M1 identify that the angle at the centre theorem applies to the reflex angle AOC
(b) A1 132 degrees cao
(b) B1 reason: the angle at the centre is twice the angle at the circumference, subtended by the same (major) arc AC oe
(b) Answer: 132 degrees

Question 16

(a) M1 angle OAP = angle OBP = 90 degrees (tangent perpendicular to radius)
(a) M1 360 - 90 - 90 - 152 oe (angle sum of quadrilateral OAPB)
(a) A1 28 degrees cao
(a) Answer: 28 degrees
(b) M1 (180 - their APB) / 2 oe, using triangle PAB isosceles since PA = PB (tangents)
(b) A1 76 degrees cao (ft)
(b) Answer: 76 degrees

Question 17

(a) M1 3x + 8 = 5x - 32 oe (equating tangent-chord angle to angle in alternate segment)
(a) A1 x = 20 cao
(a) Answer: x = 20
(b) B1 68 degrees cao (ft their x)
(b) Answer: 68 degrees

Question 18

B1 angle ABC = 90 degrees, since AC is a diameter (angle in a semicircle)
M1 angle BAC = 180 - 90 - 52 = 38 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 - 24 = 66 degrees (angle sum of triangle ACD)
A1 angle BAD = angle BAC + angle CAD = 38 + 24 = 62 degrees cao
A1 angle BCD = angle BCA + angle ACD = 52 + 66 = 118 degrees cao (ft)
Answer: Angle BAD = 62 degrees, angle BCD = 118 degrees

Question 19

B1 angle ADB = 90 degrees, since AB is a diameter (angle in a semicircle)
M1 angle DAB = 180 - 90 - 41 = 49 degrees (angle sum of triangle ABD)
M1 angle DAT = 180 - 49 = 131 degrees (angles on the straight line TAB)
B1 angle ADT = 41 degrees (alternate segment theorem: tangent-chord angle at D between DT and chord DA equals angle DBA in the alternate segment)
M1 angle DTA = 180 - 131 - 41 (angle sum of triangle ADT)
A1 8 degrees cao
Answer: 8 degrees

Question 20

B1 angle BAC = 90 degrees, since BC is a diameter (angle in a semicircle)
M1 angle ACB = 180 - 90 - a = 90 - a degrees (angle sum of triangle ABC)
M1 angle ACT = 180 - (90 - a) = 90 + a degrees (angles on the straight line BCT)
B1 angle TAC = a degrees (alternate segment theorem: tangent-chord angle at A between AT and chord AC equals angle ABC in the alternate segment)
M1 angle ATC = 180 - (90 + a) - a (angle sum of triangle ATC)
A1 angle ATC = 90 - 2a cso
Answer: Angle ATC = 90 - 2a degrees (proved)

Circle Theorems: Fluency and Exam Drill - Worksheets, Questions and Revision

6.10D Circle Theorems: Fluency and Exam Drill

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

Question 18

Question 19

Question 20