Circle Theorems Standard Questions (G10) Page 1
Q1.(a) A, B and C are points on the circumference of a circle with centre O. Not drawn accurately Work out the size of angle x. (1) Page 2
(b) P, Q and R are points on the circumference of a circle with centre O. Not drawn accurately Work out the size of angle y. Give a reason for your answer. Reason...... (2) (Total 3 marks) Page 3
Q2.AB is a tangent to the circle. Write down the value of x. Give a reason for your answer. Reason... (Total 2 marks) Page 4
Q3. (a) AB is a chord of the circle. TA and TB are tangents to the circle. Work out the size of angle x. (2) Page 5
(b) The diagram shows a circle, centre O. C and D are points on the circumference. COE is a straight line. DE is a tangent. Work out the size of angle y. You must show your working, which may be on the diagram. (3) (Total 5 marks) Page 6
Q4.The diagram shows a circle, centre O. Not drawn accurately (a) Work out the size of angle x.... (1) (b) Work out the size of angle y....... (3) (Total 4 marks) Page 7
Q5. Points A, B, C and D are on the circumference of the circle. CDE is a straight line. (a) Work out the size of angle BCD. Give a reason for your answer. Reason... (2) (b) Work out the size of angle ABC. (1) (Total 3 marks) Page 8
Q6. PAQ and PB are tangents to the circle. AC = BC Work out the size of angle x. You must show your working which may be on the diagram......................... (Total 4 marks) Page 9
Q7.(a) P, Q, R and S are points on the circumference of the circle. Work out the size of angle x. Answer... degrees (1) (b) The diagram shows a circle and a tangent at T. Write down the size of angle w. Give a reason for your answer. Answer... degrees Reason... (2) Page 10
(c) A, C and D are points on the circle, centre O. BA and BC are tangents to the circle. Work out the size of the angle y. Answer... degrees (3) (Total 6 marks) Page 11
Q8. (a) O is the centre of the circle. Write down the size of angle a. (1) (b) Write down the size of angle b. Give a reason for your answer. Reason...... (2) Page 12
(c) A, B, C and D are points on a circle, centre O. x + y + z = 290 Work out the values of x, y and z............. x =... degrees y =... degrees z =... degrees (3) (Total 6 marks) Page 13
Q9.(a) The diagram shows a circle, centre O. Work out the value of x.... (1) (b) The diagram shows a cyclic quadrilateral. Work out the value of y.......... (2) (Total 3 marks) Page 14
Q10.A, B, C, D and E are points on a circle. AC is parallel to ED. (a) Write down the size of angle x. (1) (b) Work out the size of angle y............. (3) (Total 4 marks) Page 15
Q11. P, Q and R are points on a circle. Triangle PQR is isosceles. XY is a tangent to the circle at P. Work out the size of angle RPY............. Answer... degrees (Total 2 marks) Page 16
Q12. A, B, C and D are points on the circumference of a circle centre O. AC is a diameter. Angle DAC = 42 (a) Write down the value of x. Answer... degrees (1) (b) Work out the value of y. Answer... degrees (1) (Total 2 marks) Page 17
M1.(a) 35 (b) 100 Angle at centre twice angle on circumference Must use words centre and circumference (or perimeter ) Allow poor spelling even though both words given oe (strand) (i) Q1 [3] M2.43 Alternate segment (theorem) Strand (i) Do not accept Alternate Dependent on Q1 [2] M3. (a) (180 38) 2 M1 71 May be on diagram if no contradiction A1 (b) ODE = 90 seen or implied or CDO = 25 or COD = 130 Page 18
May be on diagram DOE = 50 or CDE = 115 or 140 seen May be on diagram 40 M1 A1 Additional Guidance 40 with no working seen 115 is M1 unless from clearly incorrect working e.g. 115 leading to an answer of 65 is M1 only M1A1 [5] M4.(a) 150 (b) 360 150 or 210 or 360 their 150 oe OCA = 18 seen or implied or 180 18 75 or 87 M1 360 18 75 210 or 360 18 75 their 210 oe OCB = 75 18 or 57 seen or implied 180 93 30 or 87 30 M1dep 57 A1 [4] Page 19
M5. (a) 108 Opposite angle of a cyclic quadrilateral Strand (i) (add up to 180) Must have 108 Additional Guidance Must see opposite and cyclic (oe e.g. quadrilateral in a circle) Q1 (b) 125 [3] M6. ABC = 52 May be on diagram M1 BAC = 52 or BAQ = 104 or ACB = 76 May be on diagram M1dep PAB = 76 or PBA = 76 May be on diagram M1dep 28 Clear evidence that 28 is for angle x A1 Page 20
Additional Guidance Angles may be on diagram ACB = 52 and ABC = 52 M1 only [4] M7.(a) 56 (b) 70 Alternate segment (theorem) Strand (i) Dependent on Q1dep (c) 2 47 or 94 or Angle BOA = 47 or Angle BOC = 47 or Angle BAC = 47 or Angle BCA = 47 May be on diagram (obtuse angle) M1 90 or right angle symbol seen at A or C or 180 90 47 or (180 2 47) 2 oe M1 Page 21
43 A1 [6] M8. (a) 35 (b) 40 Opposite angles of cyclic quad (add up to) 180 oe Strand (i) No need to mention 180 if angle given as 40 Accept supplementary to mean adds to 180. Q1 (c) x = 55 y = 110 z = 125 If answer line blank mark diagram or script. B2 any two correct B2 y = 110 and x + z = 180 B2 z = 125 and y = 2x any one correct values less than 180 such that y = 2x or x + z = 180 B3 [6] M9.(a) 70 (b) 4y + y = 180 or 5y = 180 oe or 180 5 Page 22
M1 36 A1 [3] M10.(a) 70 (b) ADE = 34 or AED = 180 70 or 110 or ADC = 180 70 34 or 76 Angles seen on diagram must be in correct place M1 ADE = 34 and AED = 180 70 or 110 M1dep 36 A1 [4] M11. 180 (2 35) RQP = 110 marked on diagram 110 [2] Page 23
M12. (a) 42 (b) 48 [2] Page 24