PLC Papers Created For: ed by use of accompanying mark schemes towards the rear to attain 8 out of 10 marks over time by completing
Circle Theorems 1 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A, B, C and D are points on the circumference of a circle with centre O. Angle ABC = 116 Find the size of the angle marked x. Give reasons for your answer. (Total 3 marks)
Question 2 Diagram NOT accurately drawn B, C and D are points on the circumference of a circle, centre O. AB and AD are tangents to the circle. Angle DAB = 50 Work out the size of angle BCD. Give a reason for each stage in your working. (Total 4 marks)
Question 3 B and C are points on the circumference of a circle, centre O. AB and AC are tangents to the circle. Angle BAC = 40. Find the size of angle BCO. (Total 3 marks) Total /10
Circle Theorems 2 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A and B are points on the circumference of a circle, centre O. AC and BC are tangents to the circle. Angle ACB = 36. Find the size of angle OBA. Give reasons for your answer. (Total 3 marks)
Question 2 * A, B, C and D are points on the circumference of a circle, centre O. Angle AOC = y. Find the size of angle ABC in terms of y. Give a reason for each stage of your working. (Total 4 marks)
Question 3 B, C and D are points on the circumference of a circle, centre O. ABE and ADF are tangents to the circle. Angle DAB = 40 Angle CBE = 75 Work out the size of angle ODC. (Total 3 marks) Total /10
Circle Theorems 3 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 ADB is the tangent at D to the circle, centre O. C is a point on the circumference. Angle OCD = 25. Calculate angle CDA. Show each step of your calculation. Not to scale... [3]
Question 2 B, D and E are points on a circle, centre O. DE is a diameter. AC is the tangent to the circle at B. Angle BDE = 18. Not to scale (a) Give the reason why angle EBD is 90.... [1] (b) Calculate angle ABD. Give reasons for your answer. Angle ABD =... because............ [2]
Question 3 A, B and C are points on a circle, centre O. TCD is a tangent to the circle. Angle BAC = 52. Not to scale Find angles BOC and BCT, giving your reasons. Angle BOC =... because......... [2] Angle BCT =... because......... [2] Total / 10
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Circle Theorems 4 Grade 8 Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 (b) Work out the value of angle BAX. You must show all of your working. (5)
Question 2 Write down a value for the following angles, and give a reason. (2) (2) Total / 10
PLC Papers Created For: ed by use of accompanying mark schemes towards the rear to attain 8 out of 10 marks over time by completing
Circle Theorems 1 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A, B, C and D are points on the circumference of a circle with centre O. Angle ABC = 116 Find the size of the angle marked x. Give reasons for your answer. ABC + ADC = 180 180 116 = 64 (M1) ADC x 2 so x = 128 (A1) reasons opposite angles of a cyclic quadrilateral add to 180, the angle at the centre of a circle is twice y=the angle at the circumference (C1) Question 2 (Total 3 marks) Diagram NOT accurately drawn B, C and D are points on the circumference of a circle, centre O. AB and AD are tangents to the circle. Angle DAB = 50
Work out the size of angle BCD. Give a reason for each stage in your working. ODA = 90 (angle between radius and tangent is 90 ) DAO = 25 DOA = 180 (90 25) = 65 (triangle adds to 180 ) DOB = 65 x 2 = 130 Reasons given (M1) (M1) (C1) BCD = 130/2 = 65 (angle at centre is double angle at circumference) (A1) (Total 4 marks) Question 3 B and C are points on the circumference of a circle, centre O. AB and AC are tangents to the circle. Angle BAC = 40. Find the size of angle BCO. ACO is the angle between tangent and radius is 90 (M1) OAC = 20 or AOC = 70 or BOC = 140 or ABC = ACB or BCA = 180 40 Tangents from a point equal so BOA is congruent to AOC Base angles of an isosceles triangle BCO = 20 (A1) 2 (M1) = 70 (Total 3 marks) Total /10
Circle Theorems 2 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 A and B are points on the circumference of a circle, centre O. AC and BC are tangents to the circle. Angle ACB = 36. Find the size of angle OBA. Give reasons for your answer. OBC = 90 or AOB = 144 or both angles CAB and CBA = 72 or ACO and BOC = 18 or AOC = 72 Tangent/radius is 90 BOA an isosceles triangle tangents from a point are equal for complete correct method 90-180 36 2 one correct reason given answer 18 (M1) (C1) (A1) (Total 3 marks)
Question 2 * A, B, C and D are points on the circumference of a circle, centre O. Angle AOC = y. Find the size of angle ABC in terms of y. Give a reason for each stage of your working. ADC = (M1) 2 180 - (A1) 2 Two reasons given Angle at centre is twice the angle at the circumference (C1) Opposite angles in cyclic quadrilateral add to 180 (C1) OR Reflex AOC = 360 - y 360 2 (M1) (A1) Two reasons given Angles round a point add to 360 (C1) Angle at centre is twice the angle at the circumference (C1) (Total 4 marks)
Question 3 B, C and D are points on the circumference of a circle, centre O. ABE and ADF are tangents to the circle. Angle DAB = 40 Angle CBE = 75 Work out the size of angle ODC. ABO = 90 or OBC = 15 or ADO = 90 or FDO = 90 or EBO = 90 (M1) Reflex angle BOD = 360 (360-90-90-40) = 220 Or BCD = (360-90-90-40) = 70 Or DBO = 90 (180-40)/2 = 20 Or BOC = 180 (15 + 15) = 150 With valid reasons ODC = 55 (M1) (A1) (Total 3 marks) Total /10
Circle Theorems 3 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 ADB is the tangent at D to the circle, centre O. C is a point on the circumference. Angle OCD = 25. Calculate angle CDA. Show each step of your calculation. Not to scale [3]
Question 2 B, D and E are points on a circle, centre O. DE is a diameter. AC is the tangent to the circle at B. Angle BDE = 18. Not to scale
Question 3 A, B and C are points on a circle, centre O. TCD is a tangent to the circle. Angle BAC = 52. Not to scale Total / 10
Circle Theorems 4 Grade 8 Solutions Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results Question 1 (1) Opposite angles of a cyclic quadrilateral add up to 180 o (B1) (b) Work out the value of angle BAX. You must show all of your working. Angle CDB = 180 38 105 = 37 (angles in a triangle add up to 180) Angle DBA = 37 (alternate angles) Angle CDA = 180-75 = 105 (Opposite angles of a cyclic quadrilateral add up to 180 o ) Angle BDA = 105 37 = 68 Angle BAX = 68 o (Alternate segment theorem) M1 M1 M1 M1 A1 (5)
Question 2 Write down a value for the following angles, and give a reason. Angle DBC = 37 o - angles in the same sector, subtended from the same point, are equal B1 B1 (2) Angle ADC = 90 as angle in a semi-circle. Angles in a triangle add up to 180, so Angle ACD = 53 o B1 B1 (2) Total / 10