Grade 11 Maths · Euclidean Geometry · Paper 2

Circle Theorems Grade 11

Real Circle Theorems past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 79 past paper questions
skills: 12 skills
videos: 12 videos

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Question 8.1 · November 2025 · Paper 2 · Provincial · 5 marks

In the diagram below, O is the centre of the circle. AC is a chord. OB is drawn perpendicular to the chord. $Diagram$

8.1 Use the diagram above to prove the theorem which states that the line drawn from the centre of a circle perpendicular to a chord bisects the chord, that is, prove that AB

=

BC.(5)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

Circle Theorems skills

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More Circle Theorems Grade 11 questions and answers

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4. Question 9.1.1 · November 2025 · Paper 2 · Provincial · 2 marks
9.1 In the diagram below, O is the centre of the circle. ACDE and BCDE are cyclic quadrilaterals. GCF is a tangent to the circle at point C. Chord CE and line OC are drawn. $\hat{E}_{2} = 15°$ .

Determine the size of the following angles below, giving reasons: $Diagram for question 9.1$
9.1.1 $\hat{O}_{1}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  S $\hat{O}_{1} = 30°$
  1 mark · Use angle at the centre
2. ✓
  R[ $∠$ at centre $= 2∠$ at circumf.]
  1 mark · Use angle at the centre
5. Question 4.2.3 · June 2024 · Paper 2 · Provincial · 2 marks
4.2 In the given figure, $D \hat{C} O = 2 5^{\circ}$ , $O$ is the center of the circle. $A$ , $B$ , $E$ , $C$ and $D$ are points on the circumference. Calculate with reasons, the size of:
4.2.3 $\hat{A}_{1}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  S $\hat{A}_{1} = 2 5^{\circ}$
  1 mark · Use angles in the same segment
2. ✓
  R $∠$ s in same segment
  1 mark · Use angles in the same segment
6. Question 5.2.1(a) · June 2024 · Paper 2 · Provincial · 2 marks
5.2 In the diagram below, $O$ is the centre of the circle and $A C$ is the diameter.
$A C$ is produced to meet $E D$ produced at $G$ . $OE ∥ C D$ and $\hat{B} = 7 2^{\circ}$ .
5.2.1 Calculate, giving reasons, the size of:
5.2.1(a) $\hat{C}_{1}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Statement $\hat{C}_{1} = \hat{B} = 7 2^{\circ}$
  1 mark · Use angles in the same segment
2. ✓
  Reason(angles in the same segment)
  1 mark · Use angles in the same segment
7. Question 9.1 · November 2024 · Paper 2 · Provincial · 5 marks
In the diagram, ABCD is a cyclic quadrilateral. O is the centre of the circle. $Diagram$
9.1 Use the diagram above to prove the THEOREM which states that, the opposite angles of a cyclic quadrilateral add up to 180°, then prove that, $\hat{B} + \hat{D} = 180°$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruction: Draw AO and OC. $Memo diagram$
  1 mark · State the construction
2. ✓
  S/R $\hat{O}_{1} = 2 \hat{B}$ [ $∠$ at centre $= 2 \times ∠$ at circumf]
  1 mark · Prove a theorem or formula
3. ✓
  S $\hat{O}_{2} = 2 \hat{D}$ [ $∠$ at centre $= 2 \times ∠$ at circumf]
  1 mark · Prove a theorem or formula
4. ✓
  S/R $\hat{O}_{1} + \hat{O}_{2} = 360°$ [ $∠$ s around a point]
  1 mark · Prove a theorem or formula
5. ✓
  S $2 \hat{B} + 2 \hat{D} = 360°$
  $\hat{B} + \hat{D} = 180°$
  1 mark · Prove a theorem or formula
8. Question 10.1.1 · November 2024 · Paper 2 · Provincial · 4 marks
10.1 In the diagram, O is the centre of circle ABP. PB is extended to C. AC is drawn. PN is a tangent to the circle at P. OC intersects AP perpendicularly at point O. $\hat{P}_{2} = 37°$

Answer the questions that follow. $Diagram for question 10.1$
10.1.1 Determine, giving reasons, the size of $\hat{O}_{1}$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  S $\hat{A}_{1} = 37°$
  1 mark · Use the tan-chord theorem
2. ✓
  R[tan chord]
  1 mark · Use the tan-chord theorem
3. ✓
  S $\hat{O}_{1} = 2 \hat{A}_{1} = 74°$
  1 mark · Use angle at the centre
4. ✓
  R[ $∠$ at centre $= 2 \times ∠$ at circumf]
  1 mark · Use angle at the centre
9. Question 3.6 · November 2024 · Paper 2 · Provincial · 5 marks
In the figure quadrilateral KLMN with vertices K(7 ; -2), L(0 ; -3), M(8 ; -9) and N(9 ; -6) are given. Extend NK to intersect the $x$ -axis and form the angle $β$ .
Extend LK to intersect the $x$ -axis and form the angle $θ$ . $Diagram for this question$
3.6 Prove that KLMN is a cyclic quadrilateral.(5)
Show the memo (5 ticks)Memo
1. ✓
  $m_{L N}$ $m_{L N} = \frac{- 6 - ( - 3 )}{9 - 0}$
  $m_{L N} = - \frac{1}{3}$
  1 mark · Calculate gradient between two points
2. ✓
  $m_{MN}$ $m_{MN} = \frac{- 9 - ( - 6 )}{8 - 9}$
  $m_{MN} = 3$
  1 mark · Calculate gradient between two points
3. ✓
  $L \hat{N} M = 90°$ $3 \times - \frac{1}{3} = - 1$
  $∴ m_{L N} \times m_{MN} = - 1$
  $∴ L N ⊥ NM$
  $∴ L \hat{N} M = 90°$
  1 mark · Use parallel and perpendicular gradients
4. ✓
  Equal anglesBut $K L ⊥ K M$ (3.5)
  $∴ L \hat{K} M = 90°$
  $∴ L \hat{K} M = L \hat{N} M = 90°$
  1 mark · Prove a quadrilateral is cyclic
5. ✓
  Reason $∴$ KLMN is a cyclic quadrilateral [converse $∠$ s in the same segment]
  OR [line subtends equal $∠$ s]
  1 mark · Prove a quadrilateral is cyclic
10. Question 7.2.3 · November 2024 · Paper 2 · Provincial · 3 marks
In the diagram below, circle ACD is given where AC is a diameter. DC is produced to B and AB = 8 units.
$\hat{B} = 30°$ and BC = 5 units. $Diagram for this question$
7.2 Calculate the following:
7.2.3 AD(3)
Show the memo (3 ticks)Memo
1. ✓
  $\hat{D} = 90°$ $\hat{D} = 90°$ [ $∠$ in semi circle]
  1 mark · Use angle in a semicircle
2. ✓
  Ratio $sin 64, 27° = \frac{A D}{4 , 44}$
  1 mark · Calculate a side using trig
3. ✓
  Answer $A D = 4$
  1 mark · Calculate a side using trig

Circle Theorems Grade 11 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Euclidean geometry proofs in less than 10 minutes - Grade 11 and12 NSC & IEBMathew's Academy Of Sciences

Circle Geometry Grade 11 : Line from CenterKevinmathscience

Circle Geometry Grade 11 : Angle at centreKevinmathscience

Circle Geometry Grade 11 : Tangent Radius Theorem IntroductionKevinmathscience

Circle Geometry Grade 11 : Angle in Semi CircleKevinmathscience

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