Grade 10 Maths · Euclidean Geometry

State the construction

Say which extra lines to draw before starting a proof, for example join O to A and B, or extend a line to meet the circle.

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

Angle at Centre Theorem: ProofKevinmathscience

Formal Proof 1: Line from centre perpendicular to chord (Circle Geometry)AlphaMath'

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Question 10.1 · May/June 2025 · Paper 2 · National (DBE) · 6 marks

10.1 In the diagram,

Δ A BC

and

Δ P K L

are drawn such that

\hat{A} = \hat{P}

\hat{B} = \hat{K}

and

\hat{C} = \hat{L}

.

Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, i.e. that

\frac{A B}{P K} = \frac{A C}{P L}

(6)

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

More past paper questions: state the construction

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3. Question 9.1 · November 2024 · Paper 2 · Provincial · 5 marks
9.1 In the diagram, AB is a chord of a circle with center O. Prove the theorem that states: If M is the midpoint of AB, then OM $⊥$ AB.(5) $Diagram for question 9.1$
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruction: Connect AO and OB $Memo diagram$
  1 mark · State the construction
2. ✓
  S: three pairs of equal sidesIn $Δ A OM$ and $Δ BOM$ :
  1. $A O = BO$
  2. $OM = OM$
  3. $A M = MB$
  1 mark · Prove a theorem or formula
3. ✓
  R: reasons for the sides(Radii), (Common side), (Given)
  1 mark · Prove a theorem or formula
4. ✓
  S/R: congruence $∴ Δ A OM \equiv Δ BOM$ (S S S)
  $∴ \hat{M}_{1} = \hat{M}_{2}$
  1 mark · Prove a theorem or formula
5. ✓
  S/R: conclusionBut $\hat{M}_{1} + \hat{M}_{2} = 180°$ (AMB is a straight line)
  $∴ \hat{M}_{1} = \hat{M}_{2} = 90°$
  $∴ OM ⊥ A B$
  1 mark · Prove a theorem or formula
4. 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
5. Question 6.2 · June 2024 · Paper 2 · Provincial · 6 marks
6.2 Use the diagram below, $A$ , $B$ and $C$ are on the circle with centre $O$ . $P A$ is a tangent to the circle at $A$ . Use the diagram to prove the theorem which states that: $B \hat{A} P = \hat{C}$ (6)
Show the memo (6 ticks)Memo
1. ✓
  ConstructionDraw diameter $A D$ through $O$ and join $D C$ .
  1 mark · State the construction
2. ✓
  Angle in a semi-circle (S/R) $\hat{C}_{1} + \hat{C}_{2} = 9 0^{\circ}$ (angle in a semi-circle)
  1 mark · Prove a theorem or formula
3. ✓
  Tangent and diameter (S) $\hat{A}_{1} + \hat{A}_{2} = 9 0^{\circ}$
  1 mark · Prove a theorem or formula
4. ✓
  Reason (R)(tangent $⊥$ radius)
  1 mark · Prove a theorem or formula
5. ✓
  Angles in the same segment (S/R) $\hat{C}_{1} = \hat{A}_{2}$ (angles in the same segment)
  1 mark · Prove a theorem or formula
6. ✓
  Conclusion (S/R) $\hat{A}_{1} = 9 0^{\circ} - \hat{A}_{2} = 9 0^{\circ} - \hat{C}_{1} = \hat{C}_{2}$ , so $B \hat{A} P = \hat{C}$
  1 mark · Prove a theorem or formula
6. Question 3.2 · March 2022 · Provincial · 5 marks
Give reasons for your statements and calculations in Question 3.
3.2 In the diagram below, O is the centre of the circle. A, B and C are points on the circumference of the circle. AB, AC and OC are drawn.
Prove the theorem which states that $B \hat{O} C = 2 \times B \hat{A} C$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionJoin AO (and produce it). Label $\hat{A}_{1}$ , $\hat{A}_{2}$ at A and $\hat{O}_{1}$ , $\hat{O}_{2}$ at O.
  1 mark · State the construction
2. ✓
  Exterior angle statement with reasonIn $Δ A OB$ : $\hat{O}_{1} = \hat{A}_{1} + \hat{B}$ (ext $∠$ of $Δ$ )
  1 mark · Prove a theorem or formula
3. ✓
  Equal angles opposite radiiBut $\hat{A}_{1} = \hat{B}$ (radii, OA = OB) $∴ \hat{O}_{1} = 2 \hat{A}_{1}$
  1 mark · Prove a theorem or formula
4. ✓
  Add the two resultsSimilarly in $Δ A OC$ : $\hat{O}_{2} = 2 \hat{A}_{2}$ $∴ \hat{O}_{1} + \hat{O}_{2} = 2 \hat{A}_{1} + 2 \hat{A}_{2}$
  1 mark · Prove a theorem or formula
5. ✓
  Conclusion $B \hat{O} C = 2 (\hat{A}_{1} + \hat{A}_{2})$ $∴ B \hat{O} C = 2 \times B \hat{A} C$
  1 mark · Prove a theorem or formula

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