Grade 11 Maths · Euclidean Geometry

Use the tan-chord theorem

Use the fact that the angle between a tangent and a chord equals the angle in the circle on the other side of the chord.

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Circle Geometry Grade 11 : Tan- Chord Theorem IntroductionKevinmathscience

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Question 5.1 · March 2021 · Paper 1 · Provincial · 3 marks

M is centre. Diameter KL produced to N. MP perp KN. PRN tangent at R. MP and KR intersect at S. Let

\hat{R}_{4} = x

5.1 Write down with reasons two other angles each equal to x.(3)

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3. Question 5.2.3 · March 2022 · Provincial · 2 marks
5.2 In the diagram, O is the centre of the circle and L is a point on the circumference. RLM is a tangent at L.
5.2.3 Determine with reasons the size of $\hat{L}_{5}$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Tan-chord theorem $\hat{L}_{5} = \hat{P} = 3 2^{\circ}$ (tan-chord theorem)
  2 marks · Use the tan-chord theorem
4. Question 4.1.2(a) · March 2020 · Provincial · 2 marks
In the diagram below, points A, B, C and D lie on the circumference of a circle. FG and FD are tangents to the circle at C and D respectively. CD is produced to meet AE at E. Furthermore, $∠ GC A = 7 8^{\circ}$ , $∠ CB D = 4 1^{\circ}$ and $∠ B D A = 3 4^{\circ}$ .
4.1.2 Determine with reasons the sizes of the following angles:
4.1.2(a) $\hat{D}_{2}$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Statement and reason $\hat{D}_{2} + 3 4^{\circ} = 7 8^{\circ}$ (tan-chord theorem)
  1 mark · Use the tan-chord theorem
2. ✓
  Answer $\hat{D}_{2} = 4 4^{\circ}$
  1 mark · Use the tan-chord theorem
5. Question 9.2.1 · May/June 2025 · Paper 2 · National (DBE) · 4 marks
Provide reasons for your statements in QUESTIONS 9 and 10.
9.2 In the diagram, POL is a diameter of the larger circle with centre O. TMR is a diameter of the smaller circle with centre M. The two circles intersect at L and R.
PLK is a tangent to the smaller circle at L and TR is a tangent to the larger circle at R. OM intersects the smaller circle at N. Straight lines LT, LR, LN and PR are drawn.

Prove, giving reasons, that:
9.2.1 $L T ∥ PR$ (4)
Show the memo (1 tick)Memo
1. ✓
  METHOD 1: Using angles in a semi-circle to prove alternate angles equal $\hat{L}_{2} = 9 0^{\circ}$ [ $∠$ in semi-circle]
  $\hat{R}_{2} = 9 0^{\circ}$ [ $∠$ in semi-circle]
  $∴ \hat{L}_{2} = \hat{R}_{2}$
  $∴ L T ∥ PR$ [alt $∠$ s =]
  4 marks · Use angle in a semicircle
6. 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

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