Grade 11 Maths · Euclidean Geometry

Prove a line is a tangent

Prove a line touches a circle by showing the angle between the line and a chord equals the angle on the other side of the chord.

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Grade 11 Circle Geometry: To Prove a Line is a Tangent Pg 260 Ex 8gMaths Matters

How to prove a line is a tangent - using Euclidean techniquesTemdel

converse tan chord theoremFish Hoek Math with Mr Rob

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Question 5.2 · March 2021 · Paper 1 · Provincial · 4 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

5.2 Prove that KM is a tangent at M to the circle passing through M, P and R.(4)

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More past paper questions: prove a line is a tangent

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3. Question 5.2.3 · March 2019 · Provincial · 4 marks
5.2 In the diagram, O is the centre of the circle. Diameter BOA is produced to E such that EDG is a tangent to the circle at D. C is a point on the circle such that $B A ∣∣ C D$ . AD, BD and AC are drawn. F is a point of intersection of AC and BD. Let $\hat{B} = x$ .
5.2.3 Prove that CA is a tangent to the circle passing through A, D and E.(4)
Show the memo (3 ticks)Memo
1. ✓
  Calculate A2In $Δ A B D$ : $\hat{A}_{2} = 18 0^{\circ} - (\hat{B} + \hat{D}_{3} + \hat{A}_{1})$
  $\hat{A}_{2} = 18 0^{\circ} - (x + 9 0^{\circ} + x) = 9 0^{\circ} - 2 x$
  1 mark · Use angle sum of a triangle
2. ✓
  Compare angles $\hat{A}_{2} = \hat{E} = 9 0^{\circ} - 2 x$
  1 mark · Prove a line is a tangent
3. ✓
  Conclusion $∴$ CA is a tangent (converse: tan-chord theorem)
  2 marks · Prove a line is a tangent
4. Question 5.2.4 · March 2020 · Provincial · 2 marks
5.2 AE is a diameter of the circle. AE is produced to D. DB is a tangent to the circle at B. DC is perpendicular to AD. CBA is a straight line. BE and EC are drawn. Let $\hat{A} = x .$
5.2.4 Is CE a tangent to the circle through A, B and E? Give a reason for your answer.(2)
Show the memo (2 ticks)Memo
1. ✓
  Calculate E2 $\hat{E}_{2} = 18 0^{\circ} - (\hat{E}_{1} + \hat{E}_{3})$ ( $∠$ s on str line)
  $= 18 0^{\circ} - (9 0^{\circ} - x + 9 0^{\circ} - x) = 2 x$
  1 mark · Use angles on a straight line
2. ✓
  Compare and Conclude $\hat{E}_{2} \neq = \hat{A}$ (since $2 x \neq = x$ )
  $∴$ CE is not a tangent (converse of tan-chord theorem does not apply)
  1 mark · Prove a line is a tangent

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