Grade 11 Maths · Euclidean Geometry

Use radius and tangent facts

Use basic circle facts: all radii are equal, a tangent meets the radius at 90°, and two tangents from the same point are equal in length.

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Circle Geometry Grade 11 : Tangent Radius Theorem IntroductionKevinmathscience

Circle Geometry Grade 11 : Common TangentKevinmathscience

Tangents Drawn from the Same Point to a Circle are EqualMath Truly

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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)

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

More past paper questions: use radius and tangent facts

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3. Question 4.2.2 · March 2021 · Provincial · 6 marks
4.2 In the figure, AD, DC and BE are tangents to the circle at A, C and B respectively. O is the centre of the circle. DE and AF intersect at H. AH produced meets BE produced in F. AJ, BC and BJ are chords.
4.2.2 Prove that $O H = \frac{A O \cdot DH}{D C}$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Use similarity ratio $\frac{D A}{OC} = \frac{DH}{O H} = \frac{A H}{C H}$ ( $Δ D A H ∣∣∣Δ OC H$ )
  1 mark · Use similar triangles' sides and angles
2. ✓
  Rearrange for OH $O H = \frac{DH \cdot OC}{D A}$
  1 mark · Use similar triangles' sides and angles
3. ✓
  Statement: $D A = D C$ $D A = D C$
  1 mark · Use radius and tangent facts
4. ✓
  Reason: tangents from a common point(tangents drawn from a common point are equal)
  1 mark · Use radius and tangent facts
5. ✓
  Statement: $A O = OC$ $A O = OC$
  1 mark · Use radius and tangent facts
6. ✓
  Reason: radii, then conclude(radii of a circle) | Therefore $O H = \frac{A O \cdot DH}{D C}$
  1 mark · Use radius and tangent facts
4. Question 5.2.1 · 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.
5.2.1 Determine with reasons the size of $\hat{O}_{2}$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Use tangent-radius theorem $M \hat{L} O = 9 0^{\circ}$ (radius $⊥$ tangent)
  1 mark · Use radius and tangent facts
2. ✓
  Calculate angle in triangle $\hat{O}_{2} = 18 0^{\circ} - (9 0^{\circ} + 2 6^{\circ}) = 6 4^{\circ}$ (sum of $∠$ s in $Δ$ )
  1 mark · Use angle sum of a triangle
5. Question 10.2.1 · November 2024 · Paper 2 · Provincial · 6 marks
10.2 In the diagram, PAT is the common tangent of the smaller and larger circles at A. AMB is the diameter of the larger circle with M on the circumference of the smaller circle. ASC and BC are chords of the larger circle and MS chord of the smaller circle. M is the centre of the larger circle.

It is also given that, the length of AM $= p$ units, AS $= (p - 1)$ units and BC $= 6$ units.

Answer the questions that follow. $Diagram for question 10.2$
10.2.1 Name, giving reasons, 3 angles equal to $90°$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  S $\hat{A}_{3} = 90°$
  1 mark · Use radius and tangent facts
2. ✓
  R[tan $⊥$ rad]
  1 mark · Use radius and tangent facts
3. ✓
  S $\hat{S}_{1} = 90°$
  1 mark · Use the tan-chord theorem
4. ✓
  R[tan-chord theorem]
  1 mark · Use the tan-chord theorem
5. ✓
  S $\hat{C} = 90°$
  1 mark · Use angle in a semicircle
6. ✓
  R[ $∠$ in semi-circle]
  1 mark · Use angle in a semicircle
6. Question 9.2.2 · November 2024 · Paper 2 · Provincial · 3 marks
9.2 In the diagram below O is the center of the circle with AD = BD.
OB = 20 mm and AB = 24 mm.

Calculate the length of: $Diagram for question 9.2$
9.2.2 FB(3)
Show the memo (3 ticks)Memo
1. ✓
  FD = 36 mm $F D = 20 + 16$
  $F D = 36$ mm
  1 mark · Use radius and tangent facts
2. ✓
  Pythagoras $F B^{2} = 3 6^{2} + 1 2^{2}$ Pythagoras
  1 mark · Use the theorem of Pythagoras
3. ✓
  Answer $FB = 1440$
  $FB = 37, 95$ mm
  1 mark · Use the theorem of Pythagoras

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