Grade 11 Maths · Euclidean Geometry

Use line from centre to chord

Use the fact that a line from the centre of a circle at right angles to a chord cuts the chord into two equal parts.

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Circle Geometry Grade 11 : Line from CenterKevinmathscience

Circle Geometry Grade 11 : Line from Center : PracticeKevinmathscience

Circle Geometry Grade 11 Euclidean Geometry: Line from Centre TheoremsMiss Martins Maths and Science

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Question 5.1 · June 2024 · Paper 2 · Provincial · 4 marks

A B

and

C D

are chords of a circle with center

O

A B = 12

units,

C D = 14

units and

OE = 5

units.

OE ⊥ A B

and

OF

bisects

C D

, with

OF = x

5.1 Calculate with reasons, the radius of the circle,

r

. (Leave your answer in surd form.)(4)

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More past paper questions: use line from centre to chord

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3. Question 10.2.3 · November 2024 · Paper 2 · Provincial · 2 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. $Diagram for question 10.2$
10.2.3 Write down the ratio of AS:SC with a reason.(2)
Show the memo (2 ticks)Memo
1. ✓
  SAS : SC $= 1 : 1$
  1 mark · Use line from centre to chord
2. ✓
  R[line drawn from the centre $⊥$ chord]
  1 mark · Use line from centre to chord
4. Question 8.2.1 · November 2025 · Paper 2 · Provincial · 4 marks
8.2 In the diagram, TR is the diameter of the circle that passes through centre U. P, R, S and T are points on the circumference of the circle. UQ is a line from centre such that UQ $⊥$ PR and PR is drawn.
TS $= x$ cm, SR $= (x + 2)$ cm, PR $= 8$ cm and UQ $= 3$ cm

Calculate, giving reasons, the length of: $Diagram for question 8.2$
8.2.1 UR(4)
Show the memo (4 ticks)Memo
1. ✓
  SPQ $=$ QR $= 4$ cm
  1 mark · Use line from centre to chord
2. ✓
  R[line from centre $⊥$ to chord]
  1 mark · Use line from centre to chord
3. ✓
  Correct substitutionUR $= 4^{2} + 3^{2}$ [Pyth. theo]
  1 mark · Use the theorem of Pythagoras
4. ✓
  AnswerUR $= 5$ cm
  1 mark · Use the theorem of Pythagoras
5. Question 5.2.2 · June 2024 · Paper 2 · Provincial · 3 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 $\hat{B} = 7 2^{\circ}$ .
5.2.2 Prove that $A F = F D$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Determine angles inside triangle EFD $\hat{E}_{2} = \hat{D}_{1} = 5 4^{\circ}$ (corresponding angles, $OE ∥ C D$ )
  1 mark · Use angles on parallel lines
2. ✓
  Determine angle F2In $Δ EF D$ , $\hat{F}_{2}$ is an exterior angle.
  $\hat{F}_{2} = \hat{E}_{2} + \hat{D}_{4}$ (exterior angle of a triangle is equal to the sum of interior opposite angles)
  $\hat{F}_{2} = 5 4^{\circ} + 3 6^{\circ}$
  $\hat{F}_{2} = 9 0^{\circ}$
  1 mark · Use exterior angle of a triangle
3. ✓
  Conclude proof using perpendicular from centre theoremSince $\hat{F}_{2} = 9 0^{\circ}$ , the line $OF$ (from the centre) is perpendicular to chord $A D$ .
  $∴ A F = F D$ (perpendicular from centre to chord bisects the chord)
  1 mark · Use line from centre to chord
6. Question 3.3.4 · March 2022 · Provincial · 3 marks
Give reasons for your statements and calculations in Question 3.
3.3 In the diagram, O is centre of the circle. Points A, B, C and D lie on the circumference of the circle. BOD is a diameter. $\hat{A}_{1} = 5 1^{\circ}$ and $\hat{B}_{1} = 2 9^{\circ}$ . A line is drawn from O to F such that the length of BC = 24 m and radius of the circle centred O is 13 m.
3.3.4 Determine the length of OF.(3)
Show the memo (3 ticks)Memo
1. ✓
  Statement and reason $CF = \frac{1}{2} (24) = 12$ m (line from centre $⊥$ to chord bisects chord)
  1 mark · Use line from centre to chord
2. ✓
  Substitute into Pythagoras $O C^{2} = C F^{2} + O F^{2}$ (Pythagoras) $1 3^{2} = 1 2^{2} + O F^{2}$
  1 mark · Use the theorem of Pythagoras
3. ✓
  Answer $O F^{2} = 169 - 144 = 25$ $OF = 5$ m
  1 mark · Use the theorem of Pythagoras

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