Grade 12 Maths · Euclidean Geometry

Use the proportionality theorem

Use the fact that a line parallel to one side of a triangle cuts the other two sides in the same ratio, for example AD/DB = AE/EC.

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Proportionality Theorem Grade 12 Exam QuestionsKevinmathscience

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Question 5.1 · March 2021 · Provincial · 2 marks

In the figure

A F = 2 CG

and

FE ∥ GB

\frac{A E}{A B} = \frac{2}{5}

5.1 Determine

\frac{A F}{FG}

with reasons.(2)

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More past paper questions: use the proportionality theorem

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3. Question 10.2.3 · May/June 2025 · Paper 2 · National (DBE) · 6 marks
10.2 In the diagram, O is the centre of the circle. Points D and B lie on the circle. Points A and C lie outside the circle such that side AC of $Δ A D C$ passes through B. F is a point on BC such that $FO ∥ B D$ . $D R = RB$ and RO is drawn.
10.2.3 It is further given that $D C = 19, 2$ units, $B D = 12$ units and $\frac{RO}{R D} = \frac{3}{4}$ . Prove, with reasons, that $BF = \frac{75}{16}$ .(6)
Show the memo (3 ticks)Memo
1. ✓
  Find lengths of RD and ROGiven $D R = RB$ , so R is the midpoint of chord BD.
  $R D = \frac{1}{2} \times 12 = 6$ units.
  $\frac{RO}{R D} = \frac{3}{4}$
  $RO = \frac{3}{4} (6) = 4, 5$ units
  2 marks · Use a ratio, rate or proportion
2. ✓
  Calculate DO using PythagorasThe line from the centre (O) to the midpoint (R) of a chord (BD) is perpendicular to the chord.
  $∴ OR ⊥ B D$
  
  In $Δ D RO$ :
  $D O^{2} = R D^{2} + R O^{2}$ [Pythagoras]
  $D O = 6^{2} + 4, 5^{2}$
  $D O = 7, 5$ units
  2 marks · Use the theorem of Pythagoras
3. ✓
  Use proportionality theorem in the large triangle to find BFIn $Δ BC D$ , $FO ∥ B D$ (given):
  $\frac{BF}{BC} = \frac{D O}{D C}$ [line $∥$ one side of $Δ$ OR prop theorem]
  
  From 10.2.2, we know $BC = B D = 12$ units.
  $\frac{BF}{12} = \frac{7 , 5}{19 , 2}$
  $BF = \frac{7 , 5 \times 12}{19 , 2}$
  $BF = \frac{75}{16}$ units
  2 marks · Use the proportionality theorem
4. Question 4.2.2 · March 2022 · Provincial · 3 marks
4.2 In the figure, $Δ A BC$ has $H G ∥ A C$ and $E D ∥ BC$ . $E D$ and $H G$ intersect at $K$ . $\frac{A D}{D C} = \frac{3}{2}$ and $BG = 2 GC$ . $A B = 15$ units.
4.2.2 Determine with reasons the value of $A H$ .(3)
Show the memo (2 ticks)Memo
1. ✓
  Setup ratio using Proportionality Theorem $\frac{A H}{A B} = \frac{CG}{CB} = \frac{1}{3}$ (Prop. Thm; $H G ∥ A C$ )
  2 marks · Use the proportionality theorem
2. ✓
  Calculate length $\frac{A H}{15} = \frac{1}{3}$
  $A H = 5$ units
  1 mark · Use the proportionality theorem
5. Question 9.1.1 · March 2021 · Provincial · 3 marks
The diagram is the top view design of a new railway system. AF || BE and AC || GD. $\frac{A B}{BC} = \frac{4}{7}$ and $\frac{A G}{A F} = \frac{9}{17}$ .
9.1 Calculate:
9.1.1 $\frac{FE}{FC}$ (3)
Show the memo (2 ticks)Memo
1. ✓
  Define ratiosLet $A B = 4 k$ and $BC = 7 k$ .
  0 marks · Use a ratio, rate or proportion
2. ✓
  Apply Proportionality Theorem $\frac{FE}{FC} = \frac{A B}{A C} = \frac{4 k}{4 k + 7 k} = \frac{4}{11}$ (Prop theorem; $A F ∥ BE$ )
  3 marks · Use the proportionality theorem
6. Question 2.1.1 · March 2021 · Provincial · 2 marks
2.1 In the diagram, $△ PQR$ is given such that $QT = 3$ units, $T V = 5$ units, $TR = 8$ units and $ST ∥ P V$ . Determine the following with reasons:
2.1.1 Calculate $\frac{QS}{PS}$ (2)
Show the memo (1 tick)Memo
1. ✓
  Apply Proportionality Theorem $\frac{QS}{PS} = \frac{QT}{T V}$ (Line $∥$ to one side of $△$ OR prop. theorem, $ST ∥ P V$ )
  $\frac{QS}{PS} = \frac{3}{5}$
  2 marks · Use the proportionality theorem

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