Grade 12 Maths · Paper 2

Euclidean Geometry Grade 12

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Question 9.2 · March 2021 · Provincial · 6 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}

9.2 If the straight line distance of the track from F to C is 374 km and it takes 50 hours to build one km of the track, determine the number of hours it will take to build the section from E to D.(6)

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What's in Euclidean Geometry Grade 12

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Ratios and Proportion1 past paper questions

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More Euclidean Geometry Grade 12 questions and answers

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1. Question 9.1.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
Provide reasons for your statements in QUESTIONS 9 and 10.
9.1 In the diagram, $Δ D EF$ is drawn. Line GH intersects DF and EF at G and H respectively such that $G H ∥ D E$ and $\frac{GF}{D G} = \frac{2}{5}$ .
9.1.1 Write down, with a reason, the value of $\frac{H F}{E H}$ .(2)
Show the memo (1 tick)Memo
1. ✓
  State the ratio with the correct reason $\frac{H F}{E H} = \frac{GF}{G D} = \frac{2}{5}$
  
  Reason: line $∥$ one side of $Δ$ OR proportionality theorem; $G H ∥ D E$
  2 marks · Use the proportionality theorem
2. Question 4.2.1 · 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.
4.2.1 Determine with reasons the value of $A E$ .(3)
Show the memo (2 ticks)Memo
1. ✓
  Setup ratio using Proportionality Theorem $\frac{A E}{A B} = \frac{A D}{A C} = \frac{3}{5}$ (Prop. Thm; $E D ∥ BC$ )
  2 marks · Use the proportionality theorem
2. ✓
  Calculate length $\frac{A E}{15} = \frac{3}{5}$
  $A E = 9$ units
  1 mark · Use the proportionality theorem
3. Question 6.1 · March 2022 · Provincial · 5 marks
Given $△ A BC$ with $D E ∥ BC$ as shown in the figure.
6.1 Prove that: $\frac{A D}{D B} = \frac{A E}{EC}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruct an altitude $h$ from $E$ to $A D$ (height $h_{1}$ ) and from $D$ to $A E$ (height $h_{2}$ ). Join $D C$ and $BE$ .
  1 mark · State the construction
2. ✓
  Area ratios on LHS $\frac{Area Δ A D E}{Area Δ D BE} = \frac{\frac{1}{2} A D \cdot h _{1}}{\frac{1}{2} D B \cdot h _{1}} = \frac{A D}{D B}$
  1 mark · Prove a theorem or formula
3. ✓
  Area ratios on RHS $\frac{Area Δ A D E}{Area Δ D EC} = \frac{\frac{1}{2} A E \cdot h _{2}}{\frac{1}{2} EC \cdot h _{2}} = \frac{A E}{EC}$
  1 mark · Prove a theorem or formula
4. ✓
  Equate areasArea $Δ D BE$ = Area $Δ D EC$ (Same base $D E$ , same height between parallel lines $D E ∥ BC$ )
  1 mark · Prove a theorem or formula
5. ✓
  Conclusion $∴ \frac{A D}{D B} = \frac{A E}{EC}$
  1 mark · Prove a theorem or formula
4. Question 7.1 · March 2022 · Provincial · 4 marks
In the diagram, $A C$ and $CE$ are tangents to the circle with centre $O$ . $B, D, G$ and $H$ are points on the circumference. $H G$ is produced to $F$ . $B D = B H$ . $\hat{D}_{1} = \hat{D}_{4} = x$ .
7.1 Find four other angles equal to $x$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Find B3 $\hat{B}_{3} = x$ (Tangents from same point $C$ , so $CB = C D$ , $Δ CB D$ is isosceles / Tan-chord theorem implies $\hat{B}_{3} = \hat{D}_{1}$ but logic usually $CB = C D$ )
  1 mark · Use isosceles triangle properties
2. ✓
  Find H1 $\hat{H}_{1} = \hat{B}_{3} = x$ (Tan-chord theorem with tangent $A C$ and chord $B D$ - Wait, $A C$ touches at $B$ )
  1 mark · Use the tan-chord theorem
3. ✓
  Find D2 $\hat{D}_{2} = \hat{H}_{1} = x$ (Angles opposite equal sides, $B D = B H$ )
  1 mark · Use isosceles triangle properties
4. ✓
  Find H2 $\hat{H}_{2} = \hat{D}_{4} = x$ (Tan-chord theorem with tangent $CE$ and chord $D G$ - wait, tangent is at $D$ , chord $D G$ subtends $H_{2}$ . Correct)
  1 mark · Use the tan-chord theorem
5. Question 7.1 · March 2022 · Provincial · 4 marks
In the diagram, PQRS is a cyclic quadrilateral. PS and QR are produced to meet at T. RE is a tangent to the circle at R, with E on PT and RE || QS.
7.1 Prove that: $QR = RS$ (4)
Show the memo (3 ticks)Memo
1. ✓
  Use tan-chord theorem $\hat{R}_{2} = \hat{Q}_{2}$ (tan-chord theorem)
  2 marks · Use the tan-chord theorem
2. ✓
  Use parallel lines $\hat{S}_{2} = \hat{R}_{2}$ (alternate angles; $QS ∥ RE$ )
  1 mark · Use angles on parallel lines
3. ✓
  Conclude equality $∴ \hat{Q}_{2} = \hat{S}_{2}$ . Thus $QR = RS$ (sides opposite equal angles)
  1 mark · Use isosceles triangle properties

Euclidean Geometry Grade 12 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Euclidean geometry proofs in less than 10 minutes - Grade 11 and12 NSC & IEBMathew's Academy Of Sciences

Euclidean Geometry Grade 10: Midpoint Theorem IntroductionKevinmathscience

Grade 12 Proportionality: Area of triangle #1Kevinmathscience

Grade 12 Proportionality Theorem: #1Kevinmathscience

Gr 9 Geometry: Parallel linesKevinmathscience

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