Grade 12 Maths · Euclidean Geometry · Prove a product using similarity

Prove a product using similarity Grade 12

Prove a statement like AB^2 = AD AC or AB CD = BC AE by choosing the right pair of similar triangles, writing their sides in proportion and cross-multiplying, sometimes swapping in equal sides first.

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Similarity Grade 12 : Exam questionKevinmathscience

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Question 10.2 · May/June 2023 · Paper 2 · National (DBE) · 5 marks

In the diagram, TR is a diameter of the circle. PRKT is a cyclic quadrilateral. Chords TP and KR are produced to intersect at S. Chord PK is drawn such that PK

=

TK. $Diagram for this question$

10.2 If it is further given that SR = RK, prove that

ST = 6 R K

.(5)

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More past paper questions: prove a product using similarity

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3. Question 10.2.2 · May/June 2025 · Paper 2 · National (DBE) · 2 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.
10.2.2 If it is given that $R \hat{D} O = F \hat{C} O$ , show, with reasons, that $FO \cdot C D = CO \cdot BC$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Use the similarity ratio and given angle equalityFrom 10.2.1: $\frac{FO}{B D} = \frac{CO}{C D}$ [ $∣∣∣$ $Δ$ s]
  $∴ FO \cdot C D = CO \cdot B D$
  
  But $R \hat{D} O = F \hat{C} O$ [given]
  $∴ B D = BC$ [sides opp equal $∠$ s]
  
  Substitute BC for BD:
  $∴ FO \cdot C D = CO \cdot BC$
  2 marks · Prove a product using similarity
4. Question 11.2.4 · November 2025 · Paper 2 · National (DBE) · 3 marks
Provide reasons for your statements.
11.2 In the diagram, WVZR is a cyclic quadrilateral. RZ is produced to Q. A tangent is drawn from Q to touch the circle at S. WV is produced to E, a point on ZQ. RW produced meets ZV produced in P. PQ $∥$ WE. RS and ZS are drawn.

Prove, with reasons, that: $Diagram for question 11.2$
11.2.4 $PQ = QS$ (3)
Show the memo (3 ticks)Memo
1. ✓
  S/R $\frac{QS}{QR} = \frac{QZ}{QS}$ [ $Δ QSZ ∣∣∣ Δ QRS$ ]
  1 mark · Use similar triangles' sides and angles
2. ✓
  S $∴ Q S^{2} = QZ . QR$
  1 mark · Prove a product using similarity
3. ✓
  SBut $P Q^{2} = RQ . QZ$ [proved in 11.2.2]
  $∴ PQ = QS$
  1 mark · Prove a product using similarity
5. Question 2.2.4 · March 2021 · Provincial · 2 marks
2.2 In the diagram, a circle with a tangent CD is drawn. A, B, D, and E are points on the circumference of the circle. $A E = A B$ and $A B ∥ E D$ . $\hat{A}_{1} = x$ .
2.2.4 Hence, prove that $A E^{2} = BC \cdot E D$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute DC from (a) into (b) $E A = (\frac{BC \times A D}{D B}) (\frac{D E}{A D})$
  1 mark · Prove a product using similarity
2. ✓
  Simplify $E A = \frac{BC \times D E}{D B}$
  Since $D B = E A$ (from 2.2.3b logic)
  $E A = \frac{BC \times D E}{E A}$
  $E A^{2} = BC \times D E$
  1 mark · Prove a product using similarity
6. Question 10.5 · May/June 2024 · Paper 2 · National (DBE) · 5 marks
In the diagram, COD is the diameter of the circle with centre O. EA is a tangent to the circle at F. AO $⊥$ CE. Diameter COD produced intersects the tangent to the circle at E. OB produced intersects the tangent to the circle at A. CF intersects OB in T. CB, BD, OF and FD are drawn. $Diagram for this question$
10.5 Prove that $D O = \frac{TO . FE}{A B}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  RIn $Δ OE A$ : DB $∥$ EA [proven]
  $\frac{O D}{D E} = \frac{OB}{B A}$ [line $∥$ one side of $Δ$ ]
  1 mark · Use the proportionality theorem
2. ✓
  S $∴ D E = \frac{D O . A B}{OB}$
  1 mark · Use the proportionality theorem
3. ✓
  S/R $\frac{FO}{FE} = \frac{TO}{D E}$ [ $Δ TFO ∣∣∣ Δ D FE$ ]
  1 mark · Use similar triangles' sides and angles
4. ✓
  S $D E = \frac{TO . FE}{FO}$
  1 mark · Use similar triangles' sides and angles
5. ✓
  S $∴ \frac{D O . A B}{OB} = \frac{TO . FE}{FO}$
  $∴ \frac{D O . A B}{D O} = \frac{TO . FE}{D O}$ [DO = OB = FO]
  $∴ D O = \frac{TO . FE}{A B}$
  1 mark · Prove a product using similarity

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