# Mathematics Grade 12 P2 Eastern Cape June 2022 Memo (Afr)

> Grade 12 Mathematics Paper 2 memorandum from the Eastern Cape June 2022 (Afrikaans), 15 pages. Read online or download the free PDF.

- Page: https://www.hlayiso.com/library/maths-p2-memo-gr12-june2022-english-afrikaans/010d5d7e-d087-411a-ad67-08f865b59018
- Updated: 2026-09-27
- Subject: Mathematics
- Grade: Grade 12
- Language: Afrikaans
- Document type: Memorandum
- Year: 2022
- Exam period: Eastern Cape June
- Paper: 2
- Curriculum: CAPS
- Pages: 15

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## About this document

Mathematics · Grade 12 · Eastern Cape June · 2022 · Afrikaans. Memorandum, 15 pages. Read online or download the PDF.

Read it free online on the page above; download the PDF from there.

## Matching paper and memo

- [MATHS P2 QP GR12 JUNE 2022 Afrikaans](https://www.hlayiso.com/library/maths-p2-qp-gr12-june-2022-afrikaans/49feb536-3d0c-4e63-b857-c88def50a8c1) — QP · 2022 · Afr

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                            NATIONAL
                       SENIOR CERTIFICATE/
                            NASIONALE
                        SENIOR SERTIFIKAAT

                            GRADE/GRAAD 12

                            JUNE/JUNIE 2022

            MATHEMATICS P2/WISKUNDE V2
           MARKING GUIDELINE/NASIENRIGLYN

MARKS/PUNTE:    150

                 This marking guideline consists of 15 pages.
                 Hierdie nasienriglyn bestaan uit 15 bladsye.

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2                               MATHEMATICS P2/WISKUNDE V2                         (EC/JUNE/JUNIE 2022)

 QUESTION/VRAAG 1

 1.1   B                                                                ✓ answer/antwoord             (1)
 1.2   B                                                                ✓ answer/antwoord             (1)
 1.3   75%                                                              ✓ answer/antwoord             (1)
 1.4   Nothing. It remains the same. No change in standard deviation.   ✓reason/rede
       Niks. Dit bly dieselfde. Geen verandering in
       standaardafwyking.                                                                             (1)
 1.5
                             75 − 30
       Semi − IQR: IKV =                                                ✓ answer/antwoord
                                2

       Semi − IQR: IKV = 22,5                                                                         (1)
                                                                                                      [5]

 QUESTION/VRAAG 2

 2.1
           Time taken           No. of pupils      Cumulative frequency
           Tyd geneem          Aantal leerlinge   Kummulatiewe frekwensie
           60 ≤ t ≤ 90                  3                      3
                                                                                    ✓ for values
          90 ≤ t ≤ 120                  6                      9
                                                                                      vir waardes
          120 ≤ t ≤ 150                 7                      16
          150 ≤ t ≤ 180                 8                      24
          180 ≤ t ≤ 210                 6                      30
                                                                                                      (1)
 2.2

                                                                                    ✓ anchor
                                                                                    point/
                                                                                    ankerpunt
                                                                                    (60;0)
                                                               C
                                                                                    ✓ (120; 9)
                                                                                    ✓ (150; 16)
               A                                                                    ✓ (210; 30)

                                                           B

                                                                                                      (4)

 2.3 2.3.1       See diagram above / Sien diagram hierbo                            ✓A                (1)
     2.3.2       See diagram above / Sien diagram hierbo                            ✓B                (1)
     2.3.3       See diagram above / Sien diagram hierbo                            ✓C                (1)
                                                                                                      [8]

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                                   3

 QUESTION/VRAAG 3

 3.1          Median score / Mediaan telling = 2𝑥             ✓ answer/antwoord                         (1)

 3.2                             ∑𝑥
              Mean/Gemiddelde =
                                  𝑛
                4(𝑥 + 3) + 3(2𝑥) + 2(𝑥 − 1) + 2(6)            ✓ substitution/vervanging
              =                                               ✓ simplification/vereenvoudiging
                               11
                                                              ✓ answer/antwoord
                  12𝑥 + 22
              =
                     11
                                                                                                        (3)
 3.3          Use of a calculator where the four values are
              as follows:
              Gebruik van ’n sakrekenaar waar die vier
              waardes soos volg is:

              8 ; 10 ; 4 and/en 6                             ✓ four values/vier waardes
              𝑠𝑑(  ) = √5                                    ✓ answer/antwoord                         (2)
                                                                                                        [6]

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4                               MATHEMATICS P2/WISKUNDE V2                        (EC/JUNE/JUNIE 2022)

 QUESTION/VRAAG 4

                                                       𝑦              𝐷
                                         𝐴(−4; 5)

                                                           𝐸

                                                                    𝐵(4; 1)              𝑥

                                           𝐶(−1; −4)

 4.1          5−1                                          ✓ subst. into gradient formula/
       𝑚𝐴𝐵 =                                                 verv. in gradiënt formule
             −4 − 4
             4
           =
             −8
               1                                           ✓ 𝑚𝐴𝐵 = − 2
                                                                          1
       𝑚𝐴𝐵 = −
               2
       ∴ 𝑚𝐶𝐷 = 2
                                                                                                     (2)
 4.2   ∴ 𝑚𝐶𝐷 = 2                                           ✓ 𝑚𝐶𝐷 = 2
       ̅̅̅̅
       𝐶𝐷 ∶ 𝑦 = 2𝑥 + 𝑐                                     ✓sub of point/verv. van punt
            −4 = 2(−1) + 𝑐                                  (−1; −4)
            −2 = 𝑐                                         ✓ −2 = 𝑐
       ̅̅̅̅
       𝐶𝐷 ∶ 𝑦 = 2𝑥 − 2                                     ✓ equation of CD /
                                                             vergelyking van CD
                                                                                                     (4)
 4.3                 1
       ̅̅̅̅
       𝐴𝐵 ∶     𝑦 = − 𝑥+𝑐
                     2
                     1                                     ✓ substitution of point (4; 1)
                1 = − (4) + 𝑐
                     2                                       vervanging van punt (4;1)
                3=𝑐                                        ✓ equation of ̅̅̅̅
                                                                          𝐴𝐵
                     1                                       vergelyking van ̅̅̅̅
                                                                              𝐴𝐵
                𝑦 = − 𝑥+3
                     2                                     ✓ equating of AB and CD
        1                                                    gelykstel van AB en CD
       − 𝑥 + 3 = 2𝑥 − 2
        2
       −𝑥 + 6 = 4𝑥 − 4                                     ✓ x –value/waarde
           5𝑥 = 10                                         ✓ substitution of/vervanging van
            𝑥=2                                              𝑥 − value/waarde
            𝑦 = 2(2) − 2                                   ✓ 𝑦 − value/waarde
            𝑦=2
          𝐸(2; 2)                                                                                    (6)

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                                5

 4.4     𝐷(𝑥; 𝑦)
         𝑥−1                        𝑦−4
             =2                      =2
          2                             2
         𝑥−1=4                    𝑦−4=4
                                                            ✓𝑥=5
         𝑥=5                        𝑦=8
                                                            ✓𝑦=8                                    (2)
         𝐷(5; 8)

 4.5             5+4
         𝑚𝐴𝐶 =                                              ✓ substitution into gradient
                −4 + 1
         𝑚𝐴𝐶 = −3                                             formula/vervanging in
         Line parallel to AC has same gradient.              gradiënt formule
         Lyn ewewydig aan AC het dieselfde gradient.        ✓ 𝑚𝐴𝐶 = −3
         𝑦 = −3𝑥 + 𝑐
         8 = −3(5) + 𝑐                                      ✓ 𝑐 = 23
         𝑐 = 23
         𝑦 = −3𝑥 + 23                                       ✓ equation of line/vergelyking
                                                             van lyn                                (4)
 4.6     𝑥 intercept of CD ∶ 𝑥 𝑎𝑓𝑠𝑛𝑖𝑡 𝑣𝑎𝑛 CD
         2𝑥 − 2 = 0
         𝑥=1                                                ✓𝑥=1
         𝑚𝐵𝐶 = 1                                            ✓ 𝑚𝐵𝐶 = 1
         Equation of Altitude/Vergelyking van hoogtelyn
         𝑦 = −𝑥 + 𝑐                                         ✓ gradient of altitude −1
         5 = −(−4) + 𝑐                                        gradiënt van hoogtelyn −1
         𝑐=1                                                ✓ substitution of point
         ∴ 𝑦 = −𝑥 + 1                                         vervanging van punt
         𝑥 intercept of Altitude / x-afsnit van hoogtelyn   ✓equation of altitude
         𝑥=1                                                 vergelyking van hoogtelyn
                                                            ✓𝑥=1
         𝑥 intercept of CD = 𝑥 intercept of altitude
         x-afsnit van CD = x-afsnit van hoogtelyn                                                  (6)
                                                                                                   [24]

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6                               MATHEMATICS P2/WISKUNDE V2                        (EC/JUNE/JUNIE 2022)

    QUESTION/VRAAG 5

    5.1   𝑥 2 + 𝑦 2 = 4𝑦 − 2𝑥 + 44
          𝑥 2 + 2𝑥 + 1 + 𝑦 2 − 4𝑦 + 4 = 44 + 1 + 4 ✓✓ completing the square
          (𝑥 + 1)2 + (𝑦 − 2)2 = 49                     voltooiing van die vierkant
                                                   ✓ factorizing / faktorisering
          ∴ 𝑀(−1; 2)
                                                   ✓ 𝑀(−1; 2)                                      (4)
    5.2            𝑦−2
          𝑚𝑀𝐴 =
                   𝑥+1                             ✓ gradient of MA/gradiënt van MA
                   𝑦−2                             ✓ equating it to 1/stel dit gelyk aan 1
                         =1
                   𝑥+1                             ✓ making 𝑦 or 𝑥 the subject.
                  𝑦−2=𝑥+1                            maak y of x die onderwerp
                      𝑦 =𝑥+3                       ✓ equating the two linear functions.
                   𝑥 + 3 = −𝑥 + 5                    gelykstel van twee lineêre funksies
                      2𝑥 = 2                       ✓ coordinates of 𝐴(1; 4)
                       𝑥=1                           koördinate van 𝐴(1; 4)
                    ∴ 𝑦=4
          𝐴(1; 4)                                                                                  (5)
    5.3   (𝑥 + 1)2 + (𝑦 − 2)2 = 𝑟 2
          (1 + 1)2 + (4 − 2)2 = 𝑟 2                ✓ substitution of A/vervanging van A
                             8=𝑟  2                ✓ 8 = 𝑟2
          (𝑥 + 1)2 + (𝑦 − 2)2 = 8
                                                   ✓ equation of the circle/vergelyking van
                                                   . die sirkel                                    (3)

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(EC/JUNE/JUNIE 2022)               MATHEMATICS P2/WISKUNDE V2                                                    7

  5.4    𝐾(5; 0)                                              ✓ 𝐾(5; 0)
                                                                                                           (1)
  5.5    𝐴𝐾 = √32                                             ✓ 𝐴𝐾 = √32
                             1
         Area of/van ∆ 𝐴𝑀𝐾 = 2 𝐴𝐾 × 𝐴𝑀                        ✓ 𝐴𝑀 = √8
                                     1
         Area of/van ∆ 𝐴𝑀𝐾 = 2 √32 × √8
         Area of/van ∆ 𝐴𝑀𝐾 = 8 units 2 /𝑒𝑒𝑛ℎ𝑒𝑑𝑒 2             ✓ 8 units 2 /𝑒𝑒𝑛ℎ𝑒𝑑𝑒 2                        (3)
                                                                                                          [16]

  QUESTION/VRAAG 6

  6.1
                                                       𝑦
                                                                  𝑝
                                                                          √𝑝2 − 1
                                                              26°
                                                                1                𝑥

         6.1.1                     √𝑝2 − 1
                       sin 26° =                       ✓ sketch in Quadrant 1
                                     𝑝
                                                         skets in kwadrant 1
                                                       ✓
                                                         √𝑝2 − 1
                                                                                                            (3)
                                                       ✓ answer/antwoord
         6.1.2         cos 52° = cos 2(26° )           ✓cos 2(26° )
                                = 2 𝑐𝑜𝑠 2 26° − 1      ✓2 𝑐𝑜𝑠 2 26° − 1
                                     1 2
                                = 2( ) − 1
                                     𝑝
                                          2
                                     = 2−1             ✓ answer/antwoord
                                         𝑝                                                                  (3)
                           2   °
         6.1.3         𝑡𝑎𝑛 64 × (𝑝 + 1)
                                     2
                               1
                       =(           ) × (𝑝 + 1)                       2
                                                              1
                           √𝑝2 − 1                     ✓ (            )
                                                           √𝑝2 −1
                             1
                       = 2       × (𝑝 + 1)
                         𝑝 −1                          ✓
                                                              1
                                 1                           𝑝2 −1
                       =                   × (𝑝 + 1)   ✓ (𝑝 − 1)(𝑝 + 1)
                         (𝑝 − 1)(𝑝 + 1)
                            1
                       =                               ✓ answer/antwoord
                         𝑝−1                                                                                (4)

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8                               MATHEMATICS P2/WISKUNDE V2                           (EC/JUNE/JUNIE 2022)

 6.2     sin(−β) + sin(360° − β)
         sin(180o − β) + sin 180°
            − sin 𝛽 + (− sin 𝛽)                       ✓ − sin 𝛽
         =                                            ✓ − sin 𝛽
                 sin 𝛽 + 0
            −2 sin 𝛽                                  ✓ sin 𝛽
         =                                            ✓ simplification
             sin 𝛽                                       vereenvoudiging
         = −2                                         ✓ answer/antwoord                                (5)
 6.3              𝜃
         2𝑝 tan ( ) = sin(2𝜃)
                   2
                   82°                                ✓ substitution/vervanging
         2𝑝 tan ( ) = sin(2 × 82° )
                    2
              sin 162°                                ✓ simplification/vereenvoudiging
         𝑝=                                           ✓ answer/antwoord                                (3)
              2 tan 41°
         𝑝 = 0,16
 6.4     4 𝑠𝑖𝑛 𝜃 . 𝑐𝑜𝑠 3 𝜃 − 4 𝑐𝑜𝑠 𝜃. 𝑠𝑖𝑛3 𝜃
                          = 𝑠𝑖𝑛 4𝜃                    ✓ common factor/gemene faktor
         𝐿𝐻𝑆/𝐿𝐾 = 4 𝑠𝑖𝑛 𝜃 . 𝑐𝑜𝑠 3 𝜃                   ✓ 2 × 2 sin 𝜃 cos 𝜃
                          − 4 𝑐𝑜𝑠 𝜃. 𝑠𝑖𝑛3 𝜃           ✓ (cos 2𝜃)
               = 4 sin 𝜃 . cos 𝜃 (𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃)   ✓sin 2 𝜃.
               = 2 × 2 sin 𝜃 cos 𝜃 (cos 2𝜃)           ✓ 2. sin 2 𝜃. cos 2𝜃
               = 2. sin 2 𝜃. cos 2𝜃                   ✓ answer/antwoord
               = sin 4𝜃
               = 𝑅𝐻𝑆/𝑅𝐾                                                                               (6)
                                                                                                     [24]

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                          9

 QUESTION/VRAAG 7

 7.1    cos 3𝑥 = sin(𝑥 − 30° )
        cos 3𝑥 = cos[90° − (𝑥 − 30° )]                         ✓ Co- ratio
        cos 3𝑥 = cos[120 − 𝑥]                                    Ko-verhoud.
            3𝑥 = 120 − 𝑥 (ref angle)                           ✓ ref angle
                                                                 verwys. 
        3𝑥 = 120° − 𝑥 + 𝑘. 360° OR/OF 3𝑥 = 360° − [120° −      ✓ quadrant 1
        𝑥]𝑘. 360°                                                kwadrant 1
        4𝑥 = 120° + 𝑘. 360°           2𝑥 = 240° + 𝑘. 360°      ✓ quadrant 4
                °
         𝑥 = 30 + 𝑘. 90 °
                                      𝑥 = 120° + 𝑘. 180°         kwadrant 4
                                                               ✓ 30°
        𝑥 = 30° ; −60° ; 120°                                  ✓ −60°

                                                               ✓ 120°                         (7)
 7.2

                                        𝑔               𝑓

        ✓ Shape of 𝑓. ✓ 𝑥 intercepts ✓ start and end points
          Vorm van f x afsnitte         begin en eindpunte

        ✓ Shape of 𝑔. ✓ 𝑥 intercepts ✓ start and end points
          Vorm van g x afsnitte        begin en eindpunte                                       (6)
 7.3    −30° < 𝑥 < 30°                                                    °
                                                               ✓✓ −30 < 𝑥 < 30          °

        𝑶𝑹/𝑶𝑭 30° < 𝑥 < 90°                                    ✓ 30° < 𝑥 < 90°
        𝑶𝑹/𝑶𝑭 150° < 𝑥 < 180°                                  ✓ 150° < 𝑥 < 180°                (4)
                                                                                               [17]

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10                              MATHEMATICS P2/WISKUNDE V2                       (EC/JUNE/JUNIE 2022)

 QUESTION/VRAAG 8

                      P                  Q          R        𝑑               S
                                                  1 2

                                             𝛽
                                        𝛼     𝛼
                                         12 3

                                         T
 8.1   𝑄𝑇̂𝑅 = 𝛽 − 𝛼                                      ✓ answer/antwoord                        (1)
 8.2   𝑆̂ = 90° − 𝛽                                      ✓ answer/antwoord                        (1)
 8.3   𝑆̂ = 90° − 𝛼                                      ✓ answer/antwoord                        (1)
 8.4   In ∆ 𝑅𝑆𝑇
          𝑑         𝑅𝑇                                   ✓ use of sine rule
             =                                              gebruik van sinusreël
       sin 𝛼 sin(90° − 𝛽)
             𝑑 cos 𝛽                                     ✓ use of co-function
       𝑅𝑇 =                                                gebruik van ko-funksie
              sin 𝛼
                                                         ✓ answer/antwoord                        (3)
 8.5 𝑃𝑅           𝑅𝑇
          =
     sin 𝛽 sin(90° − 𝛼)                                  ✓ use of sine rule
          𝑅𝑇 sin 𝛽                                         gebruik van sinusreël
     𝑃𝑅 =                                                ✓ sub of RT
            cos 𝛼
          𝑑 cos 𝛽 sin 𝛽                                    vervanging van RT
     𝑃𝑅 =                                                ✓ answer/antwoord                        (3)
           sin 𝛼 cos 𝛼
                                                                                                  [9]

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                       11

 QUESTION/VRAAG 9

 9.1 centre / middelpuntshoek                                   ✓ answer/antwoord               (1)

                                                    A
                                                    𝑥

                                              O
                                              •

                                        𝑥
                                        3
                                B                               C

      ̂ C = 2𝑥
 9.2 BO               [angle at centre = 2 × angle at circum]   ✓ statement / stelling (S)
                      [Middelpuntshoek = 2  Omtrekshoek]       ✓ reason/rede (R)
               x                                                ✓statement / stelling (S)
       BĈO = 3       [angles opp = sides ; OB = OC]
                                                                ✓reason/rede (R)
                       [hoeke teenoor = sye ; OB = OC]
          x  x                                                  ✓ S/R
       ∴ 3 + 3 + 2𝑥 = 180°      [sum of angles of ∆]
                                 [som van hoeke van  ]
                  8𝑥 = 540°
                   𝑥 = 67,5°                                    ✓ answer/antwoord               (6)

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12                                 MATHEMATICS P2/WISKUNDE V2                         (EC/JUNE/JUNIE 2022)

 9.3

                                                A
                                                    2
                                                1

                                                                                            M

                                                             O

                                    1      .K
                               B     2

                                                     2
                                                         3
                                                1
                                                 C

       9.3.1     ̂
                 A1 = 30°
                 BÔ C = 60°    [angle at centre = 2 × angle at circum]             ✓ S/R
                                 [middelpuntshoek = 2 x omtrekshoek]                ✓ S/R
                 ̂     ̂    ̂
                 B2 = C1 + C2   [angles opp = sides ; OB = OC]                      ✓ S/R
                                 [hoeke teenoor = sye ; OB = OC]                    ✓S
                    ̂ 2 = 180 − 60° [sum of angles of ∆]
                 ∴ 2B         °

                                       [som van hoeke van  ]
                 ̂
                 B2 = 60°
                 ̂1 = 60° [BO bisects AB ̂ C] / [BO halveer AB
                                                             ̂ C]                   ✓ S/R
                 B                                                                                     (5)
       9.3.2     ̂
                 A1 = 30   °
                 ̂    ̂2 + M
                 B1 + B    ̂ = 180°           [opp angles of cyclic quad]
                                         [teenoorst. hoeke van koordevierhoek]      ✓ S/R
                 ̂ = 180° − 120°
                 M                                                                  ✓S
                 ̂ = 60°
                 M                                                                  ✓ answer/
                   ̂ =2×̂
                 ∴ M       A1                                                         antwoord         (3)
                                                                                                       [15]

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                                  13

 QUESTION/VRAAG 10

                                                                           R

                                                                   1 2 3
                                                       X
                                                        1 2

                        P
                                                 1 2
                                                                   2
                                             Y                 1       3
                                                                   Q

                                                                                                    S

 10.1        R1 = ̂
             ̂    R2            [YR bisects PR ̂ Q] / [YR halveer PR
                                                                   ̂ Q]        ✓ S/R
             R2 + ̂
             ̂    R3 = ̂
                       Y2       [angles opp = sides; RS = YS]                  ✓S ✓R
                                [hoeke teenoor = sye ; RS = YS]                ✓S ✓R
             R1 + ̂
             ̂    P=̂  Y2        [ext angle of ∆]/[buitehoek van  ]           ✓S
                ̂
             ∴ R3 = P̂
             ∴ SR = tangent/raaklyn
             [converse tan − chord theorem]
             [omgekeerde raaklyn-koord stelling]                                                            (6)

 10.2        Q2 = ̂
             ̂    R3            [alt angles = ; SR ∥ QX]                       ✓ S/R
                                [verw. hoeke = ; SR || QX]
             ̂
             P=̂ R3              [proven] / [bewys]                            ✓S
             Q2 = ̂
             ̂    P
             ∴ QR = tangent [converse tan − chord theorem]
               QR = raaklyn [omgekeerde raaklyn-koord stelling]                ✓S                           (3)
                                                                                                            [9]

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14                               MATHEMATICS P2/WISKUNDE V2                        (EC/JUNE/JUNIE 2022)

 QUESTION/VRAAG 11

                                                    A

                                              F
                                                   ℎ        𝑘   G

                                        D               ≫           E

                             B                         ≫                            C

 11.1   Proof / Bewys:
        Construct Perpendicular heights DG ( 𝑘) and EF (ℎ) in ∆ 𝐴𝐷𝐸. Join      ✓ constr.
        BE and DC.                                                             konstruk.
        Teken loodregte hoogtes DG (k) en EF (h) in ∆ 𝐴𝐷𝐸
        Verbind BE en DC
                                                                               ✓ S/R
                      1
        Area of ∆ADE 2 AD. h AD
                    =       =
        Area of ∆BDE 1 BD. h BD
                      2
                      1                                                        ✓S/R
        Area of ∆ADE 2 AE. k AE
                    =       =                                                  ✓S ✓ R
        Area of ∆CED 1 CE. k CE
                      2

        But/Maar Area ∆BDE = Area ∆CED (same base DE, same height)             ✓S
                                (dieselfde basis DE, dieselfde hoogte)
                                              DE ∥ BC

        Area of ∆ADE Area of ∆ADE
                    =
        Area of ∆BDE Area of ∆CED
             AD AE
         ∴     =
             BD CE
                                                                                                     (6)

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(EC/JUNE/JUNIE 2022)            MATHEMATICS P2/WISKUNDE V2                                                     15

 11.2
                                                   K

                                                  1 2

                                                Y1 2
                                   O 1                  ≫       1       P
                                        2         4 3               2

                                                  1 2                           M
                       L                                    ≫
                                                   X
         11.2.1    In ∆KOP and/𝑒𝑛 ∆KLM
                   ̂=K
                   K   ̂      [common] / [gemeen]                                        ✓ S/R
                   ̂    ̂
                   O1 = L      [corresponding angles = ; OP ∥ LM]
                               [ooreenkomstige hoeke = ; OP ∥ LM]                        ✓S/R
                   ̂    ̂
                   P1 = M      [corresponding angles =, OP ∥ LM]
                               [ooreenkomstige hoeke = ; OP ∥ LM]                        ✓R
                   ∴ ∆ KOP ||| ∆KLM [ A; A; A]                                                           (3)

         11.2.2    KO OP
                     =                  [similarity ∶ 𝑔𝑒𝑙𝑦𝑘𝑣𝑜𝑟𝑚𝑖𝑔]
                   KL LM                                                                 ✓ S/R

                   KO KY                                                                 ✓ S/R
                     =                  [ line parallel to one side of ∆]
                   KL KX
                                  [ lyn ewewydig aan een sy van  ]
                      KY OP
                   ∴     =
                      KX LM                                                                              (2)
         11.2.3    Area of ∆ KOP = Area of Quad/𝑉𝑖𝑒𝑟ℎ𝑜𝑒𝑘 OLMP
                   ∴ Area of ∆ KLM = 2 × Area of ∆ KOP                                   ✓S
                   1                  1                                                  ✓S
                     × LM × KX = 2 × × OP × KY
                   2                  2
                   1 OP. KY                    OP KY
                     =           but/𝑚𝑎𝑎𝑟         =
                   2 LM. KX                    LM KX                                     ✓S
                   OP 2 1
                        =                                                                ✓S
                   LM 2 2
                   OP     1
                       =                                                                 ✓S
                   LM √2
                   OP KO
                       =             [∆ KOP |||∆KLM]                                     ✓ S/R
                   LM KL
                     KO     1
                   ∴     =                                                                               (6)
                     KL √2
                                                                                                       [17]

                                                                            TOTAL/TOTAAL:               150

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