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Vierhoeke en poligone

Module by: Free High School Science Texts Project. E-mail the author

Inleiding

Meetkunde (Grieks: geo = aarde, metria = meet) het ontstaan as die veld van kennis wat ruimtelike verhoudings hanteer. Dit was een van die twee velde van pre-moderne wiskunde. Die ander veld was die studie van getalle. In die moderne tyd het meetkundige begrippe baie kompleks en abstrak geraak en is dit skaars herkenbaar as ʼn uitvloeisel van vroeë meetkunde.

Navorsingsprojek: Die geskiedenis van Meetkunde

Werk in pare of groepe en bestudeer die geskiedenis van die onstaan van meetkunde. Beskryf die verskillende stadiums van ontwikkeling en hoe meetkunde later gebruik is deur mense om hul lewens te verbeter. Die lys van stadiums moet dien as ʼn riglyn en hoef slegs die minimum vereistes te beskryf.

  1. Antieke Indiese meetkunde (ong. 3000 - 500 V.C.)
    1. Harappanse meetkunde
    2. Vediese meetkunde
  2. Klassieke Griekse meetkunde (ong. 600 - 300 V.C.)
    1. Thales en Pythagoras
    2. Plato
  3. Hellenistiese meetkunde (ong. 300 V.C - 500 N.C )
    1. Euclides
    2. Archimedes

Vierhoeke

In hierdie afdeling sal ons kyk na die eienskappe van sekere spesiale vierhoeke. Ons sal dan hierdie eienskappe gebruik om meetkundige probleme op te los. Dit is belangrik om daarop te let dat alhoewel al die eienskappe van ʼn figuur gegee word, benodig ons net sekere unieke eienskappe van die vierhoek om te bewys dat dit wel daardie spesifieke vierhoek is. Byvoorbeeld, as ons ʼn vierhoek het met twee pare parallellesye, dan is daardie vierhoek ʼn parallelogram. Ons kan dan die ander eienskappe van die vierhoek aflei deur ons kennis van parallellelyne en driehoeke te gebruik.

Trapesium

ʼn Trapesium is ʼn vierhoek waarvan ten minste een paar teenoorgestelde sye parallel loop. Dit word soms ook ʼn trapesoïed genoem. ʼn Spesiale tipe trapesium is die gelykbenige trapesium, waar een paar teenoorstaande sye parallel is en die ander paar ewe lank is. Die hoeke aan die eindpunte van elke parallelle sy is ewe groot. ʼn Gelykbenige trapesium het een lyn van simmetrie en sy hoeklyne is ewe lank.

Figure 1: Voorbeelde van trapesiums
Figure 1 (MG10C13_040.png)

Parallelogram

ʼn Trapesium met beide pare teenoorstaande sye parallel, word ʼn parallelogram genoem. ʼn Opsomming van die eienskappe van ʼn parallelogram is:

  • Beide pare teenoorstaande sye is parallel.
  • Beide pare teenoorstaande sye is ewe lank.
  • Beide pare teenoorstaande hoeke is ewe groot.
  • Beide hoeklyne/diagonale halveer mekaar (d.w.s. hulle sny mekaar in die helfte)
Figure 2: ʼn Voorbeeld van ʼn parallelogram
Figure 2 (MG10C13_041.png)

Reghoek

ʼn Reghoek is ʼn parallelogram met al vier hoeke ewe groot en gelyk aan 90∘90∘. ʼn Opsomming van die eienskappe van ʼn reghoek is:

  • Beide pare teenoorstaande sye is parallel.
  • Beide pare teenoorstaande sye is ewe lank.
  • Die hoeklyne halveer mekaar.
  • Die hoeklyne is ewe lank.
  • Alle hoekpunte is regte hoeke.
Figure 3: Voorbeeld van ʼn reghoek
Figure 3 (MG10C13_042.png)

Rombus / Ruit

ʼn Rombus (ruit) is ʼn parallelogram waarvan al vier sye ewe lank is. ʼn Opsomming van die eienskappe van ʼn rombus is:

  • Beide pare teenoorstaande sye is parallel.
  • Al vier sye is ewe lank.
  • Beide pare teenoorstaande hoeke is ewe groot.
  • Die diagonale halveer mekaar met hoeke van 90∘90∘.
  • Diagonale halveer die teenoorstaande hoeke.
Figure 4: ʼn Voorbeeld van ʼn ruit, ʼn parallelogram met al vier sye ewe lank
Figure 4 (MG10C13_043.png)

Vierkant

ʼn Vierkant is ʼn rombus met al vier sye ewe lank en al vier hoeke gelyk aan 90∘∘.

ʼn Opsomming van die eienskappe van ʼn vierkant:

  • Beide pare teenoorstaande sye is parallel.
  • Al vier sye is ewe lank.
  • Al vier die hoeke is 90∘90∘.
  • Beide pare teenoorstaande hoeke is ewe groot.
  • Die hoeklyne halveer mekaar met hoeke van 90∘90∘.
  • Diagonale is ewe lank.
  • Diagonale halveer beide pare teenoorstaande hoeke (d.w.s. hulle is almal 45∘45∘).
Figure 5: ʼn Voorbeeld van ʼn vierkant - ʼn rombus met al die hoeke gelyk aan 90∘∘
Figure 5 (MG10C13_044.png)

Vlieër

ʼn Vlieër is ʼn vierhoek met twee pare aangrensende sye ewe lank.

ʼn Oposmming van die eienskappe van ʼn vlieër is:

  • Twee pare aangrensende sye is ewe lank.
  • Een paar teenoorstaande hoeke (die hoeke tussen die ongelyke sye) is ewe groot.
  • Een diagonaal halveer die ander een en hierdie diagonaal halveer ook een paar teenoorstaande hoeke.
  • Diagonale sny mekaar reghoekig.
Figure 6: ʼn Voorbeeld van ʼn vlieër
Figure 6 (MG10C13_045.png)

Reghoeke is ʼn spesiale geval (ʼn deelversameling) van die parallelogramme. Reghoeke is parallelogramme met alle hoeke regte hoeke. Vierkante is ʼn spesiale geval (deelversameling) van die reghoeke. Vierkante is reghoeke met al vier sye ewe lank. So, alle vierkante is parallelogramme én reghoeke. As jy gevra word om te bewys dat ʼn vierhoek ʼn parallelogram is, is dit genoeg om aan te toon dat beide pare teenoorstaande sye parallel is. Maar, as jy gevra word om te bewys dat ʼn vierhoek ʼn vierkant is, dan moet jy ook wys dat al die hoeke regte hoeke is én dat al die sye ewe lank is.

Veelhoeke

Veelhoeke is oral rondom ons. ʼn Stopteken het die vorm van ʼn agthoek, m.a.w. ʼn agthoekige veelhoek. Die heuningkoek van ʼn bynes bestaan uit heksagonale selle. Die oppervlak van ʼn tafel is dikwels ʼn reghoek.

In hierdie afdeling sal jy leer van gelykvormige veelhoeke.

Gelykvormigheid tussen Veelhoeke

Bespreking: Gelykvormige Driehoeke

Gebruik die diagram om die tabel in te vul en beantwoord die vrae wat daarop volg.

Table 1
AB DE AB DE =...cm...cm=......cm...cm=... A^A^=...∘∘ D^D^...∘∘
BC EF BC EF =...cm...cm=......cm...cm=... B^B^=...∘∘ E^E^=...∘∘
AC DF AC DF =...cm...cm=......cm...cm=... C^C^...∘∘ F^F^=...∘∘

Figure 7
Figure 7 (MG10C14_010.png)

  1. Wat kan jy sê oor jou berekening van: AB DE AB DE , BC EF BC EF , AC DF AC DF ?
  2. Wat kan jy sê oor A^A^ en D^D^?
  3. Wat kan jy sê oor B^B^ en E^E^?
  4. Wat kan jy sê oor C^C^ en F^F^?

As twee veelhoeke gelykvormig is, is die een ʼn vergroting van die ander. Dit beteken dat die veelhoeke dieselfde grootte hoeke sal hê en dat hulle sye in verhouding tot mekaar sal wees.

Die simbool wat ons gebruik om gelykvormigheid aan te dui is ||||||.

Definition 1: Gelykvormige Veelhoeke

Twee veelhoeke is gelykvormig as:

  1. hulle ooreenstemmende hoeke ewe groot is, én
  2. hulle ooreenstemmende sye eweredig is (die verhouding van die sylengtes gelyk is.)

Exercise 1: Gelykvormigheid van Veelhoeke

Bewys dat die volgende twee veelhoeke gelykvormig is.

Figure 8
Figure 8 (MG10C14_011.png)

[ Show Solution ]

Tip:

Alle vierkante is gelykvormig.

Exercise 2: Gelykvormigheid van Veelhoeke

As twee vyfhoeke ABCDE en GHJKL gelykvormig is, bepaal die lengtes van die sye en die groottes van die hoeke wat met letters gemerk is:

Figure 9
Figure 9 (MG10C14_012.png)

[ Show Solution ]

Gelykvormigheid van Gelyksydige Driehoeke

Werk in pare en toon dat alle gelyksydige driehoeke gelykvormig is.

Veelhoeke gemeng

  1. Vind die onbekende waardes in elke geval. Gee redes.
    Figure 10
    Figure 10 (mg10c14_2.png)
    Kliek hier vir die oplossing
  2. Vind die hoeke en lengtes wat met letters gemerk is in die volgende figure:
    Figure 11
    Figure 11 (MG10C14_014.png)
    Kliek hier vir die oplossing

Ondersoek: Definieer Poligone

Ondersoek verskillende maniere om poligone te definieer. Jy behoort spesiale aandag te gee aan die volgende poligone:

  • Gelykbenige driehoeke, gelyksydige driehoeke, reghoekige driehoeke
  • Vlieërs, parallelogramme, reghoeke, rombusse, vierkante, trapesiums

Neem in oorweging hoe die figure in hierdie boek gedefinieer is en watter alternatiewe definisies daar bestaan. Byvoorbeeld, ʼn driehoek is ʼn driesydige poligoon of ʼn driehoek is ʼn figuur met drie sye en drie hoeke. Driehoeke kan geklassifiseer word volgende hulle sye of volgens hulle hoeke. Kan mens ook vierhoeke op hierdie manier klassifiseer? Watter ander name is daar vir hierdie figure? Byvoorbeeld, vierhoeke kan ook genoem word tetragone.

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