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Cartesiese vlak en die afstand tussen twee punte

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

Inleiding

Analitiese meetkunde, ook bekend as koördinaatmeetkunde en vroëer bekend as Cartesiese meetkunde, is die studie van meetkunde op grond van die beginsels van algebra en die Cartesiese koördinaatstelsel. Dit is gemoeid met die definisie van meetkundige figure op 'n numeriese wyse en onttrek numeriese inlligting uit die voorstelling. Sommige beskou die ontwikkeling van analitiese meetkunde as die begin van moderne wiskunde.

Afstand tussen Twee Punte

As ons die koördinate van die hoekpunte van 'n figuur het, dan kan ons die figuur op die Cartesiese vlak teken. Byvoorbeeld, neem die vierhoek ABCD met koördinate A(1,1), B(1,3), C(3,3) en D(1,3) en stel dit voor op die Cartesiese vlak. Dit word getoon in Figure 1.

Figure 1: Vierhoek ABCD voorgestel op die Cartesiese vlak
Figure 1 (square.png)

Om enige figuur voor te stel op die Cartesiese vlak, plaas ons 'n punt by elke gegewe koördinaat en verbind dan hierdie punte met reguitlyne. Een belangrike saak om op te let, is in die benoeming van die figuur. In bostaande voorbeeld, het ons die vierhoek ABCD genoem. Dit dui vir ons aan dat ons beweeg van punt A, na punt B, na punt C, na punt D en dan weer terug na punt A. Dus, wanneer jy gevra word om 'n figuur op die Cartesiese vlak te teken, moet jy hierdie benamingswyse gebruik. Soms word net sekere punte gegee en dan moet ons die ander punte vind deur gebruik te maak van die metodes wat ons verder in die hierdie hoofstuk gaan bespreek.

Afstand tussen Twee Punte

Een van die eenvoudigste dinge wat met analitiese meetkunde bereken kan word, is die afstand tussen twee punte. Afstand is a getal wat beskryf hoe ver twee punte van mekaar is. Byvoorbeeld, punt PP het (2,1)(2,1) as koördinate en punt QQ het (-2,-2)(-2,-2) as koördinate. Hoe ver is die punte PP en QQ van mekaar? In die figuur beteken dit, hoe lank is die stippellyn?

Figure 2
Figure 2 (MG10C14_015.png)

In die figuur kan gesien word dat lyn PRPR 3 eenhede lank is en lyn QRQR 4 eenhede. PQRPQR het 'n regte hoek RR. Dus kan die lengte van sy PQPQ bereken word deur Stelling van Pythagoras te gebruik:

P Q 2 = P R 2 + Q R 2 P Q 2 = 3 2 + 4 2 P Q = 3 2 + 4 2 = 5 P Q 2 = P R 2 + Q R 2 P Q 2 = 3 2 + 4 2 P Q = 3 2 + 4 2 = 5
(1)

Die lengte van PQPQ is gelyk aan die afstand tussen punte PP en QQ.

As 'n veralgemening van die idee, neem aan dat AA enige punt is met (x1;y1)(x1;y1) as koördinate en BB is enige ander punt met (x2;y2)(x2;y2) as koördinate.

Figure 3
Figure 3 (MG10C14_016.png)

Die formule vir die berekening van die afstand tussen twee punte word as volg afgelei. Die afstand tussen twee punte AA en BB is die lengte van die lyn ABAB. Volgens die Stelling van Pythagoras, word die lengte van ABAB gegee deur:

A B = A C 2 + B C 2 A B = A C 2 + B C 2
(2)

Ons sien

B C = y 2 - y 1 A C = x 2 - x 1 B C = y 2 - y 1 A C = x 2 - x 1
(3)

Dan is

A B = A C 2 + B C 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 A B = A C 2 + B C 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2
(4)

Gevolglik, vir enige twee punte,(x1;y1)(x1;y1) en (x2;y2)(x2;y2), is die formule:

Afstand=(x1-x2)2+(y1-y2)2(x1-x2)2+(y1-y2)2

Deur die formule te gebruik, word die afstand tussen twee punte PP en QQ met koördinate (2;1) en (-2;-2) as volg bereken. Gestel die koördinate van punt PP is (x1;y1)(x1;y1) en die koördinate van punt QQ is (x2;y2)(x2;y2). Dan is die afstand:

Afstand = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 2 - ( - 2 ) ) 2 + ( 1 - ( - 2 ) ) 2 = ( 2 + 2 ) 2 + ( 1 + 2 ) 2 = 16 + 9 = 25 = 5 Afstand = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 2 - ( - 2 ) ) 2 + ( 1 - ( - 2 ) ) 2 = ( 2 + 2 ) 2 + ( 1 + 2 ) 2 = 16 + 9 = 25 = 5
(5)

Khan Akademie video oor die afstandformule

Figure 4
Khan Akademie video oor die afstandformule

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