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11.2 Middelpunt van 'n lyn

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Middelpunt van 'n lynstuk

Soms is dit nuttig om die koördinate van 'n lyn se middel of middelpunt te hê. Byvoorbeeld: wat is die middelpunt van die lynstuk tussen punt P met koördinate ( 2 ; 1 ) en punt Q met koördinate ( - 2 ; - 2 ) ?

Die koördinate van die middelpunt van 'n lyn tussen enige twee punte A en B met koördinate ( x 1 ; y 1 ) en ( x 2 ; y 2 ) , word as volg bereken. Gestel die middelpunt van A B is die punt S met koördinate ( X ; Y ) . Die doel is om te bereken X en Y in terme van ( x 1 ; y 1 ) en ( x 2 ; y 2 ) .

X = x 1 + x 2 2 Y = y 1 + y 2 2 ∴ S x 1 + x 2 2 ; y 1 + y 2 2

Dus die koördinate van ( S ), die middelpunt van die lyn tussen punt P met koördinate ( 2 ; 1 ) en punt Q met koördinate ( - 2 ; - 2 ) , is:

X = x 1 + x 2 2 = - 2 + 2 2 = 0 Y = y 1 + y 2 2 = - 2 + 1 2 = - 1 2 ∴ S is by ( 0 ; - 1 2 )

Dit kan bewys word dat die afstande vanaf die eindpunte na die middelpunt gelyk is. Die koördinate van die middelpunt S is ( 0 ; - 0 , 5 ) .

P S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - 2 ) 2 + ( - 0 . 5 - 1 ) 2 = ( - 2 ) 2 + ( - 1 . 5 ) 2 = 4 + 2 . 25 = 6 . 25

en

Q S = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( 0 - ( - 2 ) ) 2 + ( - 0 . 5 - ( - 2 ) ) 2 = ( 0 + 2 ) ) 2 + ( - 0 . 5 + 2 ) ) 2 = ( 2 ) ) 2 + ( + 1 . 5 ) ) 2 = 4 + 2 . 25 = 6 . 25

Daar kan gesien word dat P S = Q S soos verwag is.

Die volgende video verskaf 'n opsomming oor die berekening van die middelpunt van 'n lyn.

Khan akademie video oor die middelpunt van 'n lyn

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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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