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Diketahui . Titik  pada  sehingga . Titik  pada ...

Diketahui begin mathsize 14px style triangle ABC end style.

Titik begin mathsize 14px style straight D end style pada begin mathsize 14px style BC with bar on top end style sehingga begin mathsize 14px style BD colon DC equals 2 colon 1 end style.

Titik Error converting from MathML to accessible text. pada pertengahan begin mathsize 14px style AB with bar on top end style.

Jika Error converting from MathML to accessible text. adalah titik potong begin mathsize 14px style AD with bar on top end style dan begin mathsize 14px style CE with bar on top end style, tentukan begin mathsize 14px style AZ colon ZD equals... end style dan begin mathsize 14px style CZ colon ZE equals... end style.

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Jawaban:

Untuk menyelesaikan permasalahan ini dengan menggunakan vektor posisi.

Misalkan basisnya adalah begin mathsize 14px style CA with rightwards arrow on top end style dan undefined (vektor begin mathsize 14px style CA with rightwards arrow on top equals a end style dan begin mathsize 14px style CB with rightwards arrow on top equals b end style). Berdasarkan hal tersebut diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AE with rightwards arrow on top end cell equals cell 1 half AB with rightwards arrow on top end cell row blank equals cell 1 half open parentheses AC with rightwards arrow on top plus CB with rightwards arrow on top close parentheses end cell row cell AE with rightwards arrow on top end cell equals cell 1 half open parentheses negative a plus b close parentheses space space space... open parentheses 1 close parentheses end cell end table end style 

 begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell CE with rightwards arrow on top end cell equals cell CA with rightwards arrow on top plus AE with rightwards arrow on top end cell row blank equals cell a plus 1 half open parentheses negative a plus b close parentheses end cell row blank equals cell a minus 1 half a plus 1 half b end cell row blank equals cell 1 half a plus 1 half b end cell row cell CE with rightwards arrow on top end cell equals cell 1 half open parentheses a plus b close parentheses space space... open parentheses 2 close parentheses end cell end table end style   

 begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AD with rightwards arrow on top end cell equals cell AC with rightwards arrow on top plus CD with rightwards arrow on top end cell row blank equals cell AC with rightwards arrow on top plus 1 third CB with rightwards arrow on top end cell row cell AD with rightwards arrow on top end cell equals cell negative a plus 1 third b space space... left parenthesis 3 right parenthesis end cell end table end style   

Karena begin mathsize 14px style AZ with rightwards arrow on top end style searah dengan begin mathsize 14px style AD with rightwards arrow on top end style, (begin mathsize 14px style straight lambda end style adalah skalar) maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AZ with rightwards arrow on top end cell equals cell lambda AD with rightwards arrow on top end cell row blank equals cell lambda space open parentheses negative a plus 1 third b close parentheses end cell row cell AZ with rightwards arrow on top end cell equals cell negative lambda space a plus 1 third lambda space b space space... open parentheses 4 close parentheses end cell end table end style 

Di lain pihak begin mathsize 14px style ZE with rightwards arrow on top end style  adalah kelipatan dari begin mathsize 14px style CE with rightwards arrow on top end style, (begin mathsize 14px style straight mu end style adalah skalar) sehingga begin mathsize 14px style ZE with rightwards arrow on top equals straight mu space CE with rightwards arrow on top end style, maka nilai lain dari undefined diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell AZ with rightwards arrow on top end cell equals cell AE with rightwards arrow on top plus EZ with rightwards arrow on top end cell row blank equals cell AE with rightwards arrow on top minus ZE with rightwards arrow on top end cell row blank equals cell AE with rightwards arrow on top minus straight mu space CE with rightwards arrow on top end cell row blank equals cell 1 half open parentheses negative a plus b close parentheses minus straight mu space 1 half open parentheses a plus b close parentheses end cell row blank equals cell negative 1 half a plus 1 half b minus 1 half straight mu space a minus 1 half straight mu space b end cell row cell ZE with rightwards arrow on top end cell equals cell open parentheses fraction numerator negative 1 minus straight mu over denominator 2 end fraction close parentheses a plus open parentheses fraction numerator 1 minus straight mu over denominator 2 end fraction close parentheses b space space... open parentheses 5 close parentheses end cell end table end style  

Dengan menyamakan koefisien begin mathsize 14px style a end style dan begin mathsize 14px style b end style pada persamaan (4) dan (5), diperoleh:
Koefisien begin mathsize 14px style a end stylebegin mathsize 14px style negative straight lambda equals fraction numerator negative 1 minus straight mu over denominator 2 end fraction end style  
Koefisien begin mathsize 14px style b end stylebegin mathsize 14px style 1 third straight lambda equals fraction numerator 1 minus straight mu over denominator 2 end fraction end style 

Sederhanakan koefisien begin mathsize 14px style a end style, diperoleh sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell negative lambda end cell equals cell fraction numerator negative 1 minus mu over denominator 2 end fraction end cell row lambda equals cell fraction numerator 1 plus mu over denominator 2 end fraction end cell end table end style 

Substitusikan hasil tersebut pada koefisien begin mathsize 14px style b end style, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell 1 third straight lambda end cell equals cell fraction numerator 1 minus straight mu over denominator 2 end fraction end cell row cell 1 third open parentheses fraction numerator 1 plus straight mu over denominator 2 end fraction close parentheses end cell equals cell fraction numerator 1 minus straight mu over denominator 2 end fraction end cell row cell fraction numerator 1 plus straight mu over denominator 6 end fraction end cell equals cell fraction numerator 1 minus straight mu over denominator 2 end fraction end cell row cell 1 plus straight mu end cell equals cell 3 open parentheses 1 minus straight mu close parentheses end cell row cell 1 plus straight mu end cell equals cell 3 minus 3 straight mu end cell row cell straight mu plus 3 straight mu end cell equals cell 3 minus 1 end cell row cell 4 straight mu end cell equals 2 row straight mu equals cell 1 half end cell end table end style 

Substitusikan begin mathsize 14px style straight mu equals 1 half end style ke koefisien begin mathsize 14px style a end style, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row straight lambda equals cell fraction numerator 1 plus straight mu over denominator 2 end fraction end cell row straight lambda equals cell fraction numerator 1 plus begin display style 1 half end style over denominator 2 end fraction end cell row straight lambda equals cell fraction numerator begin display style 3 over 2 end style over denominator 2 end fraction end cell row straight lambda equals cell 3 over 4 end cell end table end style 

Sehingga diperoleh begin mathsize 14px style straight mu equals 1 half end style dan begin mathsize 14px style straight lambda equals 3 over 4 end style.

Karena begin mathsize 14px style AZ with rightwards arrow on top equals straight lambda stack space AD with rightwards arrow on top end style dan begin mathsize 14px style straight lambda equals 3 over 4 end style, maka begin mathsize 14px style AZ with rightwards arrow on top equals 3 over 4 stack space AD with rightwards arrow on top end style.

Dengan menggunakan konsep panjang vektor, perhatikan perhitungan berikut:

 begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open vertical bar AZ with rightwards arrow on top close vertical bar end cell equals cell 3 over 4 open vertical bar AD with rightwards arrow on top close vertical bar end cell row AZ equals cell 3 over 4 AD end cell row cell AZ over AD end cell equals cell 3 over 4 space space open parentheses ingat space ZD equals AD minus AZ close parentheses end cell row cell AZ over ZD end cell equals cell fraction numerator 3 over denominator 4 minus 3 end fraction end cell row cell AZ over ZD end cell equals cell 3 over 1 end cell row cell AZ space colon space ZD end cell equals cell 3 space colon space 1 end cell end table end style 

Karena begin mathsize 14px style ZE with rightwards arrow on top equals straight mu space CE with rightwards arrow on top end style dan begin mathsize 14px style straight mu equals 1 half end style, maka begin mathsize 14px style ZE with rightwards arrow on top equals 1 half CE with rightwards arrow on top end style.

Dengan menggunakan konsep panjang vektor, perhatikan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open vertical bar ZE with rightwards arrow on top close vertical bar end cell equals cell 1 half open vertical bar CE with rightwards arrow on top close vertical bar end cell row ZE equals cell 1 half CE end cell row cell ZE over CE end cell equals cell 1 half space space space open parentheses ingat space CZ equals CE minus ZE close parentheses end cell row cell CZ over ZE end cell equals cell fraction numerator 2 minus 1 over denominator 1 end fraction end cell row cell CZ over ZE end cell equals cell 1 over 1 end cell row cell CZ space colon space ZE end cell equals cell 1 space colon space 1 end cell end table end style 

Jadi, diperoleh begin mathsize 14px style AZ space colon space ZD equals 3 space colon space 1 end style dan begin mathsize 14px style CZ space colon space ZE equals 1 space colon 1 end style.

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