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Diketahui segitiga ABC dengan koordinat straight A left parenthesis 2 comma negative 1 comma 0 right parenthesis comma spacestraight B left parenthesis 0 comma negative 1 comma negative 1 right parenthesis spacedan straight C left parenthesis 1 comma 1 comma negative 2 right parenthesis comma space spacemaka nilai Cosinus sudut antara sisi AB dan AC adalah ...

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai Cosinus sudut antara sisi AB dan AC adalah  fraction numerator 4 square root of 5 over denominator 15 end fraction.

Pembahasan

Vektor AB dapat ditentukan dengan cara berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards arrow on top end cell equals cell b with rightwards arrow on top minus a with rightwards arrow on top end cell row cell space space space space space end cell equals cell open parentheses table row 0 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses minus open parentheses table row 2 row cell negative 1 end cell row 0 end table close parentheses end cell row cell space space space space space end cell equals cell open parentheses table row cell negative 2 end cell row 0 row cell negative 1 end cell end table close parentheses end cell end table 

Panjang vektor AB adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A B with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 2 close parentheses squared plus 0 squared plus open parentheses negative 1 close parentheses squared end root end cell row blank equals cell square root of 4 plus 0 plus 1 end root end cell row blank equals cell square root of 5 end cell end table 

Vektor AC dapat ditentukan dengan cara berikut.

stack A C with rightwards arrow on top equals c with rightwards arrow on top minus a with rightwards arrow on top space space space space space equals open parentheses table row 1 row 1 row cell negative 2 end cell end table close parentheses minus open parentheses table row 2 row cell negative 1 end cell row 0 end table close parentheses space space space space space equals open parentheses table row cell negative 1 end cell row 2 row cell negative 2 end cell end table close parentheses 

Panjang vektor AC adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar stack A C with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative 1 close parentheses squared plus 2 squared plus open parentheses negative 2 close parentheses squared end root end cell row blank equals cell square root of 1 plus 4 plus 4 end root end cell row blank equals 3 end table 

Nilai cos antara sisi AB dan AC ditentukan dengan cara berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta end cell equals cell fraction numerator stack A B space with rightwards arrow on top times stack A C with rightwards arrow on top over denominator open vertical bar stack A B space with rightwards arrow on top close vertical bar times open vertical bar stack A C with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator open parentheses table row cell negative 2 end cell row 0 row cell negative 1 end cell end table close parentheses open parentheses table row cell negative 1 end cell row 2 row cell negative 2 end cell end table close parentheses over denominator square root of 5 times 3 end fraction end cell row blank equals cell fraction numerator 2 plus 0 plus 2 over denominator 3 square root of 5 end fraction end cell row blank equals cell fraction numerator 4 over denominator 3 square root of 5 end fraction times fraction numerator square root of 5 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 4 square root of 5 over denominator 15 end fraction end cell end table 

Dengan demikian, nilai Cosinus sudut antara sisi AB dan AC adalah  fraction numerator 4 square root of 5 over denominator 15 end fraction.

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