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Diketahui segitiga ABC dengan titik-titik sudut A(4,−6,4), B(−2,0,4) dan C(0,2,8). Dengan menggunakan rumus kosinus antara dua vektor, tentukan besar sudut-sudut berikut. b. sudut ABC

Pertanyaan

Diketahui segitiga ABC dengan titik-titik sudut straight A left parenthesis 4 comma negative 6 comma 4 right parenthesis comma space straight B left parenthesis negative 2 comma 0 comma 4 right parenthesis dan straight C left parenthesis 0 comma 2 comma 8 right parenthesis. Dengan menggunakan rumus kosinus antara dua vektor, tentukan besar sudut-sudut berikut.

b. sudut ABC  

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

  besar sudut ABC adalah 90 degree.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold 90 bold degree.

Ingat !

Jika diketahui titik straight A open parentheses table row cell x subscript 1 comma y subscript 1 comma z subscript 1 end cell end table close parentheses dan straight B open parentheses table row cell x subscript 2 comma y subscript 2 comma z subscript 2 end cell end table close parentheses, maka:

  • AB with rightwards arrow on top equals OB with rightwards arrow on top minus OA with rightwards arrow on top equals left parenthesis x subscript 2 minus x subscript 1 right parenthesis comma open parentheses y subscript 2 minus y subscript 1 close parentheses comma open parentheses z subscript 2 minus z subscript 1 close parentheses 
  • open vertical bar a with rightwards arrow on top close vertical bar equals square root of x subscript 1 squared plus y subscript 1 squared plus z subscript 1 squared end root 
  • table attributes columnalign right center left columnspacing 0px end attributes row cell cos theta end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell end table 

Sudut ABC adalah sudut yang terbentuk antara vektor BA with rightwards arrow on top dan vektor BC with rightwards arrow on top.

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space ABC end cell equals cell fraction numerator BA with rightwards arrow on top times BC with rightwards arrow on top over denominator open vertical bar BA with rightwards arrow on top close vertical bar open vertical bar BC with rightwards arrow on top close vertical bar end fraction end cell end table 

  • Vektor BA with rightwards arrow on top

table attributes columnalign right center left columnspacing 0px end attributes row cell BA with rightwards arrow on top end cell equals cell OA with rightwards arrow on top minus OB with rightwards arrow on top end cell row blank equals cell open parentheses table row cell 4 minus left parenthesis negative 2 right parenthesis end cell row cell negative 6 minus 0 end cell row cell 4 minus 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row 6 row cell negative 6 end cell row 0 end table close parentheses space end cell end table 

  • Vektor BC with rightwards arrow on top 

table attributes columnalign right center left columnspacing 0px end attributes row cell BC with rightwards arrow on top end cell equals cell OC with rightwards arrow on top minus OB with rightwards arrow on top end cell row blank equals cell open parentheses table row cell 0 minus left parenthesis negative 2 right parenthesis end cell row cell 2 minus 0 end cell row cell 8 minus 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row 2 row 2 row 4 end table close parentheses end cell end table 

  • Menentukan BA with rightwards arrow on top times BC with rightwards arrow on top 

table attributes columnalign right center left columnspacing 0px end attributes row cell BA with rightwards arrow on top times BC with rightwards arrow on top end cell equals cell open parentheses table row 6 row cell negative 6 end cell row 0 end table close parentheses open parentheses table row 2 row 2 row 4 end table close parentheses end cell row blank equals cell 12 plus left parenthesis negative 12 right parenthesis plus 0 end cell row blank equals 0 end table 

  • Panjang vektor BA with rightwards arrow on top​​​​​​​

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar BA with rightwards arrow on top close vertical bar end cell equals cell square root of 6 squared plus open parentheses negative 6 close parentheses squared plus 0 squared end root end cell row blank equals cell square root of 36 plus 36 plus 0 end root end cell row blank equals cell square root of 36 times 2 end root end cell row blank equals cell 6 square root of 2 end cell end table 

  • Panjang vektor BC with rightwards arrow on top​​​​​​​

table attributes columnalign right center left columnspacing 0px end attributes row cell BC with rightwards arrow on top end cell equals cell square root of 2 squared plus 2 squared plus 4 squared end root end cell row blank equals cell square root of 4 plus 4 plus 16 end root end cell row blank equals cell square root of 24 end cell row blank equals cell square root of 4 times 6 end root end cell row blank equals cell 2 square root of 6 end cell end table 

  • Menentukan  table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos angle ABC end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos angle ABC end cell equals cell fraction numerator BA with rightwards arrow on top times BC with rightwards arrow on top over denominator open vertical bar BA with rightwards arrow on top close vertical bar open vertical bar BC with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 0 over denominator 6 square root of 2 cross times 2 square root of 6 end fraction end cell row blank equals 0 row theta equals cell arc space cos space 0 end cell row theta equals cell 90 degree end cell end table 

Dengan demikian,  besar sudut ABC adalah 90 degree.

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