Pertanyaan

Diketahui A ( 1 , 4 , 4 ) , B ( 0 , 2 , 3 ) , dan C ( 1 , 0 , 2 ) . Hitunglah besar sudut-sudut segitiga ABC!

Diketahui $A (1, 4, 4), B (0, 2, 3), dan C (1, 0, 2)$ . Hitunglah besar sudut-sudut segitiga ABC!

P. Afrisno

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

diperoleh besar sudut segitiga ABC adalah .

diperoleh besar sudut segitiga ABC adalah 24 degree comma space 132 degree comma space dan space 24 degree

Pembahasan

Besar sudut segitiga tersebut dapat ditentukan dengan menggunakan aturan perkalian titik (dot product) vektor untuk menentukan besar sudut antara dua vektor. Jika adalah sudut yang terletak di antara dua vektor , maka diperoleh hubungan berikut. Diketahui: Misalkan: Menentukan besar sudut Menentukan besar sudut Menentukan besar sudut Karena jumlah besar sudut dalam segitiga adalah maka dapat ditentukan besar sudut sebagai berikut. Dengan demikian, diperoleh besar sudut segitiga ABC adalah .

Besar sudut segitiga tersebut dapat ditentukan dengan menggunakan aturan perkalian titik (dot product) vektor untuk menentukan besar sudut antara dua vektor. Jika alpha adalah sudut yang terletak di antara dua vektor a with rightwards arrow on top space dan space b with rightwards arrow on top , maka diperoleh hubungan berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space alpha end cell equals cell fraction numerator a with rightwards arrow on top times b with rightwards arrow on top over denominator open vertical bar a with rightwards arrow on top close vertical bar times open vertical bar b with rightwards arrow on top close vertical bar end fraction end cell row cell open vertical bar a with rightwards arrow on top close vertical bar end cell equals cell panjang space vektor space a with rightwards arrow on top end cell row cell open vertical bar b with rightwards arrow on top close vertical bar end cell equals cell panjang space vektor space b with rightwards arrow on top end cell end table$

Diketahui:

straight A open parentheses 1 comma space 4 comma space 4 close parentheses comma space straight B open parentheses 0 comma space 2 comma space 3 close parentheses comma space dan space straight C open parentheses 1 comma space 0 comma space 2 close parentheses

Misalkan:

Menentukan besar sudut

$table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards arrow on top end cell equals cell O B bond OA end cell row blank equals cell open parentheses 0 comma space 2 comma space 3 close parentheses minus sign open parentheses 1 comma space 4 comma space 4 close parentheses end cell row blank equals cell open parentheses 0 minus sign 1 comma space 2 minus sign 4 comma space 3 minus sign 4 close parentheses end cell row blank equals cell open parentheses negative sign 1 comma space minus sign 2 comma space minus sign 1 close parentheses end cell row cell open vertical bar AB with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses negative sign 1 close parentheses squared plus open parentheses negative sign 2 close parentheses squared plus open parentheses negative sign 1 close parentheses squared end root end cell row blank equals cell square root of 1 plus 4 plus 1 end root end cell row blank equals cell square root of 6 end cell row cell AC with rightwards arrow on top end cell equals cell O C bond OA end cell row blank equals cell open parentheses 1 comma space 0 comma space 2 close parentheses minus sign open parentheses 1 comma space 4 comma space 4 close parentheses end cell row blank equals cell open parentheses 1 minus sign 1 comma space 0 minus sign 4 comma space 2 minus sign 4 close parentheses end cell row blank equals cell open parentheses 0 comma space minus sign 4 comma space minus sign 2 close parentheses end cell row cell open vertical bar AC with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses 0 close parentheses squared plus open parentheses negative sign 4 close parentheses squared plus open parentheses negative sign 2 close parentheses squared end root end cell row blank equals cell square root of 0 plus 16 plus 4 end root end cell row blank equals cell square root of 20 end cell row blank equals cell 2 square root of 5 end cell row cell cos space italic alpha end cell equals cell fraction numerator AB with rightwards arrow on top middle dot AC with rightwards arrow on top over denominator open vertical bar AB with rightwards arrow on top close vertical bar middle dot open vertical bar AC with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator negative sign 1 middle dot 0 plus open parentheses negative sign 2 close parentheses middle dot open parentheses negative sign 4 close parentheses plus open parentheses negative sign 1 close parentheses middle dot open parentheses negative sign 2 close parentheses over denominator square root of 6 middle dot 2 square root of 5 end fraction end cell row blank equals cell fraction numerator 0 plus 8 plus 2 over denominator 2 square root of 30 end fraction end cell row blank equals cell fraction numerator 10 over denominator 2 square root of 30 end fraction end cell row blank equals cell fraction numerator 5 over denominator square root of 30 end fraction end cell row blank equals cell 1 over 6 square root of 30 end cell row italic alpha equals cell cos to the power of negative sign 1 end exponent space open parentheses fraction numerator square root of 30 over denominator 6 end fraction close parentheses end cell row blank almost equal to cell 24 degree end cell end table$

Menentukan besar sudut bold italic beta

$table attributes columnalign right center left columnspacing 0px end attributes row cell BA with rightwards arrow on top end cell equals cell OA minus OB end cell row blank equals cell open parentheses 1 comma space 4 comma space 4 close parentheses minus open parentheses 0 comma space 2 comma space 3 close parentheses end cell row blank equals cell open parentheses 1 minus 0 comma space 4 minus 2 comma space 4 minus 3 close parentheses end cell row blank equals cell open parentheses 1 comma space 2 comma space 1 close parentheses end cell row cell open vertical bar BA with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses 1 close parentheses squared plus open parentheses 2 close parentheses squared plus open parentheses 1 close parentheses squared end root end cell row blank equals cell square root of 1 plus 4 plus 1 end root end cell row blank equals cell square root of 6 end cell row cell BC with rightwards arrow on top end cell equals cell OC minus OB end cell row blank equals cell open parentheses 1 comma space 0 comma space 2 close parentheses minus open parentheses 0 comma space 2 comma space 3 close parentheses end cell row blank equals cell open parentheses 1 minus 0 comma space 0 minus 2 comma space 2 minus 3 close parentheses end cell row blank equals cell open parentheses 1 comma space minus 2 comma space minus 1 close parentheses end cell row cell open vertical bar BC with rightwards arrow on top close vertical bar end cell equals cell square root of open parentheses 1 close parentheses squared plus open parentheses negative 2 close parentheses squared plus open parentheses negative 1 close parentheses squared end root end cell row blank equals cell square root of 1 plus 4 plus 1 end root end cell row blank equals cell square root of 6 end cell row cell cos space beta 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 times open vertical bar BC with rightwards arrow on top close vertical bar end fraction end cell row blank equals cell fraction numerator 1 times 1 plus 2 times open parentheses negative 2 close parentheses plus 1 times open parentheses negative 1 close parentheses over denominator square root of 6 times square root of 6 end fraction end cell row blank equals cell fraction numerator 1 minus 4 minus 1 over denominator 6 end fraction end cell row blank equals cell fraction numerator negative 4 over denominator 6 end fraction end cell row blank equals cell negative 2 over 3 end cell row beta equals cell cos to the power of negative 1 end exponent space open parentheses negative 2 over 3 close parentheses end cell row blank almost equal to cell 132 degree end cell end table$

Menentukan besar sudut

Karena jumlah besar sudut dalam segitiga adalah 180 degree maka dapat ditentukan besar sudut theta sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row theta equals cell 180 degree minus left parenthesis alpha plus beta right parenthesis end cell row blank equals cell 180 degree minus open parentheses 24 plus 132 close parentheses degree end cell row blank equals cell 180 degree minus 156 degree end cell row blank equals cell 24 degree end cell end table

Dengan demikian, diperoleh besar sudut segitiga ABC adalah 24 degree comma space 132 degree comma space dan space 24 degree .

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