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Diketahui koordinat dua tempat, yaitu  dan . Hitunglah jarak kedua kota tersebut.

Pertanyaan

Diketahui koordinat dua tempat, yaitu open parentheses 5 comma space 30 degree close parentheses dan open parentheses 8 comma space 60 degree close parentheses. Hitunglah jarak kedua kota tersebut.

Pembahasan Soal:

Misalkan koordinat kartesius titik straight A open parentheses x comma space y close parentheses, dan koordinat polar titik straight A open parentheses r comma space alpha close parentheses, sehingga hubungan kedua titik tersebut:

x equals r times cos space alpha semicolon space dan space y equals r times sin space alpha  

Ingat bahwa:

  •  sin space 30 degree equals 1 half semicolon space cos space 30 degree equals 1 half square root of 3 
  • sin space 60 degree equals 1 half square root of 3 semicolon space cos space 60 degree equals 1 half

Ubah koordinat polar tersebut, menjadi koordinat kartesius. 

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 5 comma space 30 degree close parentheses end cell rightwards arrow cell x equals 5 times cos space 30 degree end cell row blank equals cell 5 times 1 half square root of 3 end cell row blank equals cell 5 over 2 square root of 3 end cell row y equals cell 5 times sin space 30 degree end cell row blank equals cell 5 times 1 half end cell row blank equals cell 5 over 2 end cell end table   

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 8 comma space 60 degree close parentheses end cell rightwards arrow cell x equals 8 times cos space 60 degree end cell row blank equals cell 8 times 1 half end cell row blank equals 4 row y equals cell 8 times sin space 30 degree end cell row blank equals cell 8 times 1 half square root of 3 end cell row blank equals cell 4 square root of 3 end cell end table  

Sehingga diperoleh koordinat kartesius open parentheses 5 over 2 square root of 3 comma space 5 over 2 close parentheses dan open parentheses 4 comma space 4 square root of 3 close parentheses

Dengan menggunakan Pythagoras, kita tentukan jaraknya.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row jarak equals cell square root of open parentheses x subscript 2 minus x subscript 1 close parentheses squared plus open parentheses y subscript 2 minus y subscript 1 close parentheses end root end cell row blank equals cell square root of open parentheses 4 minus 5 over 2 square root of 3 close parentheses squared plus open parentheses 4 square root of 3 minus 5 over 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 16 minus open parentheses 2 times 4 times 5 over 2 square root of 3 close parentheses plus 75 over 4 close parentheses plus open parentheses 48 minus open parentheses 2 times 4 square root of 3 times 5 over 2 close parentheses plus 25 over 4 close parentheses end root end cell row blank equals cell square root of open parentheses 16 minus 20 square root of 3 plus 75 over 4 close parentheses plus open parentheses 48 minus 20 square root of 3 plus 25 over 4 close parentheses end root end cell row blank equals cell square root of 64 plus 100 over 4 minus 40 square root of 3 end root end cell row blank equals cell square root of 64 plus 25 minus 40 square root of 3 end root end cell row blank equals cell square root of 89 minus 40 square root of 3 end root end cell end table end style    

Jadi, jarak kedua kota tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of 89 minus 40 square root of 3 end root close parentheses end cell end table satuan jarak.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 18 Juli 2021

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