Roboguru

Perhatikan gambar berikut. Panjang BC adalah ...

Pertanyaan

Perhatikan gambar berikut.

Panjang BC adalah ...

Pembahasan Soal:

Tentukan terlebih dahulu panjang sisi BD dengan aturan cosinus.

table attributes columnalign right center left columnspacing 0px end attributes row cell BC squared end cell equals cell AB squared plus AD squared minus 2 times AB times AD times cos space straight A end cell row cell BC squared end cell equals cell open parentheses 6 square root of 2 close parentheses squared plus 6 squared minus 2 times 6 square root of 2 times 6 times cos space 45 degree end cell row cell BC squared end cell equals cell 72 plus 36 minus 72 square root of 2 times fraction numerator square root of 2 over denominator 2 end fraction end cell row cell BC squared end cell equals cell 108 minus fraction numerator 72 times down diagonal strike 2 over denominator down diagonal strike 2 end fraction end cell row cell BC squared end cell equals cell 108 minus 72 end cell row cell BC squared end cell equals 36 row BC equals cell plus-or-minus square root of 36 end cell row BC equals cell plus-or-minus 6 end cell row BC equals cell 6 space cm end cell end table

Kemudian dengan menggunakan aturan cosinus kita dapatkan panjang BC.

table attributes columnalign right center left columnspacing 0px end attributes row cell BC squared end cell equals cell BD squared plus CD squared minus 2 times BD times CD times cos space straight D end cell row cell BC squared end cell equals cell 6 squared plus open parentheses 3 square root of 3 close parentheses squared minus 2 times 6 times 3 square root of 3 times cos space 30 degree end cell row cell BC squared end cell equals cell 36 plus 27 minus 36 square root of 3 times fraction numerator square root of 3 over denominator 2 end fraction end cell row cell BC squared end cell equals cell 63 minus fraction numerator 36 times 3 over denominator 2 end fraction end cell row cell BC squared end cell equals cell 63 minus 54 end cell row cell BC squared end cell equals 9 row BC equals cell plus-or-minus square root of 9 end cell row BC equals cell plus-or-minus 3 end cell row BC equals cell 3 space cm end cell end table

Maka, panajng BC adalah 3 space cm.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 07 Juni 2021

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table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript P A end subscript end cell equals cell V subscript p cross times t end cell row blank equals cell 8 space mil divided by jam cross times 2 space jam end cell row blank equals cell 16 space mil end cell end table

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table attributes columnalign right center left columnspacing 0px end attributes row cell Sudut space straight A end cell equals cell Sudut space straight Q minus Sudut space straight P end cell row blank equals cell 90 degree minus 30 degree end cell row blank equals cell 60 degree end cell end table

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table attributes columnalign right center left columnspacing 0px end attributes row cell P Q squared end cell equals cell S subscript P A end subscript squared plus S subscript Q A end subscript squared minus 2 open parentheses S subscript P A end subscript close parentheses open parentheses S subscript Q A end subscript close parentheses cos space 60 degree end cell row cell P Q squared end cell equals cell 16 squared plus 28 squared minus 2 open parentheses 16 close parentheses open parentheses 28 close parentheses 1 half end cell row cell P Q squared end cell equals cell 256 plus 784 minus 448 end cell row cell P Q squared end cell equals 592 row cell P Q end cell equals cell square root of 592 end cell row cell P Q end cell equals cell 4 square root of 37 end cell end table

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table attributes columnalign right center left columnspacing 0px end attributes row cell R Q squared end cell equals cell P Q squared plus P R squared minus 2 P Q times P R space cos space angle P end cell row cell open parentheses 3 square root of 3 close parentheses squared end cell equals cell open parentheses 2 square root of 5 close parentheses squared plus P R squared minus 2 cross times 2 square root of 5 cross times P R cross times cos space 60 degree end cell row 27 equals cell 20 plus P R squared minus 4 square root of 5 P R cross times 1 half end cell row 7 equals cell P R squared minus 2 square root of 5 P R end cell row cell Misal colon x end cell equals cell P R comma space maka end cell row cell straight x squared minus 2 square root of 5 straight x minus 7 end cell equals 0 row cell straight x subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative straight b plus-or-minus square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction end cell row blank equals cell fraction numerator 2 square root of 5 plus-or-minus square root of 20 plus 28 end root over denominator 2 end fraction end cell row blank equals cell square root of 5 plus-or-minus 2 square root of 3 end cell row blank rightwards arrow cell PR equals square root of 5 plus 2 square root of 3 end cell end table

Melalui aturan cosinus, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space angle R end cell equals cell fraction numerator R Q squared plus R P squared minus P Q squared over denominator 2 R Q times R P end fraction end cell row blank equals cell fraction numerator 27 plus open parentheses square root of 5 plus 2 square root of 3 close parentheses squared minus 20 over denominator 2 open parentheses 3 square root of 3 close parentheses open parentheses square root of 5 plus 2 square root of 3 close parentheses end fraction end cell row blank equals cell fraction numerator 7 plus 5 plus 12 plus 4 square root of 15 over denominator 6 square root of 15 plus 36 end fraction end cell row blank equals cell fraction numerator 24 plus 4 square root of 15 over denominator 36 plus 6 square root of 15 end fraction end cell row blank equals cell fraction numerator 4 left parenthesis 6 plus square root of 15 right parenthesis over denominator 6 left parenthesis 6 plus square root of 15 right parenthesis end fraction end cell row blank equals cell 2 over 3 end cell row blank rightwards arrow cell m plus n equals 2 plus 3 end cell row blank equals 5 end table

Oleh karena itu, nilai dari table attributes columnalign right center left columnspacing 0px end attributes row cell m plus n end cell equals 5 end table.

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Diketahui segi enam beraturan dengan panjang sisi .  Gunakan aturan cosinus untuk menghitung panjang EC.

Pembahasan Soal:

Perhatikan gambar berikut

Dari segitiga CDE 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell E C end cell equals cell square root of C D squared plus D E squared minus 2 times C D times D E times cos space 120 to the power of ring operator end root end cell row cell E C end cell equals cell square root of 10 squared plus 10 squared minus 2 times 10 times 10 times cos space left parenthesis 180 to the power of ring operator minus 60 to the power of ring operator right parenthesis end root end cell row cell E C end cell equals cell square root of 100 plus 100 minus 200 times left parenthesis negative cos space 60 to the power of ring operator right parenthesis end root end cell row cell E C end cell equals cell square root of 200 minus 200 times open parentheses negative 1 half close parentheses end root end cell row cell E C end cell equals cell square root of 200 plus 100 end root end cell row cell E C end cell equals cell square root of 300 end cell row cell E C end cell equals cell square root of 100 times 3 end root end cell row cell E C end cell equals cell 10 square root of 3 space cm end cell end table end style

Jadi panjang ECbegin mathsize 14px style 10 square root of 3 space cm end style.

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