Roboguru

Diketahui segitiga lancip dengan panjang , panjang , dan besar . Selisih besar dan adalah....

Pertanyaan

Diketahui segitiga lancip PQR dengan panjang QR equals 4 square root of 3 space cm, panjang PR equals 4 square root of 2 space cm, dan besar angle Q P R equals 60 degree. Selisih besar angle Q P R dan angle P Q R adalah....

  1. 15 degree

  2. 25 degree

  3. 45 degree

  4. 60 degree

Pembahasan Soal:

Pada segitiga sembarang, berlaku aturan sinus sebagai berikut.

fraction numerator a over denominator sin space alpha end fraction equals fraction numerator b over denominator sin space beta end fraction equals fraction numerator c over denominator sin space gamma end fraction

Dengan a comma space b comma space c adalah panjang setiap sisi segitiga tersebut.

Berdasarkan soal di atas, diketahui panjang QR equals 4 square root of 3 space cm, panjang PR equals 4 square root of 2 space cm, dan besar angle Q P R equals 60 degree, maka besar angle P Q R adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator Q R over denominator sin open parentheses angle Q P R close parentheses end fraction end cell equals cell fraction numerator P R over denominator sin space open parentheses angle P Q R close parentheses end fraction end cell row cell fraction numerator 4 square root of 3 space cm over denominator sin open parentheses 60 degree close parentheses end fraction end cell equals cell fraction numerator 4 square root of 2 space cm over denominator sin open parentheses angle P Q R close parentheses end fraction end cell row cell fraction numerator 4 square root of 3 space cm over denominator begin display style 1 half end style square root of 3 end fraction end cell equals cell fraction numerator 4 square root of 2 space cm over denominator sin open parentheses angle P Q R close parentheses end fraction end cell row cell sin open parentheses angle P Q R close parentheses end cell equals cell fraction numerator 4 square root of 2 space cm over denominator 4 square root of 3 space cm end fraction cross times 1 half square root of 3 end cell row blank blank blank row cell sin open parentheses angle P Q R close parentheses end cell equals cell 1 half square root of 2 end cell row cell angle P Q R end cell equals cell sin to the power of negative 1 end exponent open parentheses 1 half square root of 2 close parentheses end cell row cell angle P Q R end cell equals cell 45 degree end cell end table

Sehingga selisih besar angle Q P R dan angle P Q R adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell angle Q P R minus angle P Q R end cell equals cell 60 degree minus 45 degree end cell row blank equals cell 15 degree end cell end table

Dengan demikian, selisih besar angle Q P R dan angle P Q R adalah 15 degree.

Oleh karena itu, jawaban yang benar adalah A.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Rajib

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 12 Agustus 2021

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Pertanyaan yang serupa

Hitunglah nilai !

Pembahasan Soal:

Dengan  menggunakan aturan sinus diperoleh perbandingan

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator PQ over denominator sin space R end fraction end cell equals cell fraction numerator PR over denominator sin space Q end fraction end cell row cell fraction numerator 4 square root of 2 over denominator sin space R end fraction end cell equals cell fraction numerator 8 square root of 3 over denominator sin space 45 degree end fraction end cell row cell fraction numerator 4 square root of 2 over denominator sin space R end fraction end cell equals cell fraction numerator 8 square root of 3 over denominator begin display style 1 half end style square root of 2 end fraction end cell row cell sin space R end cell equals cell fraction numerator 4 square root of 2 times begin display style 1 half end style square root of 2 over denominator 8 square root of 3 end fraction end cell row cell sin space R end cell equals cell fraction numerator 4 over denominator 8 square root of 3 end fraction end cell row cell sin space R end cell equals cell fraction numerator 4 over denominator 8 square root of 3 end fraction cross times fraction numerator square root of 3 over denominator square root of 3 end fraction end cell row cell sin space R end cell equals cell fraction numerator 4 square root of 3 over denominator 8 times 3 end fraction end cell row cell sin space R end cell equals cell fraction numerator square root of 3 over denominator 6 end fraction end cell end table

Dengan menggunakan definisi sinus didapat perbandingan sisi pada segitiga siku-siku sebagai berikut

space sin space R equals depan over miring equals fraction numerator square root of 3 over denominator 6 end fraction

Sehingga dapat gambar ilustrasi sebagai berikut

Dengan menggunakan rumus Phytagoras diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell square root of 6 squared minus open parentheses square root of 3 close parentheses squared end root end cell row blank equals cell square root of 36 minus 3 end root end cell row blank equals cell square root of 33 end cell end table

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space R end cell equals cell depan over samping end cell row blank equals cell fraction numerator square root of 3 over denominator straight x end fraction end cell row blank equals cell fraction numerator square root of 3 over denominator square root of 33 end fraction end cell row blank equals cell fraction numerator square root of 1 over denominator square root of 11 end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of 11 end fraction cross times fraction numerator square root of 11 over denominator square root of 11 end fraction end cell row blank equals cell 1 over 11 square root of 11 end cell end table

Dengan demikian nilai tan space R adalah 1 over 11 square root of 11.

Roboguru

Diketahui segitiga ABC, panjang  dan besar . Tentukan besar .

Pembahasan Soal:

Tentukan besar angle ACB dengan aturan sinus.

angle BAC menghadap sisi BC, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator BC over denominator sin space BAC end fraction end cell equals cell fraction numerator AC over denominator sin space ABC end fraction end cell row cell fraction numerator 4 square root of 2 over denominator sin space 30 degree end fraction end cell equals cell fraction numerator 8 over denominator sin space ABC end fraction end cell row cell fraction numerator 4 square root of 2 over denominator begin display style 1 half end style end fraction end cell equals cell fraction numerator 8 over denominator sin space ABC end fraction end cell row cell sin space ABC end cell equals cell fraction numerator 8 cross times begin display style 1 half end style over denominator 4 square root of 2 end fraction end cell row cell sin space ABC end cell equals cell fraction numerator 1 over denominator square root of 2 end fraction end cell row cell sin space ABC end cell equals cell 1 half square root of 2 end cell row ABC equals cell 45 degree end cell end table

Karena angle BAC equals 30 degree dan angle ABC equals 45 degree maka

 table attributes columnalign right center left columnspacing 0px end attributes row cell angle ACB end cell equals cell 180 degree minus open parentheses 30 degree plus 45 degree close parentheses end cell row blank equals cell 180 degree minus 75 degree end cell row blank equals cell 105 degree end cell end table

Jadi, besar angle ACB adalah 105 degree.

Roboguru

Pada gambar di samping, KLM dengan , , dan KM = 12 cm. Hitunglah unsur-unsur yang belam diketahui pada KLM.

Pembahasan Soal:

Unsur-unsur yang belum diketahui pada segitiga KLM adalah angle K, KL dan LM.

table attributes columnalign right center left columnspacing 0px end attributes row cell angle K end cell equals cell 180 degree minus angle M minus angle L end cell row blank equals cell 180 degree minus 115 degree minus 30 degree end cell row blank equals cell 35 degree end cell end table

 

Untuk menghitung sisi KL dan LM kita gunakan aturan sinus.

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator KM over denominator sin space 30 degree end fraction end cell equals cell fraction numerator LM over denominator sin space 35 degree end fraction equals fraction numerator KL over denominator sin space 115 degree end fraction end cell row blank blank blank row cell fraction numerator KM over denominator sin space 30 degree end fraction end cell equals cell fraction numerator LM over denominator sin space 35 degree end fraction end cell row cell fraction numerator 12 over denominator 0 comma 5 end fraction end cell equals cell fraction numerator LM over denominator 0 comma 57 end fraction end cell row LM equals cell fraction numerator 0 comma 57 cross times 12 over denominator 0 comma 5 end fraction end cell row blank equals cell 13 comma 68 space cm end cell row blank blank blank row cell fraction numerator KM over denominator sin space 30 degree end fraction end cell equals cell fraction numerator KL over denominator sin space 115 degree end fraction end cell row cell fraction numerator 12 over denominator 0 comma 5 end fraction end cell equals cell fraction numerator KL over denominator 0 comma 9 end fraction end cell row KL equals cell fraction numerator 12 cross times 0 comma 9 over denominator 0 comma 5 end fraction end cell row blank equals cell 21 comma 6 space cm end cell end table

Jadi, besar angle K adalah 35 degree, panjang LM 13,68 cm dan panjang KL 21,6 cm.

Roboguru

Pada sebuah segitiga ABC, diketahui bahwa sudut  dan sudut . Jika panjang sisi , maka panjang sisi  adalah ...

Pembahasan Soal:

Dengan menggunakan aturan sinus perbandingan panjang sisi segitiga, diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator straight a over denominator sin space straight A end fraction end cell equals cell fraction numerator straight c over denominator sin space straight C end fraction end cell row cell fraction numerator straight a over denominator sin space 30 degree end fraction end cell equals cell fraction numerator straight b over denominator sin space open parentheses 180 degree minus 30 degree minus 60 degree close parentheses end fraction end cell row cell fraction numerator straight a over denominator sin space 30 degree end fraction end cell equals cell fraction numerator straight c over denominator sin space 90 degree end fraction end cell row cell fraction numerator straight a over denominator begin display style 1 half end style end fraction end cell equals cell straight c over 1 end cell row cell 2 straight a end cell equals straight c row straight c equals cell 2 straight a end cell end table 

Karena straight a plus straight c equals 9 space cm, maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell straight a plus straight c end cell equals 9 row cell straight a plus 2 straight a end cell equals 9 row cell 3 straight a end cell equals 9 row straight a equals cell 3 space cm end cell row blank blank blank row straight c equals cell 2 straight a end cell row blank equals cell 2 times 3 end cell row blank equals cell 6 space cm end cell end table 

Selanjutnya dengan menggunakan aturan cosinus perbandingan panjang sisi segitiga, diperoleh 

table attributes columnalign right center left columnspacing 0px end attributes row straight b equals cell square root of straight a squared plus straight b squared minus 2 times straight a times straight b times cos space straight B end root end cell row blank equals cell square root of 3 squared plus 6 squared minus 2 times 3 times 6 times cos space 60 degree end root end cell row blank equals cell square root of 9 plus 36 minus 2 times 3 times 6 times 1 half end root end cell row blank equals cell square root of 9 plus 36 minus 18 end root end cell row blank equals cell square root of 27 end cell row blank equals cell 3 square root of 3 space cm end cell end table 

Jadi, panjang sisi straight b adalah 3 square root of 3 space cm

Dengan demikian, jawaban yang tepat adalah A.

Roboguru

Diketahui  dengan panjang sisi , , dan . Besar

Pembahasan Soal:

Diketahui triangle ABC dengan sisi BC di depan sudut BAC. Panjang B C equals 3 square root of 2 space cm dan besar angle BAC equals 30 degree. Panjang sisi AC equals 6 space cm berada di depan sudut ABC. Dengan aturan sinus, dapat dihitung perbandingan berikut :

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin angle BAC over denominator BC end fraction end cell equals cell fraction numerator sin angle ABC over denominator AC end fraction end cell row cell fraction numerator sin space 30 degree over denominator 3 square root of 2 end fraction end cell equals cell fraction numerator sin angle ABC over denominator 6 end fraction end cell row cell fraction numerator begin display style 1 half end style over denominator 3 square root of 2 end fraction end cell equals cell fraction numerator sin angle ABC over denominator 6 end fraction end cell row cell sin angle ABC end cell equals cell fraction numerator 6 times begin display style 1 half end style over denominator 3 square root of 2 end fraction end cell row cell sin angle ABC end cell equals cell fraction numerator 3 over denominator 3 square root of 2 end fraction end cell row cell sin angle ABC end cell equals cell fraction numerator 1 over denominator square root of 2 end fraction end cell row cell sin angle ABC end cell equals cell sin space 45 degree end cell row cell angle ABC end cell equals cell 45 degree end cell end table 

Jadi, besar angle ABC adalah 45 degree.

Roboguru

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