Roboguru

Jika pada segitiga siku-siku  panjang , sudut  dan sudut . , , ,  dan seterusnya. Panjang

Pertanyaan

Jika pada segitiga siku-siku begin mathsize 14px style text ABC end text end style panjang Error converting from MathML to accessible text., sudut Error converting from MathML to accessible text. dan sudut Error converting from MathML to accessible text.begin mathsize 14px style text CC end text subscript 1 text ⊥AB end text end stylebegin mathsize 14px style text C end text subscript 1 text C end text subscript 2 text ⊥BC end text end stylebegin mathsize 14px style text C end text subscript 2 text C end text subscript 3 text ⊥AB end text end stylebegin mathsize 14px style text C end text subscript 3 text C end text subscript 4 text ⊥BC end text end style dan seterusnya. Panjang begin mathsize 14px style AC plus CC subscript 1 plus straight C subscript 1 straight C subscript 2 plus straight C subscript 2 straight C subscript 3 plus straight C subscript 3 straight C subscript 4 plus horizontal ellipsis equals horizontal ellipsis end style 

  1. begin mathsize 14px style fraction numerator x square root of 3 over denominator 2 plus square root of 3 end fraction end style 

  2.  begin mathsize 14px style 2 x square root of 3 end style 

  3. begin mathsize 14px style 2 x open parentheses 2 plus square root of 3 close parentheses end style 

  4. begin mathsize 14px style fraction numerator 2 x over denominator 2 plus square root of 3 end fraction end style 

  5. begin mathsize 14px style x open parentheses 3 plus 2 square root of 3 close parentheses end style 

Pembahasan Soal:

Pada gambar terdapat segitiga siku-siku begin mathsize 14px style text ABC end text end style dengan sudut Error converting from MathML to accessible text.. Segitiga-segitiga kecil di dalamnya saling berhimpit dan beberapa sisinya tegak lurus, sehingga sudut-sudutnya saling memengaruhi seperti gambar berikut,

Diketahui panjang begin mathsize 14px style AC end style adalah begin mathsize 14px style x end style dan besar sudut Error converting from MathML to accessible text.. Sisi begin mathsize 14px style CC subscript 1 end style tegak lurus dengan begin mathsize 14px style text AB end text end style, sehingga panjang begin mathsize 14px style CC subscript 1 end style di depan sudut begin mathsize 14px style 60 degree end style dapat dihitung mengunakan perbandingan sisi dengan trigonometri.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A end cell equals cell CC subscript 1 over AC end cell row cell sin space 60 degree end cell equals cell CC subscript 1 over x end cell row cell fraction numerator square root of 3 over denominator 2 end fraction end cell equals cell CC subscript 1 over x end cell row cell CC subscript 1 end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction x end cell end table end style 

Sisi begin mathsize 14px style C subscript 1 C subscript 2 end style tegak lurus dengan begin mathsize 14px style B C end style sehingga panjang begin mathsize 14px style C subscript 1 C subscript 2 end style di depan sudut begin mathsize 14px style 60 degree end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 60 degree end cell equals cell fraction numerator straight C subscript 1 straight C subscript 2 over denominator straight C subscript 1 straight C end fraction end cell row cell fraction numerator square root of 3 over denominator 2 end fraction end cell equals cell fraction numerator straight C subscript 1 straight C subscript 2 over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style x end fraction end cell row cell straight C subscript 1 straight C subscript 2 end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction times fraction numerator square root of 3 over denominator 2 end fraction x end cell row cell straight C subscript 1 straight C subscript 2 end cell equals cell 3 over 4 x end cell end table end style 

Sisi begin mathsize 14px style C subscript 2 C subscript 3 end style tegak lurus dengan begin mathsize 14px style AB end style maka panjang begin mathsize 14px style C subscript 2 C subscript 3 end style di depan sudut begin mathsize 14px style 60 degree end style adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 60 degree end cell equals cell fraction numerator straight C subscript 2 straight C subscript 3 over denominator straight C subscript 1 straight C subscript 2 end fraction end cell row cell fraction numerator square root of 3 over denominator 2 end fraction end cell equals cell fraction numerator straight C subscript 2 straight C subscript 3 over denominator begin display style 3 over 4 end style x end fraction end cell row cell straight C subscript 2 straight C subscript 3 end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction times 3 over 4 x end cell row cell straight C subscript 2 straight C subscript 3 end cell equals cell 3 over 8 square root of 3 x end cell end table end style 

Diperhatikan bahwa panjang sisi

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell AC comma space CC subscript 1 comma space straight C subscript 1 straight C subscript 2 comma space straight C subscript 2 straight C subscript 3 comma horizontal ellipsis end cell row blank equals cell x comma space fraction numerator square root of 3 over denominator 2 end fraction x comma space 3 over 4 x comma space 3 over 8 square root of 3 x comma horizontal ellipsis end cell end table end style 

membentuk deret geometri tak hingga dengan suku pertama begin mathsize 14px style a equals x end style dan rasio begin mathsize 14px style r equals fraction numerator square root of 3 over denominator 2 end fraction end style.

Dengan demikian hasil penjumlahan sisi begin mathsize 14px style AC plus CC subscript 1 plus straight C subscript 1 straight C subscript 2 plus straight C subscript 2 straight C subscript 3 plus straight C subscript 3 straight C subscript 4 plus horizontal ellipsis equals horizontal ellipsis end style dapat dihitung menggunakan rumus deret geometri tak hingga seperti berikut,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell S subscript infinity end cell equals cell fraction numerator a over denominator 1 minus r end fraction end cell row blank equals cell fraction numerator x over denominator 1 minus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator x over denominator begin display style fraction numerator 2 minus square root of 3 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 2 x over denominator 2 minus square root of 3 end fraction times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction end cell row blank equals cell fraction numerator 2 x open parentheses 2 plus square root of 3 close parentheses over denominator 4 minus 3 end fraction end cell row blank equals cell 2 x left parenthesis 2 plus square root of 3 right parenthesis end cell end table end style  

Dengan demikian, panjang begin mathsize 14px style AC plus CC subscript 1 plus straight C subscript 1 straight C subscript 2 plus straight C subscript 2 straight C subscript 3 plus straight C subscript 3 straight C subscript 4 plus horizontal ellipsis equals 2 x open parentheses 2 plus square root of 3 close parentheses end style.

Jadi, jawaban yang tepat adalah C.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 24 Juli 2021

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