Cermati Gambar berikut! Gambar 4.35. Kombinasi segitiga siku-siku Dengan menemukan hubungan antarsudut-sudut dan panjang sisi-sisi pada segitiga siku-siku yang ada pada gambar, hitunglah panjang AD, EC, BC, BD, AB, FB, AE, dan DE!

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Cermati Gambar berikut!


Gambar 4.35. Kombinasi segitiga siku-siku

Dengan menemukan hubungan antarsudut-sudut dan panjang sisi-sisi pada segitiga siku-siku yang ada pada gambar, hitunglah panjang AD, EC, BC, BD, AB, FB, AE, dan DE!

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F. Ayudhita

Master Teacher

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Perhatikan segitiga siku-siku ABD, berlaku:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin 30 degree end cell equals cell fraction numerator B D over denominator A D end fraction end cell row cell 1 half end cell equals cell fraction numerator B D over denominator A D end fraction end cell row blank blank blank end table end style

Jadi, begin mathsize 14px style B D equals x end style dan begin mathsize 14px style A D equals 2 x end style.

Dengan Teorema Pythagoras, diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell A B end cell equals cell square root of A D squared minus B D squared end root end cell row blank equals cell square root of left parenthesis 2 x right parenthesis squared minus x squared end root end cell row blank equals cell square root of 4 x squared minus x squared end root end cell row blank equals cell square root of 3 x squared end root end cell row blank equals cell x square root of 3 end cell row blank blank blank row blank blank blank row blank blank blank end table end style

Perhatikan!

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell angle A D B end cell equals cell 180 degree minus open parentheses angle B A D plus angle A B D close parentheses end cell row blank equals cell 180 degree minus open parentheses 30 degree plus 90 degree close parentheses end cell row blank equals cell 180 degree minus 120 degree end cell row blank equals cell 60 degree end cell end table end style

Pada segitiga ADE,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell angle A D E end cell equals cell 180 degree minus open parentheses angle D A E plus angle A E D close parentheses end cell row blank equals cell 180 degree minus open parentheses open parentheses 30 degree plus 45 degree close parentheses plus 90 degree close parentheses end cell row blank equals cell 180 degree minus 165 degree end cell row blank equals cell 15 degree end cell end table end style

Sehingga begin mathsize 14px style angle B D E equals angle A D B minus angle A D E equals 60 degree minus 15 degree equals 45 degree end style.

Perhatikan segitiga BDF, berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell size 14px sin size 14px angle size 14px B size 14px D size 14px E end cell size 14px equals cell size 14px sin size 14px 45 size 14px degree size 14px equals fraction numerator size 14px B size 14px F over denominator size 14px B size 14px D end fraction end cell row cell fraction numerator square root of size 14px 2 over denominator size 14px 2 end fraction end cell size 14px equals cell fraction numerator size 14px B size 14px F over denominator size 14px x end fraction end cell end table

Diperoleh begin mathsize 14px style B F equals fraction numerator x square root of 2 over denominator 2 end fraction end style.

Dengan Terorema Pythagoras, diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D F end cell equals cell square root of B D squared minus B F squared end root end cell row blank equals cell square root of x squared minus open parentheses fraction numerator x square root of 2 over denominator 2 end fraction close parentheses squared end root end cell row blank equals cell square root of x squared minus fraction numerator 2 x squared over denominator 4 end fraction end root end cell row blank equals cell square root of x squared over 2 end root end cell row blank equals cell fraction numerator x over denominator square root of 2 end fraction end cell row blank equals cell fraction numerator x square root of 2 over denominator 2 end fraction end cell row blank blank blank end table end style

Perhatikan segitiga ABC, berlaku:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin angle B A C end cell equals cell sin 45 degree equals fraction numerator B C over denominator A B end fraction end cell row cell fraction numerator square root of 2 over denominator 2 end fraction end cell equals cell fraction numerator B C over denominator x square root of 3 end fraction end cell row cell B C end cell equals cell fraction numerator x square root of 6 over denominator 2 end fraction end cell row blank blank blank row blank blank blank end table end style

Dengan Teorema Pythagoras, diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell A C end cell equals cell square root of A B squared minus B C squared end root end cell row blank equals cell square root of open parentheses x square root of 3 close parentheses squared minus open parentheses fraction numerator x square root of 6 over denominator 2 end fraction close parentheses squared end root end cell row blank equals cell square root of 3 x squared minus fraction numerator 6 x squared over denominator 4 end fraction end root end cell row blank equals cell square root of fraction numerator 6 x squared over denominator 4 end fraction end root end cell row blank equals cell fraction numerator x square root of 6 over denominator 2 end fraction end cell row blank blank blank row blank blank blank end table end style

Selanjutnya, karena begin mathsize 14px style A C equals A E plus C E end style, dengan begin mathsize 14px style C E equals B F equals fraction numerator x square root of 2 over denominator 2 end fraction end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell A E end cell equals cell A C minus C E end cell row blank equals cell fraction numerator x square root of 6 over denominator 2 end fraction minus fraction numerator x square root of 2 over denominator 2 end fraction end cell row blank equals cell fraction numerator x open parentheses square root of 6 minus square root of 2 close parentheses over denominator 2 end fraction end cell end table end style

Perhatikan begin mathsize 14px style E F equals B C equals fraction numerator x square root of 6 over denominator 2 end fraction end style.

Diperoleh begin mathsize 14px style D E equals D F plus E F equals fraction numerator x square root of 2 over denominator 2 end fraction plus fraction numerator x square root of 6 over denominator 2 end fraction equals fraction numerator x open parentheses square root of 6 plus square root of 2 close parentheses over denominator 2 end fraction end style.

 Jadi, berdasarkan perhitungan tersebut dapat diperoleh begin mathsize 14px style A D equals 2 x comma blank E C equals fraction numerator x square root of 2 over denominator 2 end fraction comma blank B C equals fraction numerator x square root of 6 over denominator 2 end fraction comma blank B D equals x comma blank A B equals x square root of 3 comma blank F B equals fraction numerator x square root of 2 over denominator 2 end fraction comma blank A E equals fraction numerator x open parentheses square root of 6 minus square root of 2 close parentheses over denominator 2 end fraction comma blank dan blank D E equals fraction numerator x open parentheses square root of 6 plus square root of 2 close parentheses over denominator 2 end fraction end style.

 

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