Segitiga ABC merupakan segitiga sama sisi, BC=2CD,AF=b dan DEF tegak lurus dengan AB, berapakah luas FBCE?

Pertanyaan

Segitiga $\style{font-size:14px}{ABC}$ merupakan segitiga sama sisi, $\style{font-size:14px}{BC=2CD,\;AF=b}$ dan $\style{font-size:14px}{DEF}$ tegak lurus dengan $\style{font-size:14px}{AB}$ , berapakah luas $\style{font-size:14px}{FBCE}$ ?

Pembahasan Soal:

Informasi di atas dapat digambar seperti berikut ini:

Misalkan panjang $begin mathsize 14px style C D equals x end style$ .

Segitiga $begin mathsize 14px style B D F end style$ sebangun dengan segitiga $begin mathsize 14px style A E F end style$ .

Tentukan panjang $begin mathsize 14px style A E end style$ , dimana sudut-sudut segitiga sama sisi-sisi masing- masing adalah $begin mathsize 14px style 60 degree end style$ .

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 60 degree end cell equals cell fraction numerator A F over denominator A E end fraction end cell row cell 1 half end cell equals cell fraction numerator b over denominator A E end fraction end cell row cell A E end cell equals cell 2 b end cell end table end style$

Sedangkan panjang $begin mathsize 14px style E F end style$ adalah:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 60 degree end cell equals cell fraction numerator E F over denominator A E end fraction end cell row cell 1 half square root of 3 end cell equals cell fraction numerator E F over denominator 2 b end fraction end cell row cell 2 E F end cell equals cell 2 b square root of 3 end cell row cell E F end cell equals cell b square root of 3 end cell end table end style$

Maka dengan menggunakan konsep kesebangunan, panjang $begin mathsize 14px style B F end style$ adalah:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator A F over denominator B F end fraction end cell equals cell fraction numerator A E over denominator B D end fraction end cell row cell fraction numerator b over denominator B F end fraction end cell equals cell fraction numerator 2 b over denominator 3 x end fraction end cell row cell B F times 2 b end cell equals cell 3 x b end cell row cell B F end cell equals cell fraction numerator 3 x b over denominator 2 b end fraction end cell row cell B F end cell equals cell 3 over 2 x end cell end table end style$

Sehingga didapat panjang $begin mathsize 14px style B F equals 3 over 2 x end style$ .

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell A B end cell equals cell B C end cell row cell b plus 3 over 2 x end cell equals cell 2 x end cell row b equals cell 2 x minus 3 over 2 x end cell row b equals cell 1 half x end cell row x equals cell 2 b end cell end table end style$

Karena $begin mathsize 14px style x equals 2 b end style$ , maka:

$begin mathsize 14px style A B equals B C equals A C equals 2 x space space space space space space space space space space space space space space space space space space space space space space space equals 2 left parenthesis 2 b right parenthesis space space space space space space space space space space space space space space space space space space space space space space space equals 4 b end style$

Sehingga, luas $begin mathsize 14px style F B C E end style$ adalah:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell L subscript F B C E end subscript end cell equals cell L subscript A B C end subscript minus L subscript A F E end subscript end cell row blank equals cell 1 half times A B times A C times sin space A minus 1 half times A F times E F end cell row blank equals cell 1 half times 4 b times 4 b times sin space 60 degree minus 1 half times b times b square root of 3 end cell row blank equals cell 8 b squared times 1 half square root of 3 minus 1 half b squared square root of 3 end cell row blank equals cell 4 b squared square root of 3 minus 1 half b squared square root of 3 end cell row blank equals cell 7 over 2 b squared square root of 3 end cell end table end style$

Jadi, luas $begin mathsize 14px style F B C E end style$ adalah $begin mathsize 14px style 7 over 2 b squared square root of 3 end style$ .

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Dwi

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 05 Oktober 2021