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Perhatikan gambar berikut! Limas segitiga ABCD dengan AD tegak lurus alas. Sudut antara bidang BCD dan BCA adalah , maka berapakah nilai ?

Pertanyaan

Perhatikan gambar berikut!

Limas segitiga ABCD dengan AD tegak lurus alas. Sudut antara bidang BCD dan BCA adalah alpha, maka berapakah nilai alpha?

Pembahasan Soal:

Terdapat 2 kemungkinan yakni segitiga BCD siku-siku dan sama sisi.

Kemungkinan 1

Cari panjang DO

table attributes columnalign right center left columnspacing 0px end attributes row BC equals cell square root of DC squared plus DB squared end root end cell row blank equals cell square root of 5 squared plus 5 squared end root end cell row blank equals cell square root of 2 cross times 5 squared end root end cell row blank equals cell 5 square root of 2 end cell row blank blank blank row DO equals cell square root of DB squared minus BO squared end root end cell row blank equals cell square root of 5 squared minus open parentheses fraction numerator 5 square root of 2 over denominator 2 end fraction close parentheses squared end root end cell row blank equals cell square root of 5 squared cross times 1 minus 5 squared cross times 1 half end root end cell row blank equals cell square root of 5 squared open parentheses 1 minus 1 half close parentheses end root end cell row blank equals cell fraction numerator 5 over denominator square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row blank equals cell fraction numerator 5 square root of 2 over denominator 2 end fraction end cell end table

Cari nilai tan sudut AOD menggunakan perbandingan trigonometri, kemudian cari besar sudutnya

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space straight a end cell equals cell fraction numerator panjang space sisi space depan over denominator panjang space sisi space samping end fraction end cell row blank equals cell fraction numerator 4 over denominator begin display style fraction numerator 5 square root of 2 over denominator 2 end fraction end style end fraction end cell row blank equals cell 4 cross times fraction numerator 2 over denominator 5 square root of 2 end fraction end cell row blank equals cell fraction numerator 8 over denominator 5 square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row blank equals cell fraction numerator 4 square root of 2 over denominator 5 end fraction end cell row blank blank blank row cell tan space open parentheses angle AOD close parentheses end cell equals cell fraction numerator 4 square root of 2 over denominator 5 end fraction end cell row cell angle AOD end cell equals cell tan to the power of negative 1 end exponent open parentheses fraction numerator 4 square root of 2 over denominator 5 end fraction close parentheses end cell row cell angle AOD end cell equals cell 48 comma 53 degree end cell row blank blank blank end table

Jadi sudut antara bidang BCD dan BCA adalah 48 comma 53 degree

 

Kemungkinan 2

Cari panjang DO

table attributes columnalign right center left columnspacing 0px end attributes row cell B O end cell equals cell 1 half B C end cell row blank equals cell 1 half cross times 5 end cell row blank equals cell 2 comma 5 end cell row blank blank blank row DO equals cell square root of DB squared minus BO squared end root end cell row blank equals cell square root of 5 squared minus open parentheses 2 comma 5 close parentheses squared end root end cell row blank equals cell square root of 25 minus 6 comma 25 end root end cell row blank equals cell square root of 18 comma 75 end root end cell row blank equals cell square root of 1875 over 100 end root end cell row blank equals cell square root of fraction numerator 625 cross times 3 over denominator 100 end fraction end root end cell row blank equals cell fraction numerator 25 square root of 3 over denominator 10 end fraction end cell end table

Cari nilai tan sudut AOD menggunakan perbandingan trigonometri, kemudian cari besar sudutnya

table attributes columnalign right center left columnspacing 0px end attributes row cell tan space straight a end cell equals cell fraction numerator panjang space sisi space depan over denominator panjang space sisi space samping end fraction end cell row blank equals cell fraction numerator 4 over denominator begin display style fraction numerator 25 square root of 3 over denominator 10 end fraction end style end fraction end cell row blank equals cell 4 cross times fraction numerator 10 over denominator 25 square root of 3 end fraction end cell row blank equals cell fraction numerator 40 over denominator 25 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 blank equals cell fraction numerator 40 square root of 3 over denominator 25 cross times 3 end fraction end cell row blank equals cell fraction numerator 8 square root of 3 over denominator 15 end fraction end cell row blank blank blank row cell tan space open parentheses angle AOD close parentheses end cell equals cell fraction numerator 8 square root of 3 over denominator 15 end fraction end cell row cell angle AOD end cell equals cell tan to the power of negative 1 end exponent open parentheses fraction numerator 8 square root of 3 over denominator 15 end fraction close parentheses end cell row cell angle AOD end cell equals cell 42 comma 73 degree end cell row blank blank blank end table

Jadi sudut antara bidang BCD dan BCA adalah 42 comma 73 degree

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 20 Juli 2021

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