Sebagaimana terlihat pada gambar, ambil, ,, ,, , dan, sebagai titik-titik sudut suatu tetrahedron (bidang empat) dan, ,, ,, , dan, berturut-turut adalah luas bidang yang terletak di hadapannya. Tunjukan bahwa, .

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PERTANYAAN :

Perhatikan gambar berikut.

Sebagaimana terlihat pada gambar, ambil straight Pstraight Qstraight R, dan straight O sebagai titik-titik sudut suatu tetrahedron (bidang empat) dan straight Astraight Bstraight C, dan straight D berturut-turut adalah luas bidang yang terletak di hadapannya. Tunjukan bahwa straight A squared plus straight B squared plus straight C squared equals straight D squared.

Table of Contents

Jawaban

terbukti bahwa straight A squared plus straight B squared plus straight C squared equals straight D squared.


Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah  straight A squared plus straight B squared plus straight C squared equals straight D squared terbukti benar.

Ingat!

  • Rumus untuk menentukan luas segitiga adalah sebagai berikut:

straight L equals 1 half cross times alas cross times tinggi

  • Jika koordinat titik straight A open parentheses x subscript 1 comma space y subscript 1 close parentheses dan straight B open parentheses x subscript 2 comma space y subscript 2 close parentheses maka dapat ditetapkan:

stack A B with rightwards arrow on top equals B with rightwards arrow on top minus A with rightwards arrow on top equals open parentheses table row cell x subscript 2 minus x subscript 1 end cell row cell y subscript 2 minus y subscript 1 end cell end table close parentheses 

  • Rumus perkalian titik skalar antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah sebagai berikut:

a with rightwards arrow on top space. space b with rightwards arrow on top equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space cos space theta 

  • Rumus perkalian silang vektor antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top adalah sebagai berikut:

open vertical bar a with rightwards arrow on top space cross times space b with rightwards arrow on top close vertical bar space equals open vertical bar a with rightwards arrow on top close vertical bar open vertical bar b with rightwards arrow on top close vertical bar space sin space theta

  • Rumus perkalian dua vektor jika diketahui a with rightwards arrow on top equals open parentheses table row cell x subscript 1 end cell row cell y subscript 1 end cell end table close parentheses dan vektor b with rightwards arrow on top equals open parentheses table row cell x subscript 2 end cell row cell y subscript 2 end cell end table close parentheses maka 

    a with rightwards arrow on top times b with rightwards arrow on top equals x subscript 1 x subscript 2 plus y subscript 1 y subscript 2

  • Rumus identitas trigonometri adalah sebagai berikut:

sin squared space theta plus cos squared space theta equals 1

Perhatikan gambar pada soal!

Diketahui:

Titik straight P open parentheses p comma space 0 comma space 0 close parentheses

Titik straight Q open parentheses 0 comma space q comma space 0 close parentheses

Titik straight R open parentheses 0 comma space 0 comma space r close parentheses

Dengan menggunakan rumus luas segitiga maka straight A squared plus straight B squared plus straight C squared adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight A squared plus straight B squared plus straight C squared end cell equals cell straight L subscript triangle straight A end subscript plus straight L subscript triangle straight B end subscript plus straight L subscript triangle straight C end subscript end cell row blank equals cell open parentheses fraction numerator q space. space r over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator p space. space r over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator q space. space p over denominator 2 end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator q r over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator p r over denominator 2 end fraction close parentheses squared plus open parentheses fraction numerator p q over denominator 2 end fraction close parentheses squared end cell row blank equals cell fraction numerator q squared r squared over denominator 4 end fraction plus fraction numerator p squared r squared over denominator 4 end fraction plus fraction numerator p squared q squared over denominator 4 end fraction end cell row blank equals cell 1 fourth open parentheses q squared r squared plus p squared r squared plus p squared q squared close parentheses end cell end table 

Akan dibuktikan straight A squared plus straight B squared plus straight C squared equals straight D squared

Dengan menggunakan rumus perkalian titik skalar antara vektor a with rightwards arrow on top dan vektor b with rightwards arrow on top maka

table attributes columnalign right center left columnspacing 0px end attributes row cell stack R Q with rightwards arrow on top space. space stack R P with rightwards arrow on top end cell equals cell open vertical bar stack R Q with rightwards arrow on top close vertical bar open vertical bar stack R P with rightwards arrow on top close vertical bar space cos space theta end cell row cell open parentheses Q with rightwards arrow on top minus R with rightwards arrow on top close parentheses space. space open parentheses P with rightwards arrow on top minus R with rightwards arrow on top close parentheses end cell equals cell open vertical bar open parentheses Q with rightwards arrow on top minus R with rightwards arrow on top close parentheses close vertical bar open vertical bar open parentheses P with rightwards arrow on top minus R with rightwards arrow on top close parentheses close vertical bar space cos space theta end cell row cell open parentheses table row 0 row q row cell negative r end cell end table close parentheses space. space open parentheses table row p row 0 row cell negative r end cell end table close parentheses end cell equals cell open parentheses square root of q squared plus open parentheses negative r close parentheses squared end root close parentheses open parentheses square root of p squared plus open parentheses negative r close parentheses squared end root close parentheses space cos space theta end cell row cell 0 plus 0 plus open parentheses negative r close parentheses cross times open parentheses negative r close parentheses end cell equals cell open parentheses square root of q squared plus r squared end root close parentheses open parentheses square root of p squared plus r squared end root close parentheses space cos space theta end cell row cell r squared end cell equals cell open parentheses square root of q squared plus r squared end root close parentheses open parentheses square root of p squared plus r squared end root close parentheses space cos space theta end cell row blank left right double arrow cell cos space theta equals fraction numerator r squared over denominator open parentheses square root of q squared plus r squared end root close parentheses open parentheses square root of p squared plus r squared end root close parentheses end fraction end cell end table 

Pada soal, straight D adalah gambar segitiga PQR, sesuai dengan ketentuan vektor maka luas segitiga PQR adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight L subscript triangle PQR end subscript end cell equals cell fraction numerator open vertical bar R Q cross times R P close vertical bar over denominator 2 end fraction space end cell row straight D equals cell fraction numerator open vertical bar stack R Q with rightwards arrow on top close vertical bar space. space open vertical bar stack R P with rightwards arrow on top close vertical bar space sin space theta over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses square root of q squared plus r squared end root close parentheses space open parentheses square root of p squared plus r squared end root close parentheses space sin space theta over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses square root of q squared plus r squared end root close parentheses space open parentheses square root of p squared plus r squared end root close parentheses open parentheses square root of space 1 minus cos squared space theta end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses square root of q squared plus r squared end root close parentheses space open parentheses square root of p squared plus r squared end root close parentheses open parentheses square root of space 1 minus open parentheses fraction numerator r squared over denominator open parentheses square root of q squared plus r squared end root close parentheses open parentheses square root of p squared plus r squared end root close parentheses end fraction close parentheses squared end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses square root of q squared plus r squared end root close parentheses space open parentheses square root of p squared plus r squared end root close parentheses open parentheses square root of space 1 minus fraction numerator r to the power of 4 over denominator open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses end fraction end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses square root of q squared plus r squared end root close parentheses space open parentheses square root of p squared plus r squared end root close parentheses open parentheses square root of fraction numerator open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses minus r to the power of 4 over denominator open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses end fraction end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator square root of open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses end root space open parentheses fraction numerator square root of open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses minus r to the power of 4 end root over denominator square root of open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses end root end fraction close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator square root of open parentheses q squared plus r squared close parentheses open parentheses p squared plus r squared close parentheses minus r to the power of 4 end root over denominator 2 end fraction end cell row blank equals cell fraction numerator square root of p squared q squared plus q squared r squared plus p squared r squared plus r to the power of 4 minus r to the power of 4 end root over denominator 2 end fraction end cell row blank equals cell fraction numerator square root of q squared r squared plus p squared r squared plus p squared q squared end root over denominator 2 end fraction end cell row blank left right double arrow cell straight D squared equals open parentheses fraction numerator square root of q squared r squared plus p squared r squared plus p squared q squared end root over denominator 2 end fraction close parentheses squared end cell row blank equals cell fraction numerator q squared r squared plus p squared r squared plus p squared q squared over denominator 4 end fraction end cell row blank equals cell 1 fourth open parentheses q squared r squared plus p squared r squared plus p squared q squared close parentheses end cell end table  

Dengan demikian, terbukti bahwa straight A squared plus straight B squared plus straight C squared equals straight D squared.


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