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PQRS adalah segiempat dengan koordinat titik sudut straight P open parentheses 8 comma space minus 2 comma space 6 close parenthesesstraight Q open parentheses 16 comma space 6 comma space minus 2 close parenthesesstraight R open parentheses 0 comma space 8 comma space 8 close parentheses, dan straight S open parentheses negative 8 comma space 0 comma space 16 close parentheses.

a. Tentukan koordinat M, yaitu titik tengah dari PQ.

b. Tentukan koordinat N yang membagi RM with rightwards harpoon with barb upwards on top dalam perbandingan 2 divided by 1.

c. Tunjukkanlah Q, N, dan S segaris. Kemudian, tentukanlah perbandingan N membagi QS with rightwards harpoon with barb upwards on top.space 

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

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Pembahasan

Diketahui koordinat titik sudut straight P open parentheses 8 comma space minus 2 comma space 6 close parenthesesstraight Q open parentheses 16 comma space 6 comma space minus 2 close parenthesesstraight R open parentheses 0 comma space 8 comma space 8 close parentheses, dan straight S open parentheses negative 8 comma space 0 comma space 16 close parentheses. Dapat diperoleh vektor posisi masing-masing titik yaitu

bold italic p equals open parentheses table row 8 row cell negative 2 end cell row 6 end table close parentheses comma space bold italic q equals open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses comma space bold italic r equals open parentheses table row 0 row 8 row 8 end table close parentheses space dan space bold italic s equals open parentheses table row cell negative 8 end cell row 0 row 16 end table close parentheses 

Ingat kembali bahwa untuk titik B terletak pada garis AC dengan bold italic abold italic b, dan bold italic c berturut-turut merupakan vektor posisi titik A, B, dan C, sedemikian sehingga stack A B with rightwards harpoon with barb upwards on top divided by stack B C with rightwards harpoon with barb upwards on top equals m divided by n berlaku

bold italic b equals fraction numerator n bold italic a plus m bold italic c over denominator n plus m end fraction 

a. Akan ditentukan koordinat M, yaitu titik tengah dari PQ. Karena M merupakan titik tengah, berarti M membagi PQ sama besar. Dengan kata lain, stack P M with rightwards harpoon with barb upwards on top divided by stack M Q with rightwards harpoon with barb upwards on top equals 1 divided by 1. Namakan bold italic m sebagai vektor posisi titik M, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row bold italic m equals cell fraction numerator 1 bold italic p plus 1 bold italic q over denominator 1 plus 1 end fraction end cell row blank equals cell fraction numerator 1 open parentheses table row 8 row cell negative 2 end cell row 6 end table close parentheses plus 1 open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses table row cell 1 cross times 8 end cell row cell 1 cross times open parentheses negative 2 close parentheses end cell row cell 1 cross times 6 end cell end table close parentheses plus open parentheses table row cell 1 cross times 16 end cell row cell 1 cross times 6 end cell row cell 1 cross times open parentheses negative 2 close parentheses end cell end table close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses table row 8 row cell negative 2 end cell row 6 end table close parentheses plus open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses table row cell 8 plus 16 end cell row cell negative 2 plus 6 end cell row cell 6 plus open parentheses negative 2 close parentheses end cell end table close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator open parentheses table row 24 row 4 row 4 end table close parentheses over denominator 2 end fraction end cell row blank equals cell open parentheses table row 12 row 2 row 2 end table close parentheses end cell end table 

Artinya diperoleh koordinat titik M yaitu open parentheses 12 comma space 2 comma space 2 close parentheses 

b. Akan ditentukan koordinat titik N. Diketahui bahwa titik N membagi stack R M with rightwards harpoon with barb upwards on top dalam perbandingan 2 divided by 1, berarti stack R N with rightwards harpoon with barb upwards on top divided by stack N M with rightwards harpoon with barb upwards on top equals 2 divided by 1. Namakan bold italic n sebagai vektor posisi titik N, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row bold italic n equals cell fraction numerator 1 bold italic r plus 2 bold italic m over denominator 1 plus 2 end fraction end cell row blank equals cell fraction numerator 1 open parentheses table row 0 row 8 row 8 end table close parentheses plus 2 open parentheses table row 12 row 2 row 2 end table close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses table row cell 1 cross times 0 end cell row cell 1 cross times 8 end cell row cell 1 cross times 8 end cell end table close parentheses plus open parentheses table row cell 2 cross times 12 end cell row cell 2 cross times 2 end cell row cell 2 cross times 2 end cell end table close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses table row 0 row 8 row 8 end table close parentheses plus open parentheses table row 24 row 4 row 4 end table close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses table row cell 0 plus 24 end cell row cell 8 plus 4 end cell row cell 8 plus 4 end cell end table close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses table row 24 row 12 row 12 end table close parentheses over denominator 3 end fraction end cell row blank equals cell open parentheses table row 8 row 4 row 4 end table close parentheses end cell end table 

Artinya diperoleh koordinat titik N yaitu open parentheses 8 comma space 4 comma space 4 close parentheses  

Ingat bahwa tiga buah titik A, B, dan C dikatakan segaris apabila stack A B with rightwards harpoon with barb upwards on top equals k stack B C with rightwards harpoon with barb upwards on top untuk suatu skalar k

c. Akan ditunjukkan bahwa Q, N, dan S segaris. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell stack Q N with rightwards harpoon with barb upwards on top end cell equals cell bold italic n minus bold italic q end cell row blank equals cell open parentheses table row 8 row 4 row 4 end table close parentheses minus open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 minus 16 end cell row cell 4 minus 6 end cell row cell 4 minus left parenthesis negative 2 right parenthesis end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 8 end cell row cell negative 2 end cell row 6 end table close parentheses end cell row blank blank blank row cell stack N S with rightwards harpoon with barb upwards on top end cell equals cell bold italic s minus bold italic n end cell row blank equals cell open parentheses table row cell negative 8 end cell row 0 row 16 end table close parentheses minus open parentheses table row 8 row 4 row 4 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 8 minus 8 end cell row cell 0 minus 4 end cell row cell 16 minus 4 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 16 end cell row cell negative 4 end cell row 12 end table close parentheses end cell end table 

Selanjutnya dapat diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell stack Q N with rightwards harpoon with barb upwards on top end cell equals cell open parentheses table row cell negative 8 end cell row cell negative 2 end cell row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half cross times open parentheses negative 16 close parentheses end cell row cell 1 half cross times open parentheses negative 4 close parentheses end cell row cell 1 half cross times 12 end cell end table close parentheses end cell row blank equals cell 1 half open parentheses table row cell negative 16 end cell row cell negative 4 end cell row 12 end table close parentheses end cell row blank equals cell 1 half stack N S with rightwards harpoon with barb upwards on top end cell end table

Ternyata didapat stack Q N with rightwards harpoon with barb upwards on top equals k stack N S with rightwards harpoon with barb upwards on top untuk k equals 1 half. Berarti terbukti bahwa Q, N, dan S segaris. Selanjutnya akan ditentukan perbandingan N membagi stack Q S with rightwards harpoon with barb upwards on top. Misalkan stack Q N with rightwards harpoon with barb upwards on top divided by stack N S with rightwards harpoon with barb upwards on top equals g divided by h, maka

table attributes columnalign right center left columnspacing 0px end attributes row bold italic n equals cell fraction numerator h bold italic q plus g bold italic s over denominator g plus h end fraction end cell row cell open parentheses table row 8 row 4 row 4 end table close parentheses end cell equals cell fraction numerator h open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses plus g open parentheses table row cell negative 8 end cell row 0 row 16 end table close parentheses over denominator g plus h end fraction end cell row cell open parentheses g plus h close parentheses open parentheses table row 8 row 4 row 4 end table close parentheses end cell equals cell h open parentheses table row 16 row 6 row cell negative 2 end cell end table close parentheses plus g open parentheses table row cell negative 8 end cell row 0 row 16 end table close parentheses end cell row cell open parentheses table row cell 8 cross times open parentheses g plus h close parentheses end cell row cell 4 cross times open parentheses g plus h close parentheses end cell row cell 4 cross times open parentheses g plus h close parentheses end cell end table close parentheses end cell equals cell open parentheses table row cell 16 h end cell row cell 6 h end cell row cell negative 2 h end cell end table close parentheses plus open parentheses table row cell negative 8 g end cell row 0 row cell 16 g end cell end table close parentheses end cell row cell open parentheses table row cell 8 g plus 8 h end cell row cell 4 g plus 4 h end cell row cell 4 g plus 4 h end cell end table close parentheses end cell equals cell open parentheses table row cell 16 h minus 8 g end cell row cell 6 h plus 0 end cell row cell negative 2 h plus 16 g end cell end table close parentheses end cell end table 

Berdasarkan kesamaan dua buah vektor didapat

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 g plus 4 h end cell equals cell 6 h plus 0 end cell row cell 4 g plus 4 h end cell equals cell 6 h end cell row cell 4 g end cell equals cell 6 h minus 4 h end cell row cell 4 g end cell equals cell 2 h end cell row cell g over h end cell equals cell 2 over 4 end cell row cell g over h end cell equals cell 1 half end cell end table 

Artinya, stack Q N with rightwards harpoon with barb upwards on top divided by stack N S with rightwards harpoon with barb upwards on top equals 1 divided by 2 

Dengan demikian diperoleh koordinat titik straight M open parentheses 12 comma space 2 comma space 2 close parentheses dan koordinat titik straight N open parentheses 8 comma space 4 comma space 4 close parentheses. Selain itu terbukti bahwa Q, N, dan S segaris dan perbandingan N membagi stack Q S with rightwards harpoon with barb upwards on top adalah 1 divided by 2 

 

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