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Diketahui segitiga ABC dengan koordinat titik A(5,−3,−6), B(1,1,1), dan C(−2,4,−6). Tentukan: a. proyeksi vektor AB⇀ pada AC⇀; b. tinggi segitiga ; dan c. luas segitiga .

Pertanyaan

Diketahui segitiga ABC dengan koordinat titik straight A open parentheses 5 comma space minus 3 comma space minus 6 close parenthesesstraight B open parentheses 1 comma space 1 comma space 1 close parentheses, dan straight C open parentheses negative 2 comma space 4 comma space minus 6 close parentheses. Tentukan:

a. proyeksi vektor AB with rightwards harpoon with barb upwards on top pada AC with rightwards harpoon with barb upwards on top;

b. tinggi segitiga ABC; dan

c. luas segitiga ABC.

Pembahasan:

Perhatikan perhitungan berikut.

Ingat, proyeksi vektor straight A with rightwards harpoon with barb upwards on top pada straight B with rightwards harpoon with barb upwards on top:

Proyeksi space vektor equals fraction numerator straight A with rightwards harpoon with barb upwards on top times straight B with rightwards harpoon with barb upwards on top over denominator open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar squared end fraction times straight B with rightwards harpoon with barb upwards on top

Perhatikan gambar berikut.

a. proyeksi vektor AB with rightwards harpoon with barb upwards on top pada AC with rightwards harpoon with barb upwards on top

table attributes columnalign right center left columnspacing 0px end attributes row cell AB with rightwards harpoon with barb upwards on top end cell equals cell straight B with rightwards harpoon with barb upwards on top minus straight A with rightwards harpoon with barb upwards on top end cell row blank equals cell open parentheses table row 1 row 1 row 1 end table close parentheses minus open parentheses table row 5 row cell negative 3 end cell row cell negative 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 minus 5 end cell row cell 1 minus open parentheses negative 3 close parentheses end cell row cell 1 minus open parentheses negative 6 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell row cell 1 plus 3 end cell row cell 1 plus 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell row 4 row 7 end table close parentheses end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell AC with rightwards harpoon with barb upwards on top end cell equals cell straight C with rightwards harpoon with barb upwards on top minus straight A with rightwards harpoon with barb upwards on top end cell row blank equals cell open parentheses table row cell negative 2 end cell row 4 row cell negative 6 end cell end table close parentheses minus open parentheses table row 5 row cell negative 3 end cell row cell negative 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 minus 5 end cell row cell 4 minus open parentheses negative 3 close parentheses end cell row cell negative 6 minus open parentheses negative 6 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 7 end cell row cell 4 plus 3 end cell row cell negative 6 plus 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 7 end cell row 7 row 0 end table close parentheses end cell end table

Misalkan AD with rightwards harpoon with barb upwards on top merupakan proyeksi vektor vektor AB with rightwards harpoon with barb upwards on top pada AC with rightwards harpoon with barb upwards on top, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell AD with rightwards harpoon with barb upwards on top end cell equals cell fraction numerator AB with rightwards harpoon with barb upwards on top times AC with rightwards harpoon with barb upwards on top over denominator open vertical bar AC with rightwards harpoon with barb upwards on top close vertical bar squared end fraction times AC with rightwards harpoon with barb upwards on top end cell row blank equals cell fraction numerator negative 4 times negative 7 plus 4 times 7 plus 7 times 0 over denominator open parentheses square root of negative 7 squared plus 7 squared plus 0 squared end root close parentheses squared end fraction times open parentheses negative 7 comma space 7 comma space 0 close parentheses end cell row blank equals cell fraction numerator 28 plus 28 plus 0 over denominator open parentheses square root of 49 plus 49 plus 0 end root close parentheses squared end fraction times open parentheses negative 7 comma space 7 comma space 0 close parentheses end cell row blank equals cell 56 over open parentheses square root of 98 close parentheses squared times open parentheses negative 7 comma space 7 comma space 0 close parentheses end cell row blank equals cell 56 over 98 times open parentheses negative 7 comma space 7 comma space 0 close parentheses end cell row blank equals cell open parentheses fraction numerator negative 7 times 56 over denominator 98 end fraction comma space fraction numerator 7 times 56 over denominator 98 end fraction comma space fraction numerator 0 times 56 over denominator 98 end fraction close parentheses end cell row blank equals cell open parentheses negative 4 comma space 4 comma space 0 close parentheses end cell end table

Jadi, proyeksi vektor vektor AB with rightwards harpoon with barb upwards on top pada AC with rightwards harpoon with barb upwards on top adalah AD with rightwards harpoon with barb upwards on top equals open parentheses negative 4 comma space 4 comma space 0 close parentheses.

b. tinggi segitiga ABC

Diketahui AB with rightwards harpoon with barb upwards on top equals open parentheses negative 4 comma space 4 comma space 7 close parentheses dan AD with rightwards harpoon with barb upwards on top equals open parentheses negative 4 comma space 4 comma space 0 close parentheses, maka untuk mencari tinggi segitiga tersebut dapat menggunakan teorema Pythagoras dengan terlebih dahulu mencari panjang AB with rightwards harpoon with barb upwards on top dan AD with rightwards harpoon with barb upwards on top, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar AB with rightwards harpoon with barb upwards on top close vertical bar end cell equals cell square root of negative 4 squared plus 4 squared plus 7 end root end cell row blank equals cell square root of 16 plus 16 plus 49 end root end cell row blank equals cell square root of 81 end cell row blank equals 9 end table

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar AD with rightwards harpoon with barb upwards on top close vertical bar end cell equals cell square root of negative 4 squared plus 4 squared plus 0 squared end root end cell row blank equals cell square root of 16 plus 16 plus 0 end root end cell row blank equals cell square root of 32 end cell row blank equals cell square root of 16 times 2 end root end cell row blank equals cell 4 square root of 2 end cell end table

Sehingga tinggi segitiga tersebut yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row t equals cell square root of 9 squared minus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of 81 minus 16 times 2 end root end cell row blank equals cell square root of 81 minus 32 end root end cell row blank equals cell square root of 49 end cell row blank equals 7 end table

Didapatkan panjang tinggi segitiga tersebut adalah 7 satuan.

c. luas segitiga ABC

Terlebih dahulu dicari panjang AC with rightwards harpoon with barb upwards on top, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar AC with rightwards harpoon with barb upwards on top close vertical bar end cell equals cell square root of negative 7 squared plus 7 squared plus 0 squared end root end cell row blank equals cell square root of 49 plus 49 plus 0 end root end cell row blank equals cell square root of 98 end cell row blank equals cell square root of 2 times 49 end root end cell row blank equals cell 7 square root of 2 end cell end table

Maka:

table attributes columnalign right center left columnspacing 0px end attributes row Luas equals cell fraction numerator open vertical bar AC with rightwards harpoon with barb upwards on top close vertical bar times straight t over denominator 2 end fraction end cell row blank equals cell fraction numerator 7 square root of 2 times 7 over denominator 2 end fraction end cell row blank equals cell 49 over 2 square root of 2 end cell end table

Jadi, luas segitiga tersebut adalah 49 over 2 square root of 2 satuan persegi.

Jawaban terverifikasi

Dijawab oleh:

N. Ayu

Mahasiswa/Alumni Universitas Negeri Padang

Terakhir diupdate 06 Oktober 2021

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