Selasa, 26 Maret 2013

Pembuktian Rumus Heron


Rumus Heron
Rumus Heron adalah rumus yang dipakai untuk menghitung luas segitiga yang diketahui ketiga sisinya.
Misalkan diketahui segitiga ABC dengan panjang sisi a, b, dan c. Jika s menyatakan setengah keliling segitiga ABC, atau dikatakan s=½(a+b+c) maka luas segitiga tersebut bisa dinyatakan dengan
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Rumus tersebut bisa dibuktikan sebagai berikut :
Cara pertama
misalkan terdapat sebuah segitiga ABC sebagai berikut  dengan alas segitiga adalah a, dan t adalah tinggi segitiga yang ditarik dari titik A
Description: http://olimpiadematematika.com/wp-content/uploads/2013/01/rumus-heron.jpg

Rumus pythagoras pada segitiga ADC adalah
x2 + t2 = b2 …………………………………………(1)
Rumus pythagoras pada segitiga ADB adalah
(a – x)2 + t2 = c2
a2 – 2ax + x2 + t2 = c2 …………………………..(2)
Dengan mensubstitusi persamaan (1) ke persamaan (2) maka diperoleh
a2 – 2ax + b2 = c2
2ax = a2 + b2 – c2
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Dari persamaan (1) diperoleh
t2 = b2 – x2
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karena s=½(a+b+c) maka a + b + c = 2s
Jadi
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Jika kedua ruas diakarkan maka diperoleh
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sehingga

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Jadi luas segitiga adalah
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Cara kedua
Menurut aturan cosinus :
c2 = a2 + b2 – 2ab cos C
2ab cos C = a2 + b2 – c2
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Luas segitiga bisa dinyatakan sbb :
L = ½ab sin C
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dengan mengganti a+b+c=2s maka diperoleh
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