low, low, low bitzzzzzz: proof of the Heron's formula

Wednesday, October 22, 2008

proof of the Heron's formula

sinulat at iniwasto ni ... Lobitzzz sabi sa relo ko 10/22/2008 12:18:00 AM

kailangan niyo ba ng proof para sa heron's formula para sa homework natin sa math?? eto yung nahanap ko na madaling maintindihan.... hahahaha, wala lang uli...

Proof of the heron's formula

A modern proof, which uses algebra and trigonometry and is quite unlike the one provided by Heron, follows. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. We have

$\cos(C) = \frac{a^2+b^2-c^2}{2ab}$

by the law of cosinesFrom this we get the algebraic statement:

$\sin(C) = \sqrt{1-\cos^2(C)} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}.$

The altitude of the triangle on base a has length b sin(C), and it follows

$\begin{align} A & = \frac{1}{2} (\mbox{base}) (\mbox{altitude}) \\ & = \frac{1}{2} ab\sin(C) \\ & = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2} \\ & = \frac{1}{4}\sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))} \\ & = \frac{1}{4}\sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)} \\ & = \frac{1}{4}\sqrt{(c -(a -b))((c +(a -b))((a +b) -c))((a +b) +c)} \\ & = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}. \end{align}$