mtk: bukti Rumus jarak titik ke garis

Minggu, 02 Januari 2011

bukti Rumus jarak titik ke garis

Perhatikan gambar di samping!
d adalah jarak titik A ke garis ax+by+c=0
misalkan :

$A(x_0,y_0),maka B(x_0,\frac{-ax_0-c}{b}),C(\frac{-by_0-c}{a},y_0)$

$|AB|=\sqrt{(\frac{-ax_0-c}{b}-y_0)^{2}}=|\frac{ax_0+by_0+c}{b}|$

$|AC|=\sqrt{(x_0-\frac{-by_0-c}{a})^2}=|\frac{ax_0+by_0+c}{a}|$

$|BC|=\sqrt{(x_0-\frac{-by_0-c}{a})^2+(\frac{-ax_0-c}{b}-y_0)^2}\Rightarrow$

$|BC|=\sqrt{\frac{(ax_0+by_0+c)^2}{a^2}+\frac{(ax_0+by_0+c)}{b^2}}=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}} \sqrt{(ax_0+by_0+c)^2}\Rightarrow$

$|BC|=\frac{\sqrt{a^2+b^2}}{\sqrt{a^2b^2}}|ax_0+by_0+c|=\frac{|ax_0+by_0+c|\sqrt{a^2+b^2}}{|ab|}$

Jadi,jarak titik A ke garis (d) :

$d=\frac{|AB||AC|}{|BC|}=\frac{|(ax_0+by_0+c)/a|.|(ax_0+ay_0+c)/b|}{(|ax_0+by_0+c|\sqrt{a^2+b^2})/|ab|}=\frac{|ax_0^2+by_0+c|^2}{|ab|}.\frac{|ab|}{|ax_0+by_0+c|\sqrt{a^2+b^2}}\Rightarrow$

$d=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$