無窮等比級數200810271545

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屬性

$\begin{align} & \Delta ABC=\frac{1}{2}\times \overline{BC}\times \overline{AO} \\ & =\frac{1}{2}\times 10\times 12=60 \\ \end{align}$
$s=\overline{AB}+\overline{AC}+\overline{BC}$
$\begin{align} & =\sqrt{12^{2}+5^{2}}+\sqrt{12^{2}+5^{2}}+10 \\ & =36 \\ \end{align}$
$\Delta ABC=\frac{1}{2}\times r_{1}\times s$
$\Rightarrow r_{1}=\frac{2\Delta ABC}{s}=\frac{120}{36}=\frac{10}{3}$
$S_{1}=\pi r_{1}^{2}=\frac{100}{9}\pi \#$

$\Delta ABC\cong \Delta AB_{1}C_{1}$
$\frac{r_{2}}{r_{1}}=\frac{h_{2}}{h_{1}}=\frac{\overline{AO}-2r_{1}}{\overline{AO}}$ ，註： $h\,$ 為三角形的高
$=\frac{12-2\times \frac{10}{3}}{12}=\frac{4}{9}$
$\frac{S_{2}}{S_{1}}=\frac{\pi r_{2}^{2}}{\pi r_{1}^{2}}=\left( \frac{4}{9} \right)^{2}$
$\Rightarrow r=\left( \frac{4}{9} \right)^{2}$
$\sum\limits_{k=1}^{\infty }{S_{k}=\frac{a}{1-r}}=\frac{\frac{100}{9}\pi }{1-\left( \frac{4}{9} \right)^{2}}=\frac{900\pi }{65}=\frac{180\pi }{13}\#$