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Question Number 175449 by cherokeesay last updated on 30/Aug/22

Commented by som(math1967) last updated on 31/Aug/22

$$isr=7-3\sqrt{5}\approx 0.29?$$$$\frac{S_2}{S_1}=\frac{\pi {(.29)}^2}{\pi }\times 100=8.41\mathrm{\%}?$$

Answered by mr W last updated on 03/Sep/22

Commented by mr W last updated on 03/Sep/22

$$\tan 2\theta =\frac{2}{1}=2$$$$\frac{2\tan \theta }{1-{\tan }^2\theta }=2$$$$\tan \theta =\frac{-1+\sqrt{5}}{2}=\frac{2}{\sqrt{5}+1}=\frac{1}{\phi }$$$$CD=\frac{r}{\tan \theta }=\phi r$$$$BF=DE=2-\phi r$$$$AF=2-r$$$$AB=2+r$$$${(2+r)}^2={(2-r)}^2+{(2-\phi r)}^2$$$$8r=4-4\phi r+{\phi }^2{r}^2$$$$0=4-4(\phi +2)r+{\phi }^2{r}^2$$$$r=\frac{2(\phi +2)-4\sqrt{\phi +1}}{{\phi }^2}=\frac{2(2-\phi )}{\phi +1}=7-3\sqrt{5}$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

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