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Question Number 203569 by cherokeesay last updated on 22/Jan/24

Commented by a.lgnaoui last updated on 22/Jan/24

$$2(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})=360$$$$(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})=180$$$$(\mathrm{a}+\mathrm{b})=180-(\mathrm{c}+\mathrm{d})$$$$$$$$\tan (\mathbf{a}+\mathbf{b})=-\tan (\mathbf{c}+\mathbf{d})$$$$\frac{\mathbf{t}1+\mathbf{t}2}{1-\mathbf{t}1.\mathbf{t}2}=\frac{\mathbf{t}3+\mathbf{t}4}{\mathbf{t}3\mathbf{t}4-1}$$$$\mathbf{t}1=\frac{\mathbf{R}}{20}\mathbf{t}2=\frac{\mathbf{R}}{15}\mathbf{t}3=\frac{\mathbf{R}}{12}\mathbf{t}4=\frac{\mathbf{R}}{10}$$$$\Rightarrow 20\mathbf{t}1=15\mathbf{t}2=12\mathbf{t}3=10\mathbf{t}4$$$$\mathbf{t}2=\frac{4}{3}\mathbf{t}1\mathbf{t}4=\frac{6}{5}\mathbf{t}3$$$$\frac{7\mathbf{t}1}{3-4{\left(\mathbf{t}1\right)}^2}=\frac{11\mathbf{t}3}{6{\left(\mathbf{t}3\right)}^2-5}(\mathbf{t}3=\frac{5}{3}\mathbf{t}1)$$$$\Rightarrow \frac{7\mathrm{t}1}{3-4{\left(\mathrm{t}1\right)}^2}=\frac{11\times \frac{5}{3}\mathrm{t}1}{6\left(\frac{25}{9}\right){\mathrm{t}1}^2-5}=\frac{55\mathrm{t}1}{50\left({\mathrm{t}1}^2\right)-15}$$$$7[50{\left(\mathrm{t}1\right)}^2-15]=55(3-4{\left(\mathrm{t}1\right)}^2$$$$350\left({\mathrm{t}1}^2\right)-105=165-220{\left(\mathrm{t}1\right)}^2$$$$[350+220]{\left(\mathrm{t}1\right)}^2=270$$$$\mathrm{t}1=\frac{3}{\sqrt{19}}\Rightarrow \frac{\mathbf{R}}{20}=\frac{3}{\sqrt{19}}$$$$$$$$\mathbf{R}=\frac{60}{\sqrt{19}}=13,764$$$$$$$$$$

Commented by a.lgnaoui last updated on 22/Jan/24

Commented by cherokeesay last updated on 22/Jan/24

$$perfect!$$$$thankyousir.$$

Answered by mr W last updated on 22/Jan/24

$${\tan }^{-1}\frac{10}{R}+{\tan }^{-1}\frac{12}{R}+{\tan }^{-1}\frac{15}{R}+{\tan }^{-1}\frac{20}{R}=\pi$$$$\tan ({\tan }^{-1}\frac{10}{R}+{\tan }^{-1}\frac{12}{R})=-\tan ({\tan }^{-1}\frac{15}{R}+{\tan }^{-1}\frac{20}{R})$$$$\frac{\frac{10}{R}+\frac{12}{R}}{1-\frac{10}{R}\times \frac{12}{R}}=-\frac{\frac{15}{R}+\frac{20}{R}}{1-\frac{15}{R}\times \frac{20}{R}}$$$$\frac{22}{R^2-120}=-\frac{35}{R^2-300}$$$$R^2=\frac{22\times 300+35\times 120}{22+35}=\frac{3600}{19}$$$$\Rightarrow R=\frac{60}{\sqrt{19}}\approx 13.765$$

Commented by cherokeesay last updated on 22/Jan/24

$$Sonice!$$$$thankyoumaster!$$

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