Question Number 199358 by ajfour last updated on 01/Nov/23
Commented by ajfour last updated on 01/Nov/23
$${Find}\:{R}. \\ $$
Answered by mr W last updated on 01/Nov/23
$$\left({R}+{R}+\mathrm{1}+\sqrt{\left({R}+\mathrm{1}\right)^{\mathrm{2}} +{R}^{\mathrm{2}} }\right)×\mathrm{1}={R}\left({R}+\mathrm{1}\right) \\ $$$$\sqrt{\left({R}+\mathrm{1}\right)^{\mathrm{2}} +{R}^{\mathrm{2}} }={R}^{\mathrm{2}} −{R}−\mathrm{1} \\ $$$${R}^{\mathrm{2}} −\mathrm{2}{R}−\mathrm{3}=\mathrm{0} \\ $$$$\left({R}−\mathrm{3}\right)\left({R}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow{R}=\mathrm{3} \\ $$
Answered by Frix last updated on 01/Nov/23
$$\mathrm{There}'\mathrm{s}\:\mathrm{exactly}\:\mathrm{one}\:\mathrm{rectangular}\:\mathrm{triangle} \\ $$$$\mathrm{with}\:\mathrm{sides}\:{a},\:{b}={a}+\mathrm{1}\:\mathrm{and}\:\mathrm{incircle}\:\mathrm{of}\:\mathrm{radius} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{1}: \\ $$$${a}={R}=\mathrm{3},\:{b}={R}+\mathrm{1}=\mathrm{4},\:{c}=\mathrm{5},\:{r}_{\mathrm{I}} =\mathrm{1} \\ $$
Commented by ajfour last updated on 01/Nov/23
$${yes},\:{thanks}.\:{simple}\:{one}\:{for}\:{you}. \\ $$
Answered by cortano12 last updated on 02/Nov/23
$$\left(\mathrm{2R}−\mathrm{1}\right)^{\mathrm{2}} =\:\left(\mathrm{R}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{R}^{\mathrm{2}} \\ $$$$\:\mathrm{4R}^{\mathrm{2}} −\mathrm{4R}+\mathrm{1}=\mathrm{2R}^{\mathrm{2}} +\mathrm{2R}+\mathrm{1} \\ $$$$\:\:\mathrm{2R}^{\mathrm{2}} −\mathrm{6R}=\mathrm{0} \\ $$$$\:\mathrm{2R}\left(\mathrm{R}−\mathrm{3}\right)=\mathrm{0} \\ $$$$\:\:\mathrm{R}=\mathrm{3} \\ $$