Question

Intr un triunghi dreptunghic ipotenuza a=8,iar raza cercului inscris r=1;daca b,c sunt catetele,sa se calculeze E=b/c + c/b

Answer 1

$\it r=\dfrac{b+c-a}{2} \Rightarrow 1=\dfrac{b+c-8}{2}\Rightarrow 2=b+c-8\Rightarrow b+c=10\Rightarrow\\ \\ \\ \Rightarrow(b+c)^2=10^2 \Rightarrow b^2+c^2+2bc=100\stackrel{T.Pitagora}{\Longrightarrow}\ \ a^2+2bc=100\Rightarrow\\ \\ \\ \Rightarrow 8^2+2bc=100 \Rightarrow 64+2bc=100|_{-64} \Rightarrow2bc=36|_{:2} \Rightarrow bc=18\ \ \ \ (*)$

$\it E = \dfrac{^{b)}b}{\ c}+\dfrac{^{c)}c}{\ b}=\dfrac{b^2+c^2}{bc} \stackrel{T.Pitagora}{\Longrightarrow}\ \ E = \dfrac{a^2}{bc} =\dfrac{8^2}{bc}=\dfrac{64}{bc}\stackrel{(*)}{=}\ \dfrac{\ 64^{(2}}{18}=\dfrac{32}{9}$