Question

In desenul alăturat ...​

Answer 1

a)

Notăm planul cercului cu $\alpha$ .

$\it d\ -\ tangent\breve a \Rightarrow\ d\perp BC\ \ \ \ \ (1)\\ \\ AB\perp \alpha,\ \ d\subset\ \alpha\ \Rightarrow AB\perp d \Rightarrow d\perp AB\ \ \ \ \ (2)\\ \\ AB\cap BC=\{B\}\ \ \ \ \ \ (3)\\ \\ (1),\ (2),\ (3) \Rightarrow d\perp (ABC)$

b)

$\it \Delta ABC\ -\ dreptunghic,\ \widehat{B}= 90^o\ \stackrel{T.P.}{\Longrightarrow}\ AC^2-BC^2=AB^2 \Rightarrow \\ \\ \Rightarrow (2,5r)^2-(2r)^2=20^2 \Rightarrow (2,5r-2r)(2,5r+2r)=20^2 \Rightarrow\\ \\ \Rightarrow 0,5r\cdot4,5r=20^2 \Rightarrow 2,25r^2=20^2 \Rightarrow 1,5^2r^2=20^2 \Rightarrow \\ \\ \Rightarrow 1,5r=20|_{\cdot2} \Rightarrow 3r=40 \Rightarrow r=\dfrac{40}{3}=13\dfrac{1}{3}$