9.
\((1)\) 在区间\({[}0{,}4{]}\)上随机取一个数\(x\),则事件“\({-}1{\leqslant }\log_{\frac{1}{2}}(x{+}\dfrac{1}{2}){\leqslant }1\)”发生的概率为______ .
\((2)A{,}B{,}C{,}D\)是同一球面上的四个点,\({\triangle }ABC\)中,\({∠}BAC{=}\dfrac{2\pi}{3}{,}AB{=}AC{,}AD{⊥}\)平面\({ABC}{,}AD{=}6{,}AB{=}2\sqrt{3}\),则该球的表面积为______ .
\((3)\) 已知函数\(f(x){=}\dfrac{1}{x{+}1}\),点\(O\)为坐标原点,点\(A_{n}(n{,}f(n))(n{∈}N^{{*}})\),向量\(\overrightarrow{i}{=}(0{,}1){,}\theta_{n}\)是向量\(\overrightarrow{OA_{n}}\)与\(\overrightarrow{i}\)的夹角,则\(\dfrac{\cos\theta_{1}}{\sin\theta_{1}}{+}\dfrac{\cos\theta_{2}}{\sin\theta_{2}}{+…+}\dfrac{\cos\theta_{2017}}{\sin\theta_{2017}}\)的值为______ .
\((4)\)在四边形\(ABCD\)中,若\(AB{=}2{,}BC{=}2\sqrt{2}{,}AD{=}\sqrt{2}{CD}{,}\overrightarrow{{AC}}\overrightarrow{{⋅}CD}{=}0\),则\({|}\overrightarrow{{BD}}{|}\)的最大值为______ .