5.
如图\(①\),\(\triangle BCD\)内接于直角梯形\({A}_{1}{A}_{2}{A}_{3}D \),\(A_{1}D/\!/A_{2}A_{3}\),\(A_{1}A_{2}⊥A_{2}A_{3}\),\(A_{1}D=10\),\(A_{1}A_{2}=8\),沿\(\triangle BCD\)三边将\(\triangle A_{1}BD\)、\(\triangle A_{2}BC\)、\(\triangle A_{3}CD\)翻折上去,恰好形成一个三棱锥\(ABCD\),如图\(②\).
\((1)\)求证:\(AB⊥CD\);
\((2)\)求直线\(BD\)和平面\(ACD\)所成的角的正切值;
\((3)\)求四面体\(ABCD\)的体积。