如圖所示:(1) 設(shè)AM=x>0,AB=y>0,則DM^=1+x^,CM^=1+(y-x)^,當(dāng)CM⊥DM時(shí),1+x^+1+(y-x)^=y^===>x^-yx+1=0,∵ M點(diǎn)唯一,∴ 判別式△=0,得y=2,x=1.∴當(dāng)AB=2且M是AB的中點(diǎn)時(shí),CM⊥DM.D'D⊥面ABCD,DM是D'M在面ABCD的射影,由三垂線逆定理,CM⊥D'M 又C'C⊥CM,∴ CM是異面直線C'C與D'M的距離=√2.(2) △D'DC的面積=1,D'M=√3,CM=√2,∴△D'MC的面積=√6/2,△D'MC在面D'DCC'的射影是△D'DC.二面角M-D'C-D的平面角為θ,由面積射影定理,得cosθ=△D'DC/△D'C=√6/3,∴ 二面角M-D'C-D的大小 是arccos(√6/3).