如图,在正方体ABCD-A1B1C1D1中,E,F分别是BB1,CD的中点.
Ⅰ.证明AD⊥D1F;
Ⅱ.求AE与D1F所成的角;
Ⅲ.证明面AED⊥面A1FD1;
Ⅳ.设AA1=2,求三棱锥F-A1ED1的体积VF-A1ED1.
解:
(1)∵AC1是正方体
∴AD⊥面DC1,
又D1F⊂面DC1,
∴AD⊥D1F
(2)取AB中点G,连接A1G,FG,
∵F是CD中点
∴GF∥AD又A1D1∥AD
∴GF∥A1D1
∴GFD1A1是平行四边形∴A1G∥D1F设A1G∩AE=H
则∠AHA1是AE与D1F所成的角
∵E是BB1的中点∴Rt△A1AG≌Rt△ABE
∴∠GA1A=∠GAH∴∠A1HA=90°即直线AE与D1F所成角是直角
(3)∵AD⊥D1F((1)中已证)
AE⊥D1F,又AD∩AE=A,
∴D1F⊥面AED,
又∵D1F⊂面A1FD1,
∴面AED⊥面A1FD1