Question

在三棱柱ABC-A1B1C1中,侧棱AA1垂直底面ABC.AC＝3,BC＝AA1＝4,AB＝5,点D是AB的中点（1）求

Answer 1

1、∵侧棱AA1垂直底面ABC,∴三棱柱是直三棱柱,CC1⊥平面ABC,∵AC=3,BC=4,AB=5,∴根据勾股定理逆定理,△ABC是RT△,∴〈ACB=90°,∴AC⊥BC,∵CC1⊥平面ABC,AC∈平面ABC,∴AC⊥CC1,∵CC1∩BC=C,∴AC⊥平面BCC1,∵BC1∈平面BCC1,∴AC⊥BC1.2、取A1B1中点E,连结AE、CE、DE,∵B1E=A1B1/2,AD=AB/2,AB=A1B1,∴B1E=AD,∵B1E//AD,∴四边形ADB1E是平行四边形,∴AE//DB1,∵E、D分别是A1B1和AB的中点,∴DE//BB1//AA1,且DE=AA1,∵AA1//CC1,∴DE//CC1,且DE=CC1,∴四边形CC1ED是平行四边形,∴C1E//CD,∵C1E∩AE=E,B1D∩DC=D,∴平面AEC1//平面DB1C,∵AC1∈平面AEC1,∴AC1//平面CDB1.