6.在棱长为1的正方体ABCD - A1B1C1D1中,E,F分别为棱B1C1和C1D1的中点,则直线EF到平面B1D1D的距离为　　　　　.

解析:设B1D1中点为O,EF中点为K,则KO即为EF到平面B1D1D的距离,KO=1/2 C1O=√2/4.

答案:√2/4

7.已知Rt△ABC的直角顶点C在平面α内,AB∥α,AC,BC与α所成的角分别为45°和30°,若AB=6,则AB到α的距离为　　　　　.

解析:设AB到α的距离为h,则CB=h/sin30"°" =2h,AC=h/sin45"°" =√2 h,由勾股定理AB2=AC2+CB2可得(√2 h)2+(2h)2=62,解得h=√6.

答案:√6

★8.在三棱锥P - ABC中,侧棱PA,PB,PC两两垂直,且PA=PB=PC=2,则点P到平面ABC的距离等于　　　　　.

答案:(2√3)/3

9.在三棱锥B-ACD中,平面ABD⊥平面ACD,若棱长AC=CD=AD=AB=1,且∠BAD=30°,求点D到平面ABC的距离.

解:如图所示,以AD的中点O为原点,以OD,OC所在直线为x轴、y轴,过O作OM⊥平面ACD交AB于点M,以直线OM为z轴建立空间直角坐标系,

　　则A("-" 1/2 "," 0"," 0),B((√3 "-" 1)/2 "," 0"," 1/2),C(0"," √3/2 "," 0),D(1/2 "," 0"," 0),

　　∴(AC) ⃗=(1/2 "," √3/2 "," 0),(AB) ⃗=(√3/2 "," 0"," 1/2),

　　(DC) ⃗=("-" 1/2 "," √3/2 "," 0).

　　设n=(x,y,z)为平面ABC的一个法向量,

则{■(n"·" (AB) ⃗=√3/2 x+1/2 z=0"," @n"·" (AC) ⃗=1/2 x+√3/2 y=0"," )┤