(2014?呼和浩特一模)如图,四棱锥P-ABCD中,四边形ABCD为平行四边形,面PAD⊥平面ABCD,PA=PD,Q为AD的

2025-05-07 04:45:08
推荐回答(1个)
回答1:

解答:(Ⅰ)证明:∵PA=PD,Q为AD的中点,
∴PQ⊥AD,PQ∩QB=Q,
∴AD⊥平面QPB,
∵四边形ABCD是平行四边形,∴BC∥AD,
∴BC⊥面PQB,
∴BC⊥PB.
(Ⅱ)由于面PAD⊥面ABCD,且PQ⊥AD,
∴PQ⊥面PBQ,
∴PQ的长即为四棱锥P-ABCD的高,
设M到面ABCD的距离为h,
则由

PM
MC
=
1
2
知,
h
PQ
=
2
3

h=
2
3
PQ

设四边形ABCD的面积为S,
∴VC-MQB=VM-QBC=
1
3
×
1
2
Sh
=
1
9
S?PQ

VC?MQB
VP?ABCD
=
1
6
Sh
1
3
S?PQ
=
1
3