题目内容
(本题满分12分)
如图,三棱锥A—BPC中,AP⊥PC,AC⊥BC,M为AB中点,D为PB中点,且△PMB为正三角形.
(Ⅰ)求证:DM//平面APC;
(Ⅱ)求 证:平面ABC⊥平面APC;(Ⅲ)若BC=4,AB=20,求三棱锥D—BCM的体积.
如图,三棱锥A—BPC中,AP⊥PC,AC⊥BC,M为AB中点,D为PB中点,且△PMB为正三角形.
(Ⅰ)求证:DM//平面APC;
(Ⅰ)略
(Ⅱ)略
(Ⅲ)VD-BCM=VM-BCD=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231513233401006.gif)
(Ⅱ)略
(Ⅲ)VD-BCM=VM-BCD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231513233401006.gif)
解:(Ⅰ)∵M为AB中点,D为PB中点,
∴MD//AP, 又∴MD
平面ABC
∴DM//平面APC ……………3分
(Ⅱ)∵△PMB为正三角形,且D为PB中点。
∴MD⊥PB
又由(Ⅰ)∴知MD//AP, ∴AP⊥PB
又已知AP⊥PC ∴AP⊥平面PBC,
∴AP⊥BC, 又∵AC⊥BC
∴BC⊥平面APC, ∴平面ABC⊥平面PAC ……………8分
(Ⅲ)∵AB=20
∴MB="10 " ∴PB=10
又BC=4,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151323387770.gif)
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231513234031187.gif)
又MD![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151323481809.gif)
∴VD-BCM=VM-BCD=
………………12分
∴MD//AP, 又∴MD
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151323356194.gif)
∴DM//平面APC ……………3分
(Ⅱ)∵△PMB为正三角形,且D为PB中点。
∴MD⊥PB
又由(Ⅰ)∴知MD//AP, ∴AP⊥PB
又已知AP⊥PC ∴AP⊥平面PBC,
∴AP⊥BC, 又∵AC⊥BC
∴BC⊥平面APC, ∴平面ABC⊥平面PAC ……………8分
(Ⅲ)∵AB=20
∴MB="10 " ∴PB=10
又BC=4,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151323387770.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231513234031187.gif)
又MD
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151323481809.gif)
∴VD-BCM=VM-BCD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231513233401006.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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