题目内容
如图,在三棱锥
中
底面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807221914.gif)
点
,
分别在棱
上,且
(Ⅰ)求证:
平面
;
(Ⅱ)当
为
的中点时,求
与平面
所成的角的大小;
(Ⅲ)是否存在点
使得二面角
为直二面角?并说明理由.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508075185154.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807003311.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807206246.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807221914.gif)
点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807221210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807237204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807362310.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807377439.gif)
(Ⅰ)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807393264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807409265.gif)
(Ⅱ)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807221210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807440234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807455236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807409265.gif)
(Ⅲ)是否存在点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807237204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807502326.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508075185154.jpg)
(Ⅰ)略
(Ⅱ)
与平面
所成的角的大小![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807549510.gif)
(Ⅲ)存在点E使得二面角
是直二面角.
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807455236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807409265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807549510.gif)
(Ⅲ)存在点E使得二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807502326.gif)
【解法1】本题主要考查直线和平面垂直、直线与平面所成的角、二面角等基础知识,考查空间想象能力、运算能力和推理论证能力.
(Ⅰ)∵PA⊥底面ABC,∴PA⊥BC.
又
,∴AC⊥BC.
∴BC⊥平面PAC. ……………4分
(Ⅱ)∵D为PB的中点,DE//BC,
∴
,
又由(Ⅰ)知,BC⊥平面PAC,
∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,……………6分
∵PA⊥底面ABC,∴PA⊥AB,又PA=AB,
∴△ABP为等腰直角三角形,∴
,
∴在Rt△ABC中,
,∴
.
∴在Rt△ADE中,
,
∴
与平面
所成的角的大小
……………8分.
(Ⅲ)∵DE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
平面PAC,PE
平面PAC,∴DE⊥AE,DE⊥PE,
∴∠AEP为二面角
的平面角, ……………10分
∵PA⊥底面ABC,∴PA⊥AC,∴
. ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807830332.jpg)
∴在棱PC上存在一点E,使得AE⊥PC,这时
,
故存在点E使得二面角
是直二面角. ……………12分
【解法2】如图,以A为原煤点建立空间直角坐标系
,
设
,由已知可得
.……………2分
(Ⅰ)∵
, ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807830332.jpg)
∴
,∴BC⊥AP.
又∵
,∴BC⊥AC,∴BC⊥平面PAC. ……………4分
(Ⅱ)∵D为PB的中点,DE//BC,∴E为PC的中点,
∴
,
∴又由(Ⅰ)知,BC⊥平面PAC,∴∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角, ……………6分
∵
,∴
.
∴
与平面
所成的角的大小
……………8分
(Ⅲ)解法同一 (略)
(Ⅰ)∵PA⊥底面ABC,∴PA⊥BC.
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807580492.gif)
∴BC⊥平面PAC. ……………4分
(Ⅱ)∵D为PB的中点,DE//BC,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807596526.gif)
又由(Ⅰ)知,BC⊥平面PAC,
∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,……………6分
∵PA⊥底面ABC,∴PA⊥AB,又PA=AB,
∴△ABP为等腰直角三角形,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807611446.gif)
∴在Rt△ABC中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807627491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807658510.gif)
∴在Rt△ADE中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807674951.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807455236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807409265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807549510.gif)
(Ⅲ)∵DE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807721135.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807721135.gif)
∴∠AEP为二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807502326.gif)
∵PA⊥底面ABC,∴PA⊥AC,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807799494.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807830332.jpg)
∴在棱PC上存在一点E,使得AE⊥PC,这时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807845482.gif)
故存在点E使得二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807502326.gif)
【解法2】如图,以A为原煤点建立空间直角坐标系
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807908379.gif)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807923275.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508079231807.gif)
(Ⅰ)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807939970.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807830332.jpg)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150808173376.gif)
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807580492.gif)
(Ⅱ)∵D为PB的中点,DE//BC,∴E为PC的中点,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508082041262.gif)
∴又由(Ⅰ)知,BC⊥平面PAC,∴∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角, ……………6分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508083601611.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231508083761105.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807455236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150807409265.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150808454435.gif)
(Ⅲ)解法同一 (略)
![](http://thumb2018.1010pic.com/images/loading.gif)
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