题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074635-39796.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074635-80157.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-93832.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-27284.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-55827.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-41146.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-83230.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-30611.jpg)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-75746.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-47546.gif)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-27284.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-57832.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-19523.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-47546.gif)
(Ⅲ)是否存在点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074636-55827.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-30083.gif)
答案
又
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-76039.gif)
∴BC⊥平面PAC.…………
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074639-36900.gif)
(Ⅱ)∵D为PB的中点,DE//BC,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074639-30859.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,
∵PA⊥底面ABC,∴PA⊥AB,又PA=AB,
∴△ABP为等腰直角三角形,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074639-95441.gif)
∴在Rt△ABC中,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074639-77910.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-62738.gif)
∴在Rt△ADE中,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-68347.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-10213.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-19523.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-47546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-13610.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-45535.gif)
(Ⅲ)∵AE//BC,又由(Ⅰ)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074641-20943.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074641-20943.gif)
∴∠AEP为二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-30083.gif)
∵PA⊥底面ABC,∴PA⊥AC,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074641-79287.gif)
∴在棱PC上存在一点E,使得AE⊥PC,这时
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074641-53115.gif)
故存在点E使得二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-30083.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074641-41616.gif)
解法二:如图,以A为原煤点建立空间直角坐标系
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074642-92068.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074642-25934.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074642-29920.gif)
(Ⅰ)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074642-15707.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074643-55730.gif)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-76039.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074639-36900.gif)
(Ⅱ)∵D为PB的中点,DE//BC,∴E为PC的中点,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074643-10077.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,
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074643-99540.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074643-73015.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074638-19523.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074637-47546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-13610.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074640-45535.gif)
(Ⅲ)同解法1.
解析
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074625-21981.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-40922.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-73468.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-61556.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-45905.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-42260.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-94304.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-41896.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-67624.gif)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-61556.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-27804.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074628-83527.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074627-67624.gif)
(Ⅲ)是否存在点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074626-45905.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074628-48440.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074628-63659.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074613-60458.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-26728.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-70546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-42756.gif)
求证:(I)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-67342.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-33272.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-88047.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-33272.gif)
(自编)(Ⅲ)若E为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-42756.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-70546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-49094.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-33272.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-35082.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074614-76368.jpg)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074600-15714.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074601-34165.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074601-64543.gif)
(1)证明:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074601-25569.gif)
(2)设D是
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074601-30296.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074602-52205.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074602-99241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074602-95784.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074547-95265.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-37384.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-75332.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-10091.gif)
(I)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-99092.gif)
(Ⅱ)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-62370.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074547-95265.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-24589.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074548-92622.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022074549-18888.jpg)
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