题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-17565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-53057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-47822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-66506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-78792.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-47143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-60172.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-94569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-36359.jpg)
(1)求四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-17565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-31133.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-56642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-63601.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-94347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-44914.png)
(3)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-23459.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-75041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-15923.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-83126.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-70476.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-53216.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134538-10657.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-53216.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134538-20869.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134538-27912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134538-79176.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134538-72656.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-15645.png)
(2)解法一∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-66506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-78792.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-16590.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-39824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-41600.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-67532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-81625.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-47143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-56642.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134535-60172.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-63601.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-49165.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-20589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134539-81625.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-29447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-44914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-94347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-44914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-15671.jpg)
(3)解法一:取
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-89382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-66590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-65843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-83952.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-92950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-86977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-66590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-40594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-56551.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-81324.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-92950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-40594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-75508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-70011.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-56365.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-23459.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-66590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134540-89382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134541-40594.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134542-81432.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134542-53137.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134542-26842.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134542-34648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134542-53795.png)
即二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-23459.png)
(2)解法二:建立如图所示的空间直角坐标系A-xyz ………………5分
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134543-37712.png)
A(0,0,0) B(1,0,0)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134543-52561.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134543-32971.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134543-10368.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134544-58206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134544-39562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134544-40193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134544-89807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134545-97094.png)
设平面AEF的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134545-21818.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134545-38972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134545-17684.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134545-41266.png)
又平面PAC的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134546-44644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134546-97066.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134536-94347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-44914.png)
(3)解法二:易知平面ACD的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134546-97857.png)
设平面AEF的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134546-10488.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134547-81018.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134547-90807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134547-87465.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134547-21612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134548-47359.png)
∴结合图形知二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134537-23459.png)
解析
(Ⅰ)把四边形面积分成2个直角三角形面积之和,代入棱锥体积公式进行计算.
(Ⅱ)先证 CD⊥平面PAC,由三角形中位线的性质得EF∥CD,得到EF⊥平面PAC,从而证得平面PAC⊥平面AEF.
(Ⅲ)由三垂线定理作出∠EQM为二面角E-AC-D的平面角,并证明之,解直角三角形EQM,求出∠EQM的大小.
核心考点
举一反三
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134508-82862.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134508-19878.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-33954.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-42326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-26241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-22200.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-72804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-37854.png)
(1)求棱
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134509-43946.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134510-19693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134510-30811.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134510-81530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134510-52703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134500-49299.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134500-85241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134450-28721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134450-42363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134450-93852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134450-75702.png)
(1)求异面直线AD与BC所成角大小;
(2)求二面角B-AC-D平面角的大小;
(3)求四面体ABCD外接球的体积。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021134451-48705.png)
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