题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230649-51944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230649-35350.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230649-44594.png)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-40474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-51299.png)
(Ⅱ)若直线PC与平面PDE所成角的正弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-23902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230649-51944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-32770.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
解析
试题分析:(Ⅰ)由线线垂直得到线面垂直CD⊥平面PAC,进而求证出面面垂直;(Ⅱ)设AP=h,求出平面PDE的一个法向量,再由线面成角的正弦值得到关于h的方程,解出即可.
试题解析:(Ⅰ)在正六边形ABCDEF中,CD⊥AC.
因为PA⊥底面ABCDEF,CDÌ平面ABCDEF,所以CD⊥PA.
又AC∩PA=A,所以CD⊥平面PAC.
因为CDÌ平面PCD,所以平面PAC⊥平面PCD.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230651-87058.png)
(Ⅱ)如图,分别以AC,AF,AP为x轴,y轴,z轴,建立空间直角坐标系A-xyz.
设AP=h(h>0).
则P(0,0,h),C(
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230651-68117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230651-12030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-26528.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-44895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-14633.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230651-68117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-92339.png)
设面PDE的一个法向量为n=(x,y,z),则n·
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-44895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-14633.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-44914.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
记直线PC与平面PDE所成的角为θ,则
sinθ=|cosá
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230652-26528.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230653-95202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230653-71773.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230653-71773.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-23902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
所以六棱锥P-ABCDEF高为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230650-63899.png)
核心考点
试题【如图,六棱锥的底面是边长为1的正六边形,底面。(Ⅰ)求证:平面平面;(Ⅱ)若直线PC与平面PDE所成角的正弦值为,求六棱锥高的大小。】;主要考察你对空间几何体的结构特征等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230643-30156.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230643-58060.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230643-61503.png)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230643-25317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230644-33894.png)
(Ⅱ)若直线PC与平面PDE所成角为
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230644-11094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230644-14086.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230644-46171.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230634-62455.png)
(Ⅰ) 求证:FG∥平面PDC;
(Ⅱ) 求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230634-50038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230611-29747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-58178.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-74466.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-62757.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-74466.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-95204.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230612-79818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230613-57877.png)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230613-89693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230613-47807.png)
(Ⅱ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230613-88065.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230552-15806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-77318.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-55488.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-64423.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-92288.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230552-15806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-28345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230553-44012.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230554-70484.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230554-64779.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230554-76812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230554-63537.png)
(Ⅱ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230554-68364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230555-11170.png)
(Ⅲ)求三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230555-10798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230540-85236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230540-97329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230540-18043.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230540-55179.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230541-65437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230541-95543.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230541-34134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230541-54478.png)
(Ⅱ)求平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230541-54478.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230540-85236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020230542-47682.png)
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