题目
题型:同步题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114936-62252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114936-71375.png)
(2)若AD=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114937-46036.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114937-52096.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114937-79997.png)
因PA⊥底面ABCD ,故PA ⊥AB ,
由PA=AB 知△PAB 为等腰直角三角形,
又点E 是棱PB 的中点,故AE ⊥PB.
又在矩形ABCD 中,BC ⊥AB ,而AB 是PB 在底面ABCD 内的射影,
由三垂线定理得BC⊥PB ,从而BC⊥平面PAB ,
故BC⊥AE,从而AE ⊥平面PBC ,
故AE 的长即为直线AD与平面PBC的距离.
在Rt △PAB 中,PA=AB=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114937-60960.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114938-41278.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114938-27954.png)
即直线AD与平面PBC的距离为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114938-53216.png)
(2)过点D作DF⊥CE,交CE于F,过点F作FG⊥CE,交AC于G,
则∠DFG为所求二面角的平面角.
由(1)知BC⊥平面PAB,
又AD∥BC,得AD⊥平面PAB,
故AD⊥AE,从而DE=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114938-38502.png)
在Rt△CBE中,CE=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114939-26724.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114939-47309.png)
所以△CDE为等边三角形,
故点F为CE的中点,且DF=CD·
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114939-95386.png)
因为AE⊥平面PBC,
故AE⊥CE,
又FG⊥CE,
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114939-83635.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114940-36124.png)
且点G为AC的中点.连结DC.
则在Rt△ADC中,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114940-26748.png)
所以cos∠DFG=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114940-45557.png)
即二面角A-EC-D的平面角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114940-42985.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114941-27465.png)
核心考点
试题【如图,四棱锥P-ABCD 中,底面ABCD 为矩形,PA ⊥底面ABCD ,PA=AB=,点E是棱PB的中点。(1)求直线AD与平面PBC的距离;(2)若AD=】;主要考察你对空间几何体的表面积与体积等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021114919-31668.gif)