题目
题型:不详难度:来源:
如图,四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011320-13639.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011320-77566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011320-42496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011320-11456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011321-14967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011321-30070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011321-70731.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011321-81424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011321-35716.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011322-42168.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011322-38409.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011322-57521.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011322-66009.png)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-70369.png)
(2)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-95895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-27902.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-76104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-95895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-27902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-32365.png)
解析
解法一(几何法):(Ⅰ)作ME∥CD交CD于E,由已知中,∠ADC=∠BCD=90°,PA=PD=AD=2BC=2,N是AD的中点,可得BN⊥AD,结合侧面PAD垂直于底面ABCD,及面面垂直和线面垂直的性质可得BN⊥NE,即∠DNE为二面角M-BN-C的平面角,由二面角M-BN-C为30°,可得∠DNE=30°,可求出DE=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011324-13104.png)
(2)连接BE,由(Ⅰ)可知PE⊥平面BMN,即∠PBE为直线PB与平面BMN所成的角.连接PN,则PN⊥平面ABCD,从而PN⊥BN,解△PBE可得直线PB与平面MBN所成的角。解法二(向量法):(Ⅰ)建立如图所示的坐标系N-xyz,设PM=λPC(λ>0),求出面MBN的法向量,及面BNC的法向量,由二面角M-BN-C为30°,求出λ值,即可得到值。
(2)由上可知(
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011324-59231.png)
PB,代入线面夹角公式sinθ,可得直线PB与平面MBN所成的角.
(1)建立如图所示的坐标系
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011324-77080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011324-93021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011325-60369.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011325-61517.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011325-44074.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011325-77893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011325-46554.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011326-24972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011326-69464.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011326-11114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011326-42110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011326-73104.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011327-78681.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011327-74606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011327-74560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011327-83649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-94677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-45292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-65074.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-30393.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-77460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011322-38409.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011329-93483.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011329-77914.png)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011329-43470.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-76104.png)
(2)由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-65074.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011328-77460.png)
设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-95895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011327-74606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011329-63852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011329-73810.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011330-61373.png)
所以直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-95895.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-27902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011323-32365.png)
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011311-12157.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011311-55701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011311-97280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-61442.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-98234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-57545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011311-55701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-68333.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-90785.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-68333.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011312-61442.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011313-26810.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011302-24821.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011302-51163.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011302-19343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011302-97683.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011302-44123.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011303-84017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011303-77563.jpg)
(Ⅰ)求异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011303-40872.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011303-86977.png)
(Ⅱ)BE和平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011303-82302.png)
(1)求B到平面B1ED距离
(2)求直线DC和平面B1ED所成角的正弦值. (12分)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011257-14572.jpg)
如图, 在直三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011247-76444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011247-67610.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011248-84059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011248-98140.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011248-92432.png)
(2)问:是否在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011248-93806.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011249-84194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011249-21965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011249-32469.png)
若存在,请证明;若不存在,请说明理由。
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011249-70059.png)
(本题满分12分)
如图,已知三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011234-33531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011234-62547.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011234-49912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011234-35446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011235-10179.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011235-68085.png)
(1)求异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011235-63525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011235-70900.png)
(2)求直线BE和平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011236-46311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022011236-37730.png)
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