题目
题型:不详难度:来源:
如图,在四棱锥S—ABCD中,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234534-30466.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234534-90360.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234534-28689.png)
(1)求证:平面BED
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-55375.png)
(2)求直线SA与平面BED所成角的大小.
答案
解析
(1)证明平面BED⊥平面SAB,利用面面垂直的判定定理,证明DE⊥平面SAB即可;
(2)作AF⊥BE,垂足为F,可得∠AEF是直线SA与平面BED所成的角,在Rt△AFE中,即可求得结论.
解:(1)∵SD⊥平面ABCD,∴平面SAD⊥平面ABCD,
∵AB⊥AD,∴AB⊥平面SAD,∴DE⊥AB.…………………………………………3分
∵SD=AD,E是SA的中点,∴DE⊥SA,
∵AB∩SA=A,∴DE⊥平面SAB
∴平面BED⊥平面SAB.(若用向量法请参照给分)……………………………………6分
(2)法一:作AF⊥BE,垂足为F.
由(Ⅰ),平面BED⊥平面SAB,则AF⊥平面BED,
则∠AEF是直线SA与平面BED所成的角.……………………………………………8分
设AD=2A,则AB=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
△ABE是等腰直角三角形,则AF=A.
在Rt△AFE中,sin∠AEF=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-86346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-87859.png)
故直线SA与平面BED所成角的大小45°.…………………………………………12分
(2)法二:分别以DA,DC,DS为坐标轴建立坐标系D—xyz,不妨设AD=2,则
D(0,0,0),A(2,0,0),B(2,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
C(0,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-29553.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234536-76361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234536-58391.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234536-65700.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
设m=(x1,y1,z1)是面BED的一个法向量,则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234536-94975.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234535-58743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234537-83955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234537-49202.png)
核心考点
试题【(本小题满分12分)如图,在四棱锥S—ABCD中,底面ABCD,底面ABCD是矩形,,E是SA的中点.(1)求证:平面BED平面SAB;(2)求直线SA与平面B】;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/20191020/20191020234528-91446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234528-66388.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234529-73215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234529-98337.png)
求证:(1)直线EF ∥面ACD ;(2)面EFC⊥面BCD .
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234529-73446.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234517-17564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234517-71000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234517-86245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234517-13568.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234518-13200.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234518-27822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234517-71000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234518-74817.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234518-97814.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234518-63616.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234519-39798.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234520-13623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234508-16520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234508-78685.png)
(Ⅰ)求AC与平面BCD所成的角;
(Ⅱ)求点A到BC的距离.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234508-80731.png)
(Ⅰ)求证:BE∥平面PAD;
(Ⅱ)求证:BC⊥平面PBD;
(Ⅲ)求四棱锥P-ABCD的体积。
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234502-14090.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234453-96555.png)
分别是线段AB.BC的中点,
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234453-79671.jpg)
(1)证明:PF⊥FD;
(2)在PA上找一点G,使得EG∥平面PFD;.
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234453-44579.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234454-63849.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234454-34909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020234454-68383.png)
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