题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-53689.png)
(I)在平面ABC内,试做出过点P与平面A1BC平行的直线l,说明理由,并证明直线l⊥平面ADD1A1;
(II)设(I)中的直线l交AB于点M,交AC于点N,求二面角A﹣A1M﹣N的余弦值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-35720.png)
解析
∵直线l⊄平面A1BC,BC⊂平面A1BC,
∴直线l∥平面A1BC,
∵△ABC中,AB=AC,D是BC的中点,
∴AD⊥BC,结合l∥BC得AD⊥l
∵AA1⊥平面ABC,l⊂平面ABC,∴AA1⊥l
∵AD、AA1是平面ADD1A1内的相交直线
∴直线l⊥平面ADD1A1;
(II)连接A1P,过点A作AE⊥A1P于E,过E点作EF⊥A1M于F,连接AF
由(I)知MN⊥平面A1AE,结合MN⊂平面A1MN得平面A1MN⊥平面A1AE,
∵平面A1MN∩平面A1AE=A1P,AE⊥A1P,∴AE⊥平面A1MN,
∵EF⊥A1M,EF是AF在平面A1MN内的射影,
∴AF⊥A1M,可得∠AFE就是二面角A﹣A1M﹣N的平面角
设AA1=1,则由AB=AC=2AA1,∠BAC=120°,可得∠BAD=60°,AB=2且AD=1
又∵P为AD的中点,∴M是AB的中点,得AP=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-16820.png)
Rt△A1AP中,A1P=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-79955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-99171.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-43926.png)
∴AE=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-63502.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-62824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-11752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231203-61516.png)
∴Rt△AEF中,sin∠AFE=
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231204-66022.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231204-11154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231204-61998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-35720.png)
即二面角A﹣A1M﹣N的余弦值等于
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231202-35720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231204-96672.png)
核心考点
试题【(12分)如图,在三棱柱ABC﹣A1B1C1中,侧棱AA1⊥底面ABC,AB=AC=2AA1,∠BAC=120°,D,D1分别是线段BC,B1C1的中点,P是线】;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/20191020/20191020231149-34265.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231149-42908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231149-42908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231149-79140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231149-74329.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231149-43942.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-73008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-28922.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-17013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-80981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231151-55442.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231151-74832.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231151-20011.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231151-14545.png)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-28922.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231150-80981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-21964.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-59695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-27249.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-42468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-59695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231137-84186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231137-68965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231137-50143.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231137-97443.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231137-43948.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231138-43114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231138-63839.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231138-52977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-59695.png)
(Ⅲ)求多面体
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231136-21964.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231138-25859.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231126-22340.png)
(Ⅰ)当E为BC中点时,求证:CP∥平面ABEF;
(Ⅱ)设BE=x,当x为何值时,三棱锥A-CDF的体积有最大值?并求出这个最大值.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231126-69469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231110-58356.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231110-71477.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231110-86057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231110-74766.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231111-35738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231111-23095.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231111-40687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231111-19367.png)
(Ⅱ)求棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231112-42800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231112-39972.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231050-94821.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231051-19755.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231051-23444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231051-23992.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231051-66593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231051-23444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231052-29085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231052-29045.png)
(I) 求证:平面OEF//平面APD;
(II)求直线CD
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231052-39486.png)
(III)在棱PC上是否存在一点
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231052-24559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020231052-24559.png)
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