题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031902-58717.jpg)
如图,圆柱OO1内有一个三棱柱ABC-A1B1C1,
三棱柱的底面为圆柱底面的内接三角形,且AB是圆O的直径。
(Ⅰ)证明:平面A1ACC1⊥平面B1BCC1;
(Ⅱ)设AB=AA1。在圆柱OO1内随机选取一点,记该点取自于
三棱柱ABC-A1B1C1内的概率为P。
(i) 当点C在圆周上运动时,求P的最大值;
记平面A1ACC1与平面B1OC所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031902-53517.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031902-53517.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031902-49554.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031902-53517.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031903-91605.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031903-32106.jpg)
解法一 :
(I)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031903-81014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031903-62528.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-54080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031903-62528.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-21248.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-35569.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-83259.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-14973.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-75013.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-77225.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031904-54080.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-79749.gif)
所以平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-77225.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-22247.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-79749.gif)
(II)(i)设圆柱的底面半径为r,则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-97392.gif)
故三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-59302.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-68178.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-81485.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-29620.gif)
当且仅当
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-56373.gif)
从而,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-63844.gif)
而圆柱的体积
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-81460.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-71075.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-56373.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-94881.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-79401.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-98050.gif)
(ii)由(i)可知,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-79401.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-94881.gif)
于是,以O为坐标原点,建立空间直角坐标系
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-66981.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031909-71646.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031909-60792.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031909-25699.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031909-25163.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-77225.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031910-10346.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031910-17305.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031905-77225.gif)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031910-47270.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031910-81360.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031911-69833.jpg)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031911-91351.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031910-47270.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031911-92609.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031911-95403.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031912-74610.jpg)
解法二:
(I)同解法一
(II)(i)设圆柱的底面半径为r,则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-97392.gif)
故三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-59302.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031912-97582.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-68178.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031912-70416.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031912-51835.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031912-77332.gif)
由于
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-82663.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-49227.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-36634.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-63844.gif)
而圆柱的体积
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-81460.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-71075.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-49227.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-36634.gif)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-79401.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-98050.gif)
(ii)同解法一
解法三:
(I)同解法一
(II)(i)设圆柱的底面半径
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-50842.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-97392.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031907-81460.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031913-30713.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031914-56254.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-79401.gif)
又因为点C在圆周上运动,所以当
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-94881.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031914-97216.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031906-59302.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031914-84558.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-79401.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031908-98050.gif)
(ii)同解法一
解析
核心考点
试题【(本小题满分13分)如图,圆柱OO1内有一个三棱柱ABC-A1B1C1,三棱柱的底面为圆柱底面的内接三角形,且AB是圆O的直径。(Ⅰ)证明:平面A1ACC1⊥平】;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
如图,在长方体ABCD-A1B1C1D1中,E,H分别是棱A1B1,D1C1上的点(点E与B1不重合),且EH∥A1 D1. 过EH的平面与棱BB1,CC1相交,交点分别为F,G。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031856-73620.jpg)
(I) 证明:AD∥平面EFGH;
(II) 设AB=2AA1 ="2" a .在长方体ABCD-A1B1C1D1内随机选取一点。记该点取自几何体A1ABFE-D1DCGH内的概率为p,当点E,F分别在棱A1B1上运动且满足EF=a时,求p的最小值.
如图,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031841-69652.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-78008.gif)
平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-58400.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-41939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-78539.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-41939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-42634.gif)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031843-28239.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031843-42836.gif)
(2)求平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031843-58435.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031842-41939.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031843-54541.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-18406.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-84446.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-25583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-79286.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-97869.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-66693.jpg)
上的点,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-25152.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-70162.gif)
(1) 求异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-87543.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-62126.gif)
(2) 证明
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-91096.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-49591.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-82091.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-39262.gif)
(3) 求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-88707.gif)
A.(0,![]() | B.(1,![]() |
C.(![]() ![]() | D.(0,![]() |
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031813-74300.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031814-95511.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031814-44140.gif)
A.若![]() ![]() ![]() | B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() | D.若![]() ![]() ![]() |
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