题目
题型:不详难度:来源:
一个几何体是由圆柱
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011417-38520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-24488.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-65813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-42238.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-25995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-62315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-60296.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-23775.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-55786.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-47767.gif)
答案
方法1:(1)证明:因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-62315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-31637.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-20663.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-79424.gif)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-74114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-35616.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-95189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-23767.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-98003.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
(2)解:因为点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-65813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-42238.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-25995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-84378.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-33631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
设圆
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-39261.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-38489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-94980.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-43583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-97488.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-43909.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-96644.gif)
过点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-25995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-98718.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-72430.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-21132.gif)
由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-64532.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-14668.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-37128.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-76155.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-37128.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-66984.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-87424.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-95189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-76155.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-22771.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-68448.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-43740.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-60539.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-90086.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-54403.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-39963.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-64200.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011424-80020.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-35152.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-87424.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-77968.gif)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-78023.gif)
方法2:(1)证明:因为点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-65813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-42238.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-25995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-84378.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-33631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
设圆
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-39261.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-38489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-94980.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-43583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-97488.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-43909.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-96644.gif)
以点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-19531.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-11600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-17285.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-48021.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-96580.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-50868.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-45509.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-40490.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-47512.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-44228.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-12920.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-58645.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-98760.gif)
………………………5分
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-17589.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011429-85773.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011429-74915.gif)
(2)解:设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011429-74574.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-73198.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-64846.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-87345.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-41647.gif)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-98587.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-47885.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-73198.gif)
由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-74114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-31734.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-95189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-58645.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-52991.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011432-77848.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011432-85934.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-78023.gif)
方法3:(1)证明:因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-62315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-31637.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-20663.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-79424.gif)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-74114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-35616.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-95189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-23767.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-98003.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
(2)解:因为点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-65813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-42238.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-25995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-84378.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-33631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
设圆
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011418-21623.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-39261.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-38489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-94980.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011421-43583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-97488.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-43909.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011422-96644.gif)
以点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-19531.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-11600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-17285.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-48021.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011426-96580.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-50868.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-45509.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-40490.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011427-47512.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-44228.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-12920.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-64846.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-98760.gif)
…………………………9分
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011429-74574.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-73198.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-87345.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-41647.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011429-74915.gif)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-98587.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-47885.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011430-73198.gif)
由(1)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-81024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-74114.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011431-31734.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011420-95189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011428-58645.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011423-89930.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011432-33011.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011432-77848.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011432-85934.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011419-53809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011425-78023.gif)
解析
核心考点
试题【(本小题满分14分)一个几何体是由圆柱和三棱锥组合而成,点、、在圆的圆周上,其正(主)视图、侧(左)视图的面积分别为10和12,如图3所示,其中,,,.(1)求】;主要考察你对平面的法向量等知识点的理解。[详细]
举一反三
如图8,在直角梯形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011357-65164.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-48296.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-95339.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-56573.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-45104.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-46628.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-45104.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-46628.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-46628.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011357-65164.gif)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-91431.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-67373.gif)
(2)求平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011357-65164.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011358-78947.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011359-50809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011340-55367.jpg)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011340-65841.gif)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011340-90291.gif)
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011341-48853.gif)
(4)求CB1与平面A1ABB1所成的角的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011333-15059.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011333-88260.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011333-93207.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011333-24135.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011334-49957.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011334-10621.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011334-13860.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
四、附加题:本大题共2小题,每小题10分,共20分。
(20)(本小题满分10分)
已知
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011322-99862.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011323-31962.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011323-62340.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011323-16409.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011324-48111.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011324-38712.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011324-37518.gif)
(I)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011324-95297.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011325-79586.gif)
(II)设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011325-99995.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011325-62936.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011326-63341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011326-64504.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011326-26807.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011326-64504.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106011317-31032.gif)
(1)求证AC⊥平面DEF;
(2)若M为BD的中点,问AC上是否存在一点N,使MN∥平面DEF?若存在,说明点N的位置;若不存在,试说明理由.
(3)求平面ABD与平面DEF所成锐二面角的余弦值。
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