题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005222-19887.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005222-87828.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-99361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-12160.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-62857.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-16044.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-93181.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005224-92729.png)
答案
解析
试题分析:因为在直三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005222-19887.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005222-87828.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-99361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-12160.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-62857.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-16044.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-93181.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-93181.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005223-16044.png)
试题解析:方法一:如图1以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005224-47122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005224-18736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005224-53073.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005225-17575.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005225-16829.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005225-70496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005225-66775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005226-65257.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005226-77664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005226-62264.png)
设平面A1BC1的一个法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005226-10436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005226-85781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005227-10232.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005227-53701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005227-80153.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005227-67018.png)
设AD与平面A1BC1所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005228-15205.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005228-25764.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005228-88468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005228-85890.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005228-30509.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005229-81604.png)
方法二:由题意知四边形AA1B1B是正方形,故AB1⊥BA1.
由AA1⊥平面A1B1C1得AA1⊥A1C1.
又A1C1⊥A1B1,所以A1C1⊥平面AA1B1B,故A1C1⊥AB1.
从而得 AB1⊥平面A1BC1. 4分
设AB1与A1B相交于点O,则点O是线段AB1的中点.
连接AC1,由题意知△AB1C1是正三角形.
由AD,C1O是△AB1C1的中线知:AD与C1O的交点为重心G,连接OG.
知AB1⊥平面A1BC1,故OG是AD在平面A1BC1上的射影,
于是∠AGO是AD与平面A1BC1所成的角. 6分
在直角△AOG中,AG=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005229-78859.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005229-40397.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005229-69433.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005230-78220.png)
所以sin∠AGO=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005230-42558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005230-32291.png)
故∠AGO=60°,即AD与平面A1BC1所成的角为60°. 12分
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005204-10255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005205-63750.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005205-57781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005205-65952.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005205-82591.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005205-19203.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005206-36576.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005206-31506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005206-51447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005206-99341.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005207-89538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005152-68558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005152-15059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005153-65977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005153-77944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005153-97824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005153-52054.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005154-80288.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005154-46211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005154-39084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005154-53288.png)
(2)当三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005154-72991.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005153-52054.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005142-84273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005142-98720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005142-14998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005142-54474.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005142-46529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005143-68730.png)
(1)求证:;
(2)求二面角B-AC-D的大小.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005127-80226.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005127-60903.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005127-61582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005127-69884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005128-11124.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005107-17167.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022005107-95488.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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