题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130423-11328.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130423-87613.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130423-59293.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-10984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-59557.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-43122.jpg)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-31351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-70302.png)
(2)求四面体
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-81821.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-54215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130424-38277.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-95509.png)
解析
试题分析:(1)设正方形ABCD的中心为O,取BE中点G,连接FG,OG,由中位线定理,我们易得四边形AFGO是平行四边形,即FG∥OA,由直线与平面平行的判定定理即可得到AC∥平面BEF;
(2)由已知中正方形ABCD与直角梯形ADEF所在平面互相垂直,∠ADE=90°,我们可以得到AB⊥平面ADEF,结合DE=DA=2AF=2.分别计算棱锥的底面面积和高,代入棱锥体积公式即可求出四面体BDEF的体积.(1)的关键是证明出FG∥OA,(2)的关键是得到AB⊥平面ADEF,即四面体BDEF的高为AB.
试题解析:(1)证明:设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-16069.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-38526.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-38876.png)
连结
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-91343.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-66528.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-60721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-19474.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-43637.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-20366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-16217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-60721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-66528.png)
从而四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-65966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-53045.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-30780.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-58688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-57458.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-58688.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-96565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130426-58688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-50649.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-47031.png)
(2)解:因为平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-92571.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-51977.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130427-34644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130428-51865.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130428-52962.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130428-84202.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130425-43637.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130428-84987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130428-74313.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130429-54504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130429-66809.png)
所以四面体
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130429-28529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130429-82782.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130430-62318.png)
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130410-95511.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130410-68134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130411-28612.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130411-11946.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130411-91040.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130401-18157.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130401-31688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130401-61534.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130401-31688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130401-42217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130402-57489.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130402-41963.png)
求证:(1)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130402-90285.png)
(2)求三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130402-93172.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130343-10970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130343-38801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130343-31048.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130344-35384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130344-34689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130344-35344.jpg)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130344-63206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130345-14607.png)
(2)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130344-63206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130345-32699.png)
(3)求三棱锥的体积
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130345-24617.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130335-64999.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130336-93059.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130336-44924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130302-91970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130303-17350.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130303-93939.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130303-22262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130304-65503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130304-72839.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130304-79304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130304-84228.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130304-49555.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130305-68205.png)
(1)求三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130302-91970.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021130305-32709.png)
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