2010·河北·一模
真题
解题方法
1 . 在正三角形
中,
、
、
分别是
、
、
边上的点,满足
(如图1),将三角形
沿
折起到
的位置,使二面角
成直二面角,连结
(如图2)
![](https://img.xkw.com/dksih/QBM/2010/4/23/1569702465716224/1569702471057408/STEM/d2d6d525-cba5-4d34-b69b-9aba5c099ab1.png?resizew=353)
(Ⅰ)求证:
平面
;
(Ⅱ)求直线
与平面
所成角的大小;
(Ⅲ)求二面角
的大小(用反三角函数表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c06ee44206d4e110610bc412f11f2ab5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb2d555062f34d5a74f6d47da4ea8888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc14ed237a4bcc35cbd1f5f1321b3718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35b09d684cdab04c0b8960952a27553b.png)
![](https://img.xkw.com/dksih/QBM/2010/4/23/1569702465716224/1569702471057408/STEM/d2d6d525-cba5-4d34-b69b-9aba5c099ab1.png?resizew=353)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4525c00ed908bed8ba8d353e747a858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24fac8b38a9cf7602391f6d6ca933bd2.png)
(Ⅱ)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1b066f8869d0ff4513f7a99745125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d808a1351940a41a2ba27ab26d7fc680.png)
(Ⅲ)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fd0d9f130f3b43e745d99090cfa1b2e.png)
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2010·河南·一模
2 . 已知如图(1),正三角形
的边长为
,
是
边上的高,
、
分别是
和
边上的点,且满足
,现将
沿
翻折成直二面角
,如图(2).
(Ⅰ) 试判断翻折后直线
与平面
的位置关系,并说明理由;
(Ⅱ) 求二面角
的平面角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878e89b6eca35e34c863e832a2c661db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be505a5cbc0bd3abc570d440979c4d12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6dfb039aafe2d2841f8c28b117cf741.png)
(Ⅰ) 试判断翻折后直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
(Ⅱ) 求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c909cd1b6f3fa1ec39eb245e8f5c11c.png)
![](https://img.xkw.com/dksih/QBM/2010/4/7/1569694219395072/1569694271168512/STEM/2e7434b9-a568-4611-af37-322477eb536d.png?resizew=264)
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