Question

如圖，正四棱柱中，，點(diǎn)在上．
（1）證明：平面；（2）求二面角的大小．

Answer 1

解法一：                                   ，　           
依題設(shè)知，．
（Ⅰ）連結(jié)交于點(diǎn)，則．
由三垂線(xiàn)定理知，．······························································· 3分
在平面內(nèi)，連結(jié)交于點(diǎn)，
由于，
故，，
與互余．
于是．
與平面內(nèi)兩條相交直線(xiàn)都垂直，
所以平面．········································································· 6分
（Ⅱ）作，垂足為，連結(jié)．由三垂線(xiàn)定理知，
故是二面角的平面角．··············································· 8分
，
，．
，．
又，．
．
所以二面角的大小為．··············· 12分
解法二：
以為坐標(biāo)原點(diǎn)，射線(xiàn)為軸的正半軸，
建立如圖所示直角坐標(biāo)系．
依題設(shè)，．
，
．······························································· 3分
（Ⅰ）因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823124213687519.gif" style="vertical-align:middle;" />，，
故，．
又，
所以平面．········································································· 6分
（Ⅱ）設(shè)向量是平面的法向量，則
，．
故，．
令，則，，．·············································· 9分
等于二面角的平面角，
．
所以二面角的大小為．

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