Question

如圖，正四棱柱中，，點在上．
（1）證明：平面；（2）求二面角的大小．

Answer 1

解法一：                                   ，　           
依題設知，．
（Ⅰ）連結交于點，則．
由三垂線定理知，．······························································· 3分
在平面內，連結交于點，
由于，
故，，
與互余．
于是．
與平面內兩條相交直線都垂直，
所以平面．········································································· 6分
（Ⅱ）作，垂足為，連結．由三垂線定理知，
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：
以為坐標原點，射線為軸的正半軸，
建立如圖所示直角坐標系．
依題設，．
，
．······························································· 3分
（Ⅰ）因為，，
故，．
又，
所以平面．········································································· 6分
（Ⅱ）設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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