Question

如图，正四棱柱中，，点在上．
（1）证明：平面；（2）求二面角的大小．

Answer 1

解法一：                                   ，　           
依题设知，．
（Ⅰ）连结交于点，则．
由三垂线定理知，．······························································· 3分
在平面内，连结交于点，
由于，
故，，
与互余．
于是．
与平面内两条相交直线都垂直，
所以平面．········································································· 6分
（Ⅱ）作，垂足为，连结．由三垂线定理知，
故是二面角的平面角．··············································· 8分
，
，．
，．
又，．
．
所以二面角的大小为．··············· 12分
解法二：
以为坐标原点，射线为轴的正半轴，
建立如图所示直角坐标系．
依题设，．
，
．······························································· 3分
（Ⅰ）因为，，
故，．
又，
所以平面．········································································· 6分
（Ⅱ）设向量是平面的法向量，则
，．
故，．
令，则，，．·············································· 9分
等于二面角的平面角，
．
所以二面角的大小为．

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