摘要：12． 如图.正四棱柱中..点在上且． (Ⅰ)证明:平面, (Ⅱ)求二面角的大小． 解法一: 依题设..． (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．·························································· 12分

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