12.(全国Ⅱ20)(本小题满分12分)

如图,正四棱柱中,,点上且

(Ⅰ)证明:平面

(Ⅱ)求二面角的大小.

解法一:

依题设,

(Ⅰ)连结于点,则

由三垂线定理知,.····················································································· 3分

在平面内,连结于点

由于

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

是二面角的平面角.································································· 8分

所以二面角的大小为.··························································· 12分

解法二:

为坐标原点,射线轴的正半轴,

建立如图所示直角坐标系

依题设,

.··································· 3分

(Ⅰ)因为

所以平面.································································································ 6分

(Ⅱ)设向量是平面的法向量,则

,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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