如圖,在底面為平行四邊形的四棱錐
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中,
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,
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平面
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,且
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,點
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是
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的中點.
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(1)求證:
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;
(2)求證:
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平面
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;
(3)求二面角
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的大小.
試題分析:(1)利用三垂線定理可證;(2)直線與平面平行的判定定理(Ⅲ)證
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,進而找出二面角的平面角
試題解析:(1)
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AB是PB在平面ABCD上的射影,
又
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AB
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AC,AC
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平面ABCD,
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AC
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PB.
(2)連接BD,與AC相交與O,連接EO,
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ABCD是平行四邊形
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O是BD的中點又E是PD的中點,
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EO
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PB.又PB
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平面AEC,EO
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平面AEC,
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PB
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平面AEC,
(3)如圖,取AD的中點F,連EF,F(xiàn)O,則
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EF是△PAD的中位線,
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EF
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PA又
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平面
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,
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同理FO是△ADC的中位線,
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FO
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AB
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FO^AC,由三垂線定理可知
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ÐEOF是二面角E-AC-D的平面角.又FO=
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AB=
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PA=EF。
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ÐEOF=45°而二面角
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與二面角E-AC-D互補,
故所求二面角
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的大小為135°.
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,則
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;
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,
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,
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;
(3)若
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,
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,則
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;
(4)若
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,
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,
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,
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.
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