如圖,正四棱柱ABCD-A
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B
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C
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D
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中,底面邊長為2
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,側(cè)棱長為4,點E、F分別為棱AB、BC的中點,EF∩BD=G,求點D
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到平面B
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EF的距離d。
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如圖,建立空間直角坐標(biāo)系D-xyz。易得D
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(0,0,4),B
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(2
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,2
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,4),
E(2
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,
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,0),F(xiàn)(
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,2
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,0),
故
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=(-
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,
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,0),
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=(0,
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,4),
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=(2
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,2
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,0),
設(shè)
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=(x,y,z)是平面B
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EF的法向量,
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,令x=1,得
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=(1,1,-
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)。則|
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·
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|=4
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,∴d=
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。
故點D
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到平面EFB
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的距離為
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。
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(1)求證:
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