题目内容

如图,在四棱锥PABCD中,底面ABCD是菱形,∠BADAB=2,PA=1,PA⊥平面ABCDEPC的中点,FAB的中点.

(1)求证:BE∥平面PDF
(2)求证:平面PDF⊥平面PAB
(3)求三棱锥PDEF的体积.
⑴略;⑵略;⑶解析:
(1)取PD的中点为M,连结MEMF,因为EPC的中点,所以ME是△PCD的中位线.所以MECDME.又因为FAB的中点,且由于ABCD是菱形,ABCDABCD,所以MEFB,且MEFB.所以四边形MEBF是平行四边形,所以BEMF
连结BD,因为BE平面PDFMF平面PDF,所以BE∥平面PDF
(2)因为PA⊥平面ABCDDF平面ABCD,所以DFPA
连结BD,因为底面ABCD是菱形,∠BAD,所以△DAB为正三角形.
因为FAB的中点,所以DFAB
因为PAAB是平面PAB内的两条相交直线,所以DF⊥平面PAB
因为DF平面PDF,所以平面PDF⊥平面PAB
(3)因为EPC的中点,所以点P到平面EFD的距离与点C到平面EFD的距离相等,故,又×2×E到平面DFC的距离h,所以××
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【小题1】Alexandra became well-known all over the world for her paintings at the age of _______.
A.18monthsB.threeC.eightD.ten
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【小题5】 From the passage, we can learn ________.
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【小题8】 A. visit                       B. go                    C. call
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