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    精英家教网 > 初中英语 > 题目详情
    如图，在四棱锥P－ABCD中，底面ABCD是菱形，∠BAD＝，AB＝2，PA＝1，PA⊥平面ABCD，E是PC的中点，F是AB的中点．

    （1）求证：BE∥平面PDF；
    （2）求证：平面PDF⊥平面PAB；
    （3）求三棱锥P－DEF的体积．
    ⑴略；⑵略；⑶解析:
    （1）取PD的中点为M，连结ME，MF，因为E是PC的中点，所以ME是△PCD的中位线．所以ME∥CD，ME＝．又因为F是AB的中点，且由于ABCD是菱形，AB∥CD，AB＝CD，所以ME∥FB，且ME＝FB．所以四边形MEBF是平行四边形，所以BE∥MF．
    连结BD，因为BE平面PDF，MF平面PDF，所以BE∥平面PDF．
    （2）因为PA⊥平面ABCD，DF平面ABCD，所以DF⊥PA．
    连结BD，因为底面ABCD是菱形，∠BAD＝，所以△DAB为正三角形．
    因为F是AB的中点，所以DF⊥AB．
    因为PA，AB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E是PC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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