题干

如图,在四棱锥P﹣ABCD中,PA⊥平面ABCD,AD∥BC,BC=2AD,PB⊥AC,Q是线段PB的中点.

(Ⅰ)求证:AB⊥平面PAC;

(Ⅱ)求证:AQ∥平面PCD.


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证明:(Ⅰ)∵PA⊥平面ABCD,AC,AB⊂平面ABCD,∴PA⊥AC,PA⊥AB,∵PB⊥AC,AP⊥AC,PA,PB⊂平面PAB,PA∩PB=P,∴AC⊥平面PAB,∵AB⊂平面PAB,∴AC⊥AB,PA⊥AB,PA,AC⊂平面PAC,PA∩AC=A;∴AB⊥平面PAC.(Ⅱ)取PC中点E,连结QE,ED,∵Q是线段PB的中点,E是PC的中点,∴QE∥BC,BC=2AD,∴QE∥AD,QE=AD,∴四边形AQED是平行四边形,∴AQ∥DE,∵AQ∥ED,ED⊂平面PCD,∴AQ∥平面PCD.

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    John Smith was a very handsome young man but he was lazy. He had finished his education and was happy to sit in his room and listen to the radio all day long. In fact, he had been doing nothing for almost six months, which worried his father a lot. One day Mr. Smith decided that he had to do something. “Johnny, When I was your age I was working and supporting my ten brothers and sisters. I want you to go out and get a job.” So that was what he did. In fact, in the next three months, Johnny started ten jobs and was fired from ten jobs. He explained to his father.

    “They wanted me to be at work at 8:00 a. m. and told me to do all sorts of unpleasant things.”

    “I don't care how you do it,” Mr. Smith said. “Either you earn some money or move out of the house. I gave you one week's time.”

    In the next few days Johnny began to change. He still stayed in his room all day, but spent his time writing letters and reading through the newspapers. More and more posts began to arrive for him. He bought a new suit for himself and invited his parents to the theatre and for dinner afterwards at the most expensive restaurant in town. When his proud parents arrived home after their evening hour, a policeman was waiting at the front door.

    “John Smith,” he said as he handed him a piece of paper, “I'll see you in the court tomorrow.”

    When he got outside, Johnny told his parents everything. “When you told me to earn some money, I decided to put an advertisement in the newspaper saying, 'New way to money fast! Send me $ 5 and I'll tell you my secret.' When I received the money I wrote back telling people to do as I do.”

    Johnny was fined $250 by the court and was ordered to pay all the people back. As he left the court house feeling very ashamed, a newspaper man came up to him, “Young man, would you like to tell your story to my newspaper for $2,500?”