题干

正方形ABCD中,点O是对角线DB的中点,点P是DB所在直线上的一个动点,PE⊥BC于E,PF⊥DC于F.

(1)当点P与点O重合时(如图①),猜测AP与EF的数量及位置关系,并证明你的结论;

(2)当点P在线段DB上(不与点D、O、B重合)时(如图②),探究(1)中的结论是否成立?若成立,写出证明过程;若不成立,请说明理由;

(3)当点P在DB的长延长线上时,请将图③补充完整,并判断(1)中的结论是否成立?若成立,直接写出结论;若不成立,请写出相应的结论.

 

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答案(点此获取答案解析)

解:(1)AP=EF,AP⊥EF,理由如下:

连接AC,则AC必过点O,延长FO交AB于M;

∵OF⊥CD,OE⊥BC,且四边形ABCD是正方形,

∴四边形OECF是正方形,

∴OM=OF=OE=AM,

∵∠MAO=∠OFE=45°,∠AMO=∠EOF=90°,

∴△AMO≌△FOE(AAS),

∴AO=EF,且∠AOM=∠OFE=∠FOC=45°,即OC⊥EF,

故AP=EF,且AP⊥EF.

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阅读下列短文,从每题所给的A、B、C和D四个选项中,选出最佳选项。

    China has been drinking tea since the time of Shennong, 5,000 years ago. Britain's relationship with tea is much shorter, but tea enjoys the pride as the UK's national drink. According to the UK Tea Council, British people drink an average of three cups a day or a national total of 165 million cups every day.

    With figures like these, it is no surprise that time spent taking tea affects the working day in Britain. A recent survey found that 24 minutes a day are lost to making, buying and drinking tea and coffee. That is, £400 a year is lost in working hours per employee, or 190 days over a lifetime. So, should employers be worried about this lost working time, or does the tea break make up in other ways?

    One argument is that caffeine(咖啡因) improves mental state: a drink of tea or coffee can make you active and focus on work.

    Professor Rogers of the University of Bristol disagrees. After years of studying caffeine he sees nothing can prove that. “Workers would perform equally well if not drinking it at all,” he says. “But if they're often drinking it and then go without, they'll feel tired and won't perform well.”

    Occupational Psychologist Cooper instead emphasizes the role tea breaks play in sedentary (久坐的) office life, and in building social relationships. “We need to make people more active and see other people. The tea break is one way of doing this,” says Cooper.

    And Professor Rogers also points out the comfort effect of a hot drink: “We warm our hands on them on a cold day; they're comforting and play a big role in our everyday life. Whatever the caffeine's doing, I'd say these 24 minutes aren't wasted.”