题干

如图,边长为2的正方形ABCD中,点E在AB边上,点F在BC边上,

(Ⅰ)若点E是AB的中点,点F是BC的中点,将△AED,△DCF分别沿DE,DF折起,使A,C两点重合于点A′.求证:A′D⊥EF.

(Ⅱ)当BE=BF=

1
4
BC时,求三棱锥A′﹣EFD的体积.
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答案(点此获取答案解析)

解:(Ⅰ)证明:∵边长为2的正方形ABCD中,E、F是AB、BC的中点,将△AED、△DCF分别沿DE、DF折起,使AC两点重合于点A′,

∴A'D⊥A'E,A'D⊥A'F,A'E∩A'F=A',

∴A'D⊥面A'EF,

∵EF⊂面A'EF,∴A'D⊥EF;

同类题1

阅读理解

    Your New Year's plan to move more is one of the toughest to stick to, and a group of scientists working with obese(过度肥胖的) mice think they're starting to understand why

    Rather than our sedentary(久坐的) lives causing weight gain, says Alexxai Kraviz, the National Institutes of Health neuroscientist who led the study, changes in brain chemistry after we start gaining weight affect our capacity to move.

    “Obses mice can move just fine,” says Kravitz, who published the work with his team in Cell Metabolism “They just don't.”

    What Kravitz's team found is that the activity of a particular dopamine(多巴胺) receptor linked to movement goes down as mice gain weight on a high-fat diet. So the nice slow down and they move less. And when the researchers restored the activity of that dopamine receptor—DR2—the mice started moving more, even though they were still obese. The team also saw that lean mice missing the DR2 receptor acted like obese mice. This is the target, says Kravitz—restoring that dopamine receptor function. “Maybe 20 or 30 years down the road, we could do that in people,” he says.

    And there is one more thing: The scientists fed normal mice and the mice lacking DR2 the same high-fat diet. Both sets of mice gained weight at the same rate.

    Kravitz says this is important because mice lacking DR2 move less from the get-go, whereas a normal mouse takes a little time to start seeing that dopamine receptor-related loss of exercise. The ability to exercise seems to be disconnected from weight gain, he says.

    “Exercise is a healthy thing to do, but its impact on weight loss has been exaggerated,” he says. “We have to be realistic about the size of the effect of exercise on weight, as opposed to health benefits.”

    Still, before you abandon your New Year's exercise plan, keep in mind that this study was done using a high-fat diet, and not the normal calorie restriction that people maintain when they diet.

    That's a big drawback to the work, says Vicki Vieira-Potter, a University of Missouri physiologist not involved in the study.

    “They feed the mice with high-fat diet, it damages the receptor, and that decreases activity. Those who plan to lost weight should remember high-fat diet is a nice way to cause obesity in the lab, but it's not the same as the normal situation of obesity,” she says.

    Sine also says that a lot of the weight gain in the mice came after they stopped moving around, which indicates that the loss of movement did impact obesity.

同类题3

阅读下面的文字,完成下面各题。

秋来查干湖(李旭光)

    ①查干湖美,美在秋天。秋湖的长天,极为疏朗。站在环湖路上,依稀看得见天宇的穹顶与湖水交割的弧线。惊鸿照影,间或掠过几朵祥云。夜幕乍落,伫立在湖岸高坡上的妙因寺,不时传出阵阵晚课的钟鼓和诵经的吟唱,剪影里,空气也凝固了一般。到了晓夜当空,繁星拱月时,天上湖中,已辨不出哪里是真实的,哪里是虚幻的。

    ②查干湖美,美在秋水。秋水如娴静而又落落大方的少妇,深邃的眼睛定定地凝视着你。从松花江引入湖中的百里运河,已放慢了流速,使得这时的湖水波澜不惊,清澈见底。泥沙已经淘尽,水中的游鱼和水草分明可见。夏日肥得如油的湖水,已经显得清瘦,是湖水中的鱼虾和健硕的蒲苇把它的营养吸吮殆尽。湖水从暑热变为秋温,鱼儿多半向湖的深处游去,垂钓的人们便懒于来此蹲守。随着秋一天天走向肃穆,湖区的游人也少了许多,但仍有不少老人、情侣、画家、摄影家、迁客骚人来此小住。与其说是游湖,莫若说是人生的一次悄然回眸。

    ③查干湖美,美在秋草。轻露染过的蒲草依然葱郁滴翠,很深的一截还在水里,蒲棒却透出橙黄。渔民介绍,蒲棒采摘下来,可以作为驱赶蚊蝇的香炷,一支蒲棒,可以燃上几介小时。有些地方用蒲草做床垫,铺在下面厚厚的、暖暖的,还散发着蒲香,销路很被看好。秋苇比起蒲草来,要更美些,更张扬些,更浪漫些。一枝枝、一簇簇,泛着轻黄的舒展的苇叶,就像跳藏族舞的姑娘,张开双臂,弓起腰肢,献出哈达。而泛着油光的银白色的芦苇花,在风中柔软地散落开,烈火呼啸般地狂舞。修长的苇秆,接踵摩肩,竹林般地森然列阵。待到湖上结冰时,这些蒲草和芦苇都要被渔民割掉。割去固然可惜,但为了将一腔热血回报养育它们的大地,为了来年春天新生命的再一次萌发,想一想,也就释然了。湖中还有铺天盖地的菱角与荷花。花期过后,尚有一片片圆得可人的叶子浮在湖面,新绿里透着油一般的光泽,圆叶的下面,是一枝细蔓连到湖床。这时,肥硕的菱角,黑黝黝的已经成熟。记得小时候开那达慕大会时,总有一些小商贩,将菱角煮熟后对半切开,用粗瓷碗盛着叫卖。孩子们用菱角那略朝里弯曲的锋利的尖儿,把果肉从菱角壳里抠出来,送到嘴里,吃起来真香。荷花的莲蓬籽,连同泥里鲜嫩稚白的藕都已成熟,渔民们却不舍得去捡拾,任由它们在湖中自在地散落,肆意地蔓生。

    ④秋天的查干湖,是水鸟的王国。从8月底到10月初,在北国圣水中完成季节性侨居、顺便又繁衍后代的候鸟,几十万只之众,从四面八方向湖中翔集。对水禽颇有研究的鲍鹏友局长告诉我,这里面有天鹅,有丹顶鹤、白鹤、灰鹤、苍鹭、白鹳,有大雁以及其他近百种水禽,其中,光是野鸭,就有针尾鸭、绿头鸭、赤颈鸭等20多种。一时间,湖中用来接待“客人”的蒲苇显得容量不足,鸟儿们吵闹着、争抢着、嬉戏着,一边庆幸着对圣湖的占领,一边抓紧时间在这里捕鱼啄虾,补充能量,同时讨论着生活琐事,谋划着南归的路线。

    ⑤秋天,湖中的鱼也肥了。南湖的水草茂密,草根鱼、青根鱼长得特别快。夏天到南湖垂钓,听到“咔哧”“咔哧”,那是草根鱼进食的声音,“咯嘣”“咯嘣”的声响则是青根鱼的。青根鱼被渔民归类为牙鱼,它们非常爱吃田螺、河蚌等贝类,因此青根鱼的牙床后面有个冲压床一样的凹凸槽,骨头自然是极硬的。渔民在食用青根鱼前,都要先把这块骨头取出来,做成小饰件。青根鱼的肉雪白、细腻,口感极爽,是湖中的上佳之品。每年冬捕时,南湖的青根鱼一出水就极畅销,大的四五十斤重,一两米长,一斤就要卖到100多元钱。在北湖,鲢鱼、鳙鱼、鲤鱼、湖虾,俱为上佳之品。尤其是北湖的鳙鱼,俗称胖头鱼,是经过国家认证的绿色有机食品。冬捕时,湖面上人潮涌动,车如穿梭,冰晶雪国里蒸腾着节庆的热浪。

    ⑥秋天的查干湖,最美的还是渔民。春天投放鱼苗,夏天管护,到了秋天,终于眼见着湖里的鱼肥了、大了,将整个湖都胀满了,渔民便美在了心里。他们这时开始收拾冬网,做着冬捕的准备。干活累了,几杯酒落肚儿,他们在一起海吹神聊,什么泥鳅用皮肤、肠道呼吸啦,胡罗鱼要在河蚌里产卵啦,老头鱼会冬眠啦,更别有一番情趣。

(选自《2011年中国散文精选》)