题干

如图1,在高为2的梯形ABCD中,AB∥CD,AB=2,CD=5,过A、B分别作AE⊥CD,BF⊥CD,垂足分别为E、F.已知DE=1,将梯形ABCD沿AE、BF同侧折起,得空间几何体ADE﹣BCF,如图2.

(Ⅰ)若AF⊥BD,证明:△BDE为直角三角形;

(Ⅱ)若DE∥CF, CD=3 ,求平面ADC与平面ABFE所成角的余弦值.

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答案(点此获取答案解析)

解:证明:连接BE,

由已知可知四边形ABFE是正方形,∴AF⊥BE,

又AF⊥BD,BE∩DE=E,

∴AF⊥平面BDE,又DE⊂平面BDE,

∴AF⊥DE,

又DE⊥AE,AE∩AF=F,

∴DE⊥平面ABFE,又BE⊂平面ABFE,

∴DE⊥BE,即△BDE为直角三角形.

(Ⅱ)取CF的中点M,连结DM,则四边形DEFM是平行四边形,

∴DM=EF=2,CM=

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    A new study suggests that washing dishes by hand is healthier than using a dishwasher.

    Nobody likes doing the dishes, but it turns out that doing this task might pay off in an unexpected way. According to a new study published in the Journal of Pediatrics, washing dishes by hand instead of using a dishwasher might prevent the development of allergies (过敏).

    Researchers in Sweden surveyed the parents of 1,029 children aged 7 and 8. They discovered that children whose families hand-washed the dishes instead of using a machine were less likely to have allergies.

    Earlier research has shown that dishes washed by machine are cleaner than those washed by hand. So why would kids who eat with slightly dirtier plates be better when it comes to preventing allergies? One explanation is based on a theory known as the “hygiene hypothesis,” which says the reason why kids develop allergies is that their environment is actually too clean. Your immune system keeps you healthy by fighting germs like viruses and bacteria. But when you have allergies, it overreacts and tries to fight ordinary things like pollen(花粉) or certain foods.

    Being exposed to germs, especially early in life is good training for the immune system, says the lead author of the study, Dr. Bill Hesselmar of Queen Silvia Hospital in Sweden. “You stimulate the immune system in various ways and it becomes tolerant.”

    This study shows that while using the dishwasher might be easier, the old-fashioned method of cleaning up could be better for your health.

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