题干

如图l,在正方形ABCD中,AB=2,E是AB边的中点,F是BC边上的一点,对角线AC分别交DE、DF于M、N两点.将ADAE,CDCF折起,使A、C重合于A点,构成如图2所示的几何体.

(I)求证:A′D⊥面A′EF;

(Ⅱ)试探究:在图1中,F在什么位置时,能使折起后的几何体中EF∥平面AMN,并给出证明.

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答案(点此获取答案解析)

证明:(Ⅰ)∵A′D⊥A′E,A′D⊥A′F,

又A′E∩A′F=A′,A′E⊂面A′EF,A′F⊂面A′EF,

∴A′D⊥面A′EF.

(Ⅱ)当点F为BC的中点时,EF∥面A′MN.

证明如下:当点F为BC的中点时,

在图(1)中,E,F分别是AB,BC的中点,

所以EF∥AC,

即在图(2)中有EF∥MN.

又EF⊄面A′MN,MN⊂面A′MN,

所以EF∥面A′MN.

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    When I spent the summer with my grandmother, she always set me down to the general store with a list. Behind the counter was a lady like no one I'd ever seen.

    “Excuse me,” I said. She looked up and said, “I'm Miss Bee.”

    “I need to get these.” I said, holding up my list. “So? Go get them. ” Miss Bee pointed to a sign. “There's no one here except you and me and I'm not your servant, so get yourself a basket from that pile.”

    I visited Miss Bee twice a week that summer. Sometimes she shortcharged me. Other times she overcharged. Going to the store was like going into battle. All summer long she found ways to trick me. No sooner had I learned how to pronounce “bicarbonate of soda” and memorized its location on the shelves than she made me hunt for it all over again. But by summer's end the shopping trip that had once taken me an hour was done in 15 minutes.

    “All right, little girl,” she said. “What did you learn this summer?” “That you're a meanie!” I replied. Miss Bee just laughed and said, “I know what you think of me. Well, I don't care! My job is to teach every child I meet life lessons. When you get older you'll be glad!” Glad I met Miss Bee? Ha! The idea was absurd…

    Until one day my daughter came to me with homework troubles. “It's too hard,” she said. “Could you finish my math problems for me?”

    “If I do it for you, how will you ever learn to do it yourself?” I said. Suddenly, I was back at that general store where I had learned the hard way to add up my bill by myself. Had I ever been overcharged since?