如圖,在邊長(zhǎng)為4的菱形
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中,
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.點(diǎn)
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分別在邊
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上,點(diǎn)
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與點(diǎn)
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不重合,
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.沿
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將
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翻折到
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的位置,使平面
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平面
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.
(1)求證:
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平面
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;
(2)設(shè)點(diǎn)
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滿足
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,試探究:當(dāng)
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取得最小值時(shí),直線
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與平面
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所成角的大小是否一定大于
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?并說明理由.
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(1)證明:∵ 菱形
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的對(duì)角線互相垂直,∴
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,∴
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,
∵
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,∴
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.
∵ 平面
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⊥平面
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,平面
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平面
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,且
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平面
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,
∴
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平面
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, ∵
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平面
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,∴
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……………4分
(2)如圖,以
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為原點(diǎn),建立空間直角坐標(biāo)系
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.
設(shè)
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因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823211249068636.png" style="vertical-align:middle;" />,所以
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為等邊三角形,
故
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,
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.又設(shè)
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,則
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,
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.
所以
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,
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,
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,
故
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,
所以
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,
當(dāng)
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時(shí),
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.此時(shí)
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,………………………………6分
設(shè)點(diǎn)
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的坐標(biāo)為
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,由(1)知,
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,則
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,
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,
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,
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.所以
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,
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,
∵
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, ∴
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.
∴
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,∴
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. 10分
設(shè)平面
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的法向量為
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,則
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.
∵
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,
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,∴
取
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,解得:
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, 所以
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.……………………………… 8分
設(shè)直線
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與平面
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所成的角
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,
∴
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.……………………………………………… 10分
又∵
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∴
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. ∵
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,∴
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.
因此直線
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與平面
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所成的角大于
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,即結(jié)論成立
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如圖,在直四棱柱
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ABCD為等腰梯形,
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如圖1, 在直角梯形

中,

,

,

,

為線段

的中點(diǎn). 將

沿
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折起,使平面
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平面
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,得到幾何體
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,如圖2所示.
(1)求證:
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平面

;
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的余弦值.
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科目:高中數(shù)學(xué)
來源:不詳
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(本大題12分)如圖,在棱長(zhǎng)為ɑ的正方體ABCD-A
1B
1C
1D
1中,E、F、G分別是CB、CD、CC
1的中點(diǎn).
(1)求直線
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C與平面ABCD所成角的正弦的值;
(2)求證:平面A B
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1∥平面EFG;
(3)求證:平面AA
1C⊥面EFG .
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,正方形AA
1D
1D與矩形ABCD所在平面互相垂直,AB=2AD=2,點(diǎn)E為AB上一點(diǎn)
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(I) 當(dāng)點(diǎn)E為AB的中點(diǎn)時(shí),求證;BD
1//平面A
1DE
(II)求點(diǎn)A
1到平面BDD
1的距離;
(III) 當(dāng)
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時(shí),求二面角D
1-EC-D的大小.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,點(diǎn)P是正方形ABCD外一點(diǎn),PA
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平面ABCD,PA=AB=2,且E、F分別是AB、PC的中點(diǎn).
(1)求證:EF//平面PAD;
(2)求證:EF
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平面PCD;
(3)求:直線BD與平面EFC所成角的大小.
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
已知
ABCD是正方形,
PA⊥平面
ABCD,且
PA=AB=2,
E、
F是側(cè)棱
PD、
PC的中點(diǎn)。
(1)求證:
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平面
PAB;
(2)求直線
PC與底面
ABCD所成角
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的正切值。
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科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,四棱錐P-ABCD的底面ABCD為矩形,且PA="AD=1,AB=2,"
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,
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.
(1)求證:平面
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平面
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;
(2)求三棱錐D-PAC的體積;
(3)求直線PC與平面ABCD所成角的正弦值.
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
正方體
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的棱長(zhǎng)為1,
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是
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的中點(diǎn),則
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是平面
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的距離是( �。�
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