如圖,在直四棱柱
ABCD-
A1B1C
1D1中,底面
ABCD是梯形
BC∥
AD,∠
DAB=90°,
AB=
BB1=4,
BC=3,
AD=5,
AE=3,
F、
G分別為
CD、
C1D1的中點.
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(1)求證:
EF⊥平面
BB1G;
(2)求二面角
E-
BB1-
G的大�。�
(1)
略(2)
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(1)
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連接
FG ∵
F、
G分別為
CD、
C1D1的中點,
∴
FG
CC1 從而
FG
BB1∴
B、
B1、
F、
G四點共面.
連接
BF并延長與
AD的延長線交于點
H.
∵
F為
CD的中點,且
BC∥
A D.∴△
HFD
△
BFC ∴
DH=
BC=3
∴
EH=
DE+
DH=5. 又∵
BE=5,且
F為
BH的中點.
∴
EF⊥
BF,又∵
BB1⊥平面
ABCD,且
EF
平面
ABCD內(nèi).
∴
BB1⊥
EF ∴
EF⊥平面
BB1GF. 從而
EF⊥平面
BB1G.
(2)二面角
E-
BB1-
G的大小等于二面角
F-
BB1-
E的大小
∵
EF⊥平面
FBB1 且
EB⊥
BB1 FB⊥
BB1即∠
EBF為二面角
F-
BB1-
E的平面角
在△
EFB中,
EB=5,
EF=
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. ∴
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∴∠
EBF=
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∴二面角
E-
BB1-
G的大小為
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解法2:以
A為坐標原點,
AB為
x軸,
AA1為
y軸,
AD為
Z軸建立空間直角坐標系,
則
E(0,0,3)、
F(2,0,4)、
G(2,4,4)、
B(4,0,0)、
B1(4,4,0)
(1)
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、
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、
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∵
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,
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∴
EF⊥
BB1,
EF⊥
B1G ∴
EF⊥平面
BB1G(2)∵
EF⊥平面
BB1G ∴

為平面
BB1G的一個法向量
設(shè)平面
EBB1的一個

法向量為
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則
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解得
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,取
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∴
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
∴二面角
E-
BB1-
G的大小為

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
與平面

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

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如下圖所示,在等腰梯形

中,

為

邊上一點,


且

將

沿
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折起,使平面

⊥平面
.(1)求證:

⊥平面

;
(2)若

是側(cè)棱

中點,求截面

把幾何體分成的兩部分的體積之比。
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科目:高中數(shù)學(xué)
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題型:解答題

在三棱錐

中,

是邊長為

的正三角形,平面
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平面
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,
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,

、
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分別為
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、

的中點,
(1)證明:
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;
(2)求二面角

的大�。�
(3)求點

到平面

的距離.
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四面體ABCD中,共頂點A的三條棱兩兩相互垂直,且其長分別為

,若四面體的四個頂點同在一個球面上,則這個球的表面積為 。
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如圖,正三棱錐
A-
BCD中,
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在棱
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上,
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在棱
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上.并且
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(0<
l<+∞),設(shè)
a為異面直線
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與
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所成的角,
b 為異面直線
EF與
BD所成的角,則
a+
b的值是
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知二面角
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的大小為
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,
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為空間中任意一點,則過點
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且與平面
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和平面
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所成的角都是
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的直線的條數(shù)為( )
A.1 | B.2 | C.3 | D.4 |
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