(本小題滿分14分)

如圖,在四棱錐
E—
ABCD中,底面
ABCD為矩形,平面
ABCD⊥平面
ABE,∠
AEB=90°,
BE=
BC,
F為
CE的中點(diǎn),求證:
(1)
AE∥平面
BDF;
(2) 平面
BDF⊥平面
BCE.
本試題主要是考查了立體幾何中線面的平行的判定和面面垂直的證明的運(yùn)用。
(1)根據(jù)已知條件,底面ABCD為矩形,平面ABCD⊥平面ABE,∠AEB=90°,BE=BC,F(xiàn)為CE的中點(diǎn),設(shè)AC∩BD=G,連結(jié)FG,易知G是AC的中點(diǎn),
因?yàn)?F是EC中點(diǎn),所以在△ACE中,F(xiàn)G∥AE可知結(jié)論。
(2)因?yàn)?平面ABCD⊥平面ABE,BC⊥AB,
平面ABCD∩平面ABE=AB,所以 BC⊥平面ABE,從而得到BC⊥AE,再利用線面垂直得到面面垂直的判定。
證明:(1) 設(shè)AC∩BD=G,連結(jié)FG,易知G是AC的中點(diǎn),
因?yàn)?F是EC中點(diǎn),所以 在△ACE中,FG∥AE.………2分
因?yàn)?AE?平面BDF,FG?平面BDF,
所以 AE∥平面BDF. ………………………………………6分
(2) 因?yàn)?平面ABCD⊥平面ABE,BC⊥AB,
平面ABCD∩平面ABE=AB,所以 BC⊥平面ABE.………8分
因?yàn)?AE?平面ABE,所以 BC⊥AE.…………………………………………………………10分
又AE⊥BE,BC∩BE=B,所以 AE⊥平面BCE,又FG∥AE,
所以FG⊥平面BCE,……………………………………………………………………………12分
因?yàn)?FG?平面BDF,所以平面BDF⊥平面BCE.………………………………………………14分
練習(xí)冊(cè)系列答案
相關(guān)習(xí)題
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
如圖,四棱錐

的底面是正方形,

,點(diǎn)E在棱PB上.

(Ⅰ)求證:平面

;
(Ⅱ)當(dāng)

且

時(shí),求AE與平面PDB所成的角的正切值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題滿分12分)
已知

是矩形,

平面

,

,

,

為

的中點(diǎn).

(1)求證:

平面

;
(2)求直線

與平面

所成的角.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題滿分12分)
如圖,已知

,

分別是正方形

邊

、

的中點(diǎn),

與

交于點(diǎn)

,

、

都垂直于平面

,且

,

,

是線段

上一動(dòng)點(diǎn).

(Ⅰ)求證:平面

平面

;
(Ⅱ)試確定點(diǎn)

的位置,使得

平面

;
(Ⅲ)當(dāng)

是

中點(diǎn)時(shí),求二面角

的余弦值.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(本小題滿分12分)
如圖,四棱錐

的底面為正方形,側(cè)棱

底面

,且

,

分別是線段

的中點(diǎn).

(Ⅰ)求證:

//平面

;
(Ⅱ)求證:

平面

;
(Ⅲ)求二面角

的大�。�
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
已知直線

,平面

,且

,

,給出下列四個(gè)命題:
①若

∥

,則

;②若

,則

∥

;
③若

,則

∥

;④若

∥

,則

;
其中為真命題的序號(hào)是_______
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知a、b是不重合的兩個(gè)平面,
m、
n是直線,下列命題中不正確的是( )
A.若m∥n,m^a,則n^a | B.若m^a,mÌb,則a^b |
C.若m^a,a∥b,則m^b | D.若a^b,mÌa,則m^b |
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
如果直線l,m與平面α、β、γ滿足β∩γ=l,,

,

,那么必有( �。�
A.m//β且l⊥m | B.α//β且α⊥γ |
C.α⊥β且m//γ | D.α⊥γ且l⊥m |
查看答案和解析>>