(本小題滿分12分)
如圖,

平面ABC,EB//DC,AC=BC=EB=2DC=2,

,P、Q分別為DE、AB的中點(diǎn)。

(Ⅰ)求證:PQ//平面ACD;
(Ⅱ)求幾何體B—ADE的體積;
(Ⅲ)求平面ADE與平面ABC所成銳二面角的正切值。
(Ⅰ)證明:取

的中點(diǎn)

,連接

,易證平面


又

………………………… (4分)
(Ⅱ)

…(6分)

……………………………………… (8分)
(Ⅲ)


…(10分)

………………………… (12分)
注:用向量法請對應(yīng)給分.
(法2)解:以

C為原點(diǎn),CA、CB、CD所在直線為x、y、z軸建立空間直角坐標(biāo)系C-xyz,則A(2,0,0)B(0,2,0)C(0,0,0)D(0,0,1)E(0,2,2)
則

設(shè)面ADE法向量為

則

可取

即面ADE與面ABC所成的二面角余弦值為

易得面ADE與面ABC所成二面角的正切值為

……………………(12分)
練習(xí)冊系列答案
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,棱錐P—ABCD的底面ABCD是矩形,PA⊥平面ABCD,PA=AD=2,BD=

.
(Ⅰ)求證:BD⊥平面PAC;
(Ⅱ)求點(diǎn)C到平面PBD的距離.

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S—ABCD的底面是邊長為1的正方形,SD垂直于底面ABCD,SB=

.

(Ⅰ)求面ASD與面BSC所成二面角的大小;
(Ⅱ)設(shè)棱SA的中點(diǎn)為M,求異面直線DM與
SB所成角的大小;
(Ⅲ)求點(diǎn)D到平面SBC的距離.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
正△

的邊長為4,

是

邊上的高,

分別是

和

邊的中點(diǎn),現(xiàn)將△

沿

翻折成直二面角

.
(1)試判斷直線

與平面

的位置關(guān)系,并

說明理由;
(2)求二面角

的余弦值;
(3)在線段

上是否存在一點(diǎn)

,使

?證明你的結(jié)論.
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:解答題
(12分)
如圖,正方形ABCD-A1B1C1D1中,E、F、G分別是AB,AD,AA1的中點(diǎn),

(1)求證AC1⊥平面EFG,
(2)求異面直線EF與CC1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
、如圖所示,棱長為1的正方體

中,

,

(1)建立適當(dāng)?shù)淖鴺?biāo)系,求M、N點(diǎn)的坐標(biāo)。(2)求

的長度。(12分)

查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
已知線段

面

,

,

,

面

于點(diǎn)

,

,且

在平面

的同側(cè),若

,則

的長為
查看答案和解析>>
科目:高中數(shù)學(xué)
來源:不詳
題型:填空題
在正方體

中,異面直線

與

的夾角的大小為__________
查看答案和解析>>