(本題12分)如圖,斜三棱柱
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的底面是直角三角形,
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,點(diǎn)
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在底面
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上的射影恰好是
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的中點(diǎn),且
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.
(Ⅰ)求證:平面
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平面
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;
(Ⅱ)求證:
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;
(Ⅲ)求二面角
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的大小.
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(Ⅰ)證明:設(shè)
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的中點(diǎn)為
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.
在斜三棱柱
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中,點(diǎn)
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在底面
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上的射影恰好是
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的中點(diǎn),
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平面ABC. ……………………1分
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平面
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,
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. ……………………2分
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,
∴
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.
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,
∴
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平面
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. ……………………3分
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平面
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,
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平面
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平面
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. ………………4分
解法一:(Ⅱ)連接
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,
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平面
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,
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是直線
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在平面
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上的射影. ………………5分
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,
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四邊形
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是菱形.
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.
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. ……………6分
(Ⅲ)過(guò)點(diǎn)
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作
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交
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于點(diǎn)
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,連接
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,
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平面
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.
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.
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是二面角
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的平面角. …………9分
設(shè)
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,則
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,
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.
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.
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.
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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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.
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.
在
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中,可求
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.∵
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,∴
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.
∴
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.
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. ……………………………………10分
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.
∴二面角
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的大小為
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. ………………12分
解法二:(Ⅱ)因?yàn)辄c(diǎn)
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在底面
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上的射影是
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的中點(diǎn),設(shè)
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的中點(diǎn)為
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,則
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平面ABC.以
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為原點(diǎn),過(guò)
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平行于
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的直線為
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軸,
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所在直線為
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軸,
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所在直線為
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軸,建立如圖所示的空間直角坐標(biāo)系.
設(shè)
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,由題意可知,
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.設(shè)
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,由
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,得
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.
又
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.
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.
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. ……………………6分
(Ⅲ)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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.
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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. ……………………10分

二面角

的大小為

. ………………………………12分
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題12分)如圖,在長(zhǎng)方體

中,點(diǎn)

在棱

的延長(zhǎng)線上,且

.
下標(biāo)
(1)求證:

∥平面

;
(2)求證:平面


平面

;
(3)求四面體

的體積.
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分12分)
已知四棱錐

的三視圖如下圖所示,其中主視圖、側(cè)視圖是直角三角形,俯視圖是有一條對(duì)角線的正方形.

是側(cè)棱

上的動(dòng)點(diǎn).
(1)求證:

(2)若五點(diǎn)

在同一球面上,求該球的體積.
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(12分)如圖,在棱長(zhǎng)為2的正方體
ABCD -A1B1C1D1中,
E、F分別為
A1D1和
CC1 的中點(diǎn).

(1)求證:
EF∥平面
ACD1;
(2)求三棱錐
E-ACD1的體積與正方體
ABCD -A1B1C1D1的體積之比.
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:解答題
(本小題滿分14分

)
如圖,已知正三棱柱

的底面邊長(zhǎng)是

,

、E是

、BC的中點(diǎn),AE=DE

(1)求此正三棱柱的側(cè)棱長(zhǎng);
(2)求正三棱柱

表面積.
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
如圖,正三棱錐
A-
BCD中,

在棱

上,

在棱

上.并且

(0<
l<+∞),設(shè)
a為異面直線

與

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(lái)源:不詳
題型:單選題
若a、b是異面直線,

、

是兩個(gè)不同平面,

,則( )
A.l與a、b分別相交 |
B.l與a、b都不相交 |
C.l至多與a、b中一條相交 |
D.l至少與a、b中的一條相交 |
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
已知二面角

的大小為

,

為空間中任意一點(diǎn),則過(guò)點(diǎn)

且與平面

和平面

所成的角都是

的直線的條數(shù)為( )
A.1 | B.2 | C.3 | D.4 |
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科目:高中數(shù)學(xué)
來(lái)源:不詳
題型:單選題
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