已知:如圖,等腰直角三角形

的直角邊

,沿其中位線

將平面

折起,使平面

⊥平面

,得到四棱錐

,設(shè)
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、

、

、

的中點分別為
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、

、
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、

.



(1)求證:
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、

、
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、
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四點共面;
(2)求證:平面

平面

;
(3)求異面直線

與

所成的角.
(1)見解析;(2)見解析;(3)

.
試題分析:(1)要證四點共面,只需找到一個平面,這四個點都在這個平面內(nèi),用確定平面的方法,兩條平行線確定一個平面,即可證出;(2)要證明兩個平面垂直,只需證明其中一個平面經(jīng)過另一個平面的一條垂線即可,也就是只需證線面垂直即可,而要證線面垂直,只需證明這條直線垂直平面內(nèi)的兩條相交直線,這樣,一步步尋找成立的條件即可;(3)求異面直線所成角,先平移兩條異面直線中的一條,使它們成為相交直線,則相交直線所成角就是異面直線所成角或其補角,再放入三角形中計算即可.
試題解析:(1)由條件有

為

的中位線,
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為梯形

的中位線
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
∥
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,
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∥
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四點共面 3分
(2)證明:由等腰直角三角形
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有
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,
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又
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,
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面
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又
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∥
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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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平面

6分
(3)由條件知
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延長
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到
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,使
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,連結(jié)

8分
則

,故

為平行四邊形 10分


,又
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


為異面直線BE與QM所成的角
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(或
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的補角) 11分


,且三線兩兩互相垂直
∴由勾股定理得

12分

ACR為正三角形,
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
=
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,
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異面直線

與
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所成的角大小為

13分.
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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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如圖,在四棱柱

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.
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(1)求證:
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(2)若
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為棱
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的中點,求證:
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平面
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.
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
已知m、n是兩條不同的直線,α、β是兩個不同的平面,給出下列命題:
①若
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,
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,則
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;②若
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,
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,且
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,則
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;③若
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,
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,則
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; ④若
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,
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,且
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,則
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.其中正確命題的序號是( )
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科目:高中數(shù)學(xué)
來源:不詳
題型:單選題
設(shè)
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表示直線
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表示不同的平面,則下列命題中正確的是( )
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