試題分析:(Ⅰ)證明線面平行,關(guān)鍵在于找出線線平行.本題條件含中點(diǎn),故從中位線上找線線平行.
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,
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分別為
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,
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中點(diǎn),在△
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中,
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是
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中點(diǎn),
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是
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中點(diǎn),所以
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∥
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.又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045712214462.png" style="vertical-align:middle;" />平面
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,
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平面
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,所以
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∥平面
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.(Ⅱ)求二面角的大小,有兩個(gè)思路,一是作出二面角的平面角,這要用到三垂線定理及其逆定理,利用側(cè)面
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底面
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,可得底面
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的垂線,再作DF的垂線,就可得二面角的平面角,二是利用空間向量求出大小.首先建立空間坐標(biāo)系. 取
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中點(diǎn)
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.由側(cè)面
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底面
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易得
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面
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.以
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為原點(diǎn),
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分別為
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軸建立空間直角坐標(biāo)系.再利用兩平面法向量的夾角與二面角的平面角的關(guān)系,求出結(jié)果,(Ⅲ)存在性問(wèn)題,一般從假設(shè)存在出發(fā),構(gòu)造等量關(guān)系,將存在是否轉(zhuǎn)化為方程是否有解.
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證明:(Ⅰ)如圖,連結(jié)
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.
因?yàn)榈酌?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711684526.png" style="vertical-align:middle;" />是正方形,
所以
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與
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互相平分.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711762302.png" style="vertical-align:middle;" />是
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中點(diǎn),
所以
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是
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中點(diǎn).
在△
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中,
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是
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中點(diǎn),
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是
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中點(diǎn),
所以
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∥
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.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045712214462.png" style="vertical-align:middle;" />平面
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,
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平面
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,
所以
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∥平面
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. 4分
(Ⅱ)取
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中點(diǎn)
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.在△
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中,因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045712948492.png" style="vertical-align:middle;" />,
所以
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.
因?yàn)槊?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711700468.png" style="vertical-align:middle;" />底面
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,
且面
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面
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,
所以
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面
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.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045713072388.png" style="vertical-align:middle;" />
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平面
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所以
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.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711762302.png" style="vertical-align:middle;" />是
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中點(diǎn),
所以
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.
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如圖,以
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為原點(diǎn),
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分別為
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軸建立空間直角坐標(biāo)系.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711809642.png" style="vertical-align:middle;" />,所以
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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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,
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,
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.
于是
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,
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,
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.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045713431391.png" style="vertical-align:middle;" />面
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,所以
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是平面
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的一個(gè)法向量.
設(shè)平面
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的一個(gè)法向量是
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.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045713525984.png" style="vertical-align:middle;" />所以
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即
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令
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則
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.
所以
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.
由圖可知,二面角
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為銳角,所以二面角
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的余弦值為
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. 10分
(Ⅲ)假設(shè)在棱
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上存在一點(diǎn)
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,使
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面
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.設(shè)
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,
則
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. 由(Ⅱ)可知平面
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的一個(gè)法向量是
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.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045711918435.png" style="vertical-align:middle;" />面
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,所以
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.
于是,
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,即
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.
又因?yàn)辄c(diǎn)
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在棱
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上,所以
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與
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共線.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824045713930751.png" style="vertical-align:middle;" />,
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,
所以
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.
所以
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,無(wú)解.
故在棱
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上不存在一點(diǎn)
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,使
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面
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成立. 14分