(本小題主要考查空間線線、線面關(guān)系,二面角,三視圖等知識(shí),考查化歸與轉(zhuǎn)化數(shù)學(xué)思想方法,以及空間想象能力、推理論證能力、運(yùn)算求解能力.)
方法1:(1)證明:因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175252569536.gif" style="vertical-align:middle;" />,

,所以

,即

.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175252959317.gif" style="vertical-align:middle;" />,

,所以

平面

.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175253021450.gif" style="vertical-align:middle;" />,所以

.………………………………………………………………4分
(2)解:因?yàn)辄c(diǎn)

、

、

在圓

的圓周上,且

,所以

為圓

的直徑.
設(shè)圓

的半徑為

,圓柱高為

,根據(jù)正(主)視圖、側(cè)(左)視圖的面積可得,

…………………………………………6分
解得


所以

,

.………………………………………………………………………7分
過點(diǎn)

作

于點(diǎn)

,連接

,
由(1)知,

,

,所以

平面

.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175254519266.gif" style="vertical-align:middle;" />平面

,所以

.
所以

為二面角

的平面角.…………………………………………………………9分
由(1)知,

平面

,

平面

,
所以

,即△
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為直角三角形.
在

△

中,

,

,則

.
由

,解得

.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175255720749.gif" style="vertical-align:middle;" />.…………………………………………………………………………13分
所以


.
所以二面角

的平面角大小為

.………………………………………………………14分
方法2:(1)證明:因?yàn)辄c(diǎn)

、

、

在圓

的圓周上,且

,所以

為圓

的直徑.
設(shè)圓

的半徑為

,圓柱高為

,根據(jù)正(主)視圖、側(cè)(左)視圖的面積可得,

…………………………………………2分
解得


所以

,

.………………………………………………………………………3分
以點(diǎn)

為原點(diǎn),

、

所在的射線分別為

軸、

軸建立如圖的空間直角坐標(biāo)系

,則

,

,

,

,

,

,

.
………………………5分
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175256344866.gif" style="vertical-align:middle;" />,
所以

.
所以

.…………………………………………………9分

(2)解:設(shè)

是平面

的法向量,因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175256438562.gif" style="vertical-align:middle;" />,
所以

即
取

,則

是平面

的一個(gè)法向量.……………………………………………11分
由(1)知,

,又

,

,所以

平面

.
所以

是平面

的一個(gè)法向量.……………………………………………………12分
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231752567961215.gif" style="vertical-align:middle;" />,
所以

.
而

等于二面角

的平面角,
所以二面角

的平面角大小為

.………………………………………………………14分
方法3:(1)證明:因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175252569536.gif" style="vertical-align:middle;" />,

,所以

,即

.
又因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175252959317.gif" style="vertical-align:middle;" />,

,所以

平面

.
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823175253021450.gif" style="vertical-align:middle;" />,
所以

.…………………………………………………………………………………………4分
(2)解:因?yàn)辄c(diǎn)

、

、

在圓

的圓周上,且

,所以

為圓

的直徑.
設(shè)圓

的半徑為

,圓柱高為

,根據(jù)正(主)視圖、側(cè)(左)視圖的面積可得,

…………………………………………6分
解得


所以

,

.………………………………………………………………………7分
以點(diǎn)

為原點(diǎn),

、

所在的射線分別為

軸、

軸建立如圖的空間直角坐標(biāo)系

,則

,

,

,

,

,

,

.
…………………………9分
設(shè)

是平面

的法向量,
則

即

取

,則

是平面

的一個(gè)法向量.………11分
由(1)知,

,又

,

,
所以

平面

.
所以

是平面

的一個(gè)法向量.……………………………………………………12分
因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408231752580601216.gif" style="vertical-align:middle;" />,
所以

.
而

等于二面角

的平面角,
所以二面角

的平面角大小為

.………………………………………………………14分