如圖,△
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106135473.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106150396.png)
是它的角平分線,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106166560.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106182315.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106213396.png)
邊上,以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106228368.png)
為直徑的半圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106228292.png)
經(jīng)過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106244318.png)
,交
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106260398.png)
于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106275302.png)
。
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231062914878.png)
(1)求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106306401.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106322373.png)
的切線;
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106338527.png)
,連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106353386.png)
,求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106353386.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106213396.png)
;
(3)在(2)的條件下,若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106416621.png)
,求圖中陰影部分的面積。
(1)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
,則可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106462521.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106478642.png)
,結(jié)合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106150396.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106509531.png)
的平分線,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106260398.png)
,再根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106166560.png)
即可證得結(jié)論;(2)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106556388.png)
,則可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231067431116.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106759683.png)
,即可得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106774675.png)
,從而證得結(jié)論;(3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106790493.png)
試題分析:(1)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
,則可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106462521.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106478642.png)
,結(jié)合
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106150396.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106509531.png)
的平分線,可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106260398.png)
,再根據(jù)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106166560.png)
即可證得結(jié)論;
(2)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106556388.png)
,則可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231069151122.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106759683.png)
,即可得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106774675.png)
,從而證得結(jié)論;
(3)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106353386.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106962374.png)
,可得S
△![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106977368.png)
= S
△![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106993377.png)
,即可得到S
陰影=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107008590.png)
,設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106322373.png)
的半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107040260.png)
,在Rt△
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107055473.png)
中,求得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107071385.png)
,即可求得結(jié)果.
(1)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231070864264.jpg)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106462521.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106478642.png)
,
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106150396.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106509531.png)
的平分線,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107164665.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107164695.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106431396.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106260398.png)
,
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106166560.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107274676.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106306401.png)
是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106322373.png)
的切線;
(2)連接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106556388.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231073364302.jpg)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231067431116.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106759683.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106774675.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/201408230231074141191.png)
,
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106353386.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106962374.png)
;
(3)∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106353386.png)
∥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106962374.png)
,
∴S
△![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106977368.png)
= S
△![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106993377.png)
,
∴S
陰影=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107008590.png)
設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106322373.png)
的半徑為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107040260.png)
,在Rt△
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107055473.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107071385.png)
,
∴S
陰影=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107008590.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023107586732.png)
=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140823/20140823023106790493.png)
點評:解答此類切線的判定的問題,一般是先連接切點和圓心,再證垂直.
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