试题分析:(1)连结OC,由OC=OB得∠2=∠B,DQ=DC得∠1=∠Q,根据QP⊥PB得到∠Q+∠B=90°,则∠1+∠2=90°,再利用平角的定义得到∠DCO=90°,然后根据切线的判定定理得到CD为⊙O的切线;
(2)连结AC,由AB为⊙O的直径得∠ACB=90°,根据余弦的定义得cosB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317536561010.png)
,可计算出BC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753672414.png)
,在Rt△BPQ中,利用余弦的定义得cosB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753687680.png)
,可计算出BQ=10,然后利用QC=BQ-BC进行计算即可.
试题解析:证明:连接
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753703379.png)
,
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753719489.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753719560.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753734550.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317537503328.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753765620.png)
.
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753781581.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753797692.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753812726.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753828858.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317538431572.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753859506.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753703379.png)
是⊙
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753875291.png)
的半径,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753890377.png)
是⊙
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753875291.png)
的切线.
(2)连接
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753921406.png)
,
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753921674.png)
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317539371106.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317539531049.png)
,
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753968391.png)
是⊙
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031753875291.png)
的直径,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031754093647.png)
.
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823031754109634.png)
中,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317541241798.png)
.
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230317541401107.png)
.
考点: 1.切线的判定;2.解直角三角形.