题目内容
(2003•安徽)如图是五角星,已知AC=a,求五角星外接圆的直径(结果用含三角函数的式子表示).![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_ST/images0.png)
【答案】分析:连AO,并延长交圆O于点F,连接CF.根据直径所对的圆周角是直角,得到直角三角形ACF.根据锐角三角函数进行求解.
解答:
解:连AO,并延长交圆O于点F,连接CF,
则∠ACF=90°;
∵A,B,C,D,E是圆O的五等分点.
∴∠CAD=∠DBE=∠ACE=∠ADB=∠BEC,
又∠CAD+∠DBE+∠ACE+∠ADB+∠BEC=180°,
∴∠CAD=
×180°=36°,
∴
=18°,
在Rt△ACF中,AC=a,
∴AF=
=
.
点评:综合运用了圆周角定理的推论和正五边形的性质.熟练运用锐角三角函数进行求解.
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_DA/images0.png)
则∠ACF=90°;
∵A,B,C,D,E是圆O的五等分点.
∴∠CAD=∠DBE=∠ACE=∠ADB=∠BEC,
又∠CAD+∠DBE+∠ACE+∠ADB+∠BEC=180°,
∴∠CAD=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_DA/0.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_DA/1.png)
在Rt△ACF中,AC=a,
∴AF=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021232426423476976/SYS201310212324264234769027_DA/3.png)
点评:综合运用了圆周角定理的推论和正五边形的性质.熟练运用锐角三角函数进行求解.
![](http://thumb2018.1010pic.com/images/loading.gif)
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