若正△ABC外接圓的半徑為R,則△ABC的面積為 .
【答案】
分析:連接OB,OA,延長AO交BC于D,根據(jù)等邊三角形性質(zhì)得出AD⊥BC,BD=CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/0.png)
BC,∠OBD=30°,求出OD,根據(jù)勾股定理求出BD,即可求出BC,根據(jù)三角形的面積公式求出即可.
解答:解:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/images1.png)
連接OB,OA,延長AO交BC于D,
∵正△ABC外接圓是⊙O,
∴AD⊥BC,BD=CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/1.png)
BC,∠OBD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/2.png)
∠ABC=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/3.png)
×60°=30°,
即OD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/4.png)
OB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/5.png)
R,
由勾股定理得:BD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/6.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/7.png)
R,
即BC=2BD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/8.png)
R,AD=AO+OD=R+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/9.png)
R=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/10.png)
R,
則△ABC的面積是
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/11.png)
BC×AD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/12.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/13.png)
R×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/14.png)
R=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/15.png)
R
2,
故答案為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131103201533164426981/SYS201311032015331644269001_DA/16.png)
R
2.
點評:本題考查了等邊三角形、等腰三角形的性質(zhì),勾股定理,三角形的外接圓,三角形的面積等知識點的應(yīng)用,關(guān)鍵是能正確作輔助線后求出BD的長,題目具有一定的代表性,主要考查學(xué)生運用定理進(jìn)行推理和計算的能力.