题目内容
已知等边三角形ABC的边长是2,以BC边上的高AB1为边作等边三角形,得到第一个等边三角形AB1C1,再以等边三角形AB1C1的B1C1边上的高AB2为边作等边三角形,得到第二个等边三角形AB2C2,再以等边三角形AB2C2的边B2C2边上的高AB3为边作等边三角形,得到第三个等边AB3C3;…,如此下去,这样得到的第n个等边三角形ABnCn的面积为 .![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_ST/images0.png)
【答案】分析:由AB1为边长为2等边三角形ABC的高,利用三线合一得到B1为BC的中点,求出BB1的长,利用勾股定理求出AB1的长,进而求出第一个等边三角形AB1C1的面积,同理求出第二个等边三角形AB2C2的面积,依此类推,得到第n个等边三角形ABnCn的面积.
解答:解:∵等边三角形ABC的边长为2,AB1⊥BC,
∴BB1=1,AB=2,
根据勾股定理得:AB1=
,
∴第一个等边三角形AB1C1的面积为
×(
)2=
(
)1;
∵等边三角形AB1C1的边长为
,AB2⊥B1C1,
∴B1B2=
,AB1=
,
根据勾股定理得:AB2=
,
∴第二个等边三角形AB2C2的面积为
×(
)2=
(
)2;
依此类推,第n个等边三角形ABnCn的面积为
(
)n.
故答案为:
(
)n
点评:此题考查了等边三角形的性质,属于规律型试题,熟练掌握等边三角形的性质是解本题的关键.
解答:解:∵等边三角形ABC的边长为2,AB1⊥BC,
∴BB1=1,AB=2,
根据勾股定理得:AB1=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/0.png)
∴第一个等边三角形AB1C1的面积为
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/4.png)
∵等边三角形AB1C1的边长为
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/5.png)
∴B1B2=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/7.png)
根据勾股定理得:AB2=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/8.png)
∴第二个等边三角形AB2C2的面积为
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/9.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/10.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/11.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/12.png)
依此类推,第n个等边三角形ABnCn的面积为
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/13.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/14.png)
故答案为:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/15.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193529178948876/SYS201311011935291789488009_DA/16.png)
点评:此题考查了等边三角形的性质,属于规律型试题,熟练掌握等边三角形的性质是解本题的关键.
![](http://thumb2018.1010pic.com/images/loading.gif)
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