【答案】
分析:根據(jù)面積比等于相似比的平方,可得出
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/1.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/2.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/3.png)
,再由平行線的性質(zhì)可得出
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/5.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/6.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/9.png)
,從而可推出相鄰兩個陰影部分的相似比為1:2,面積比為1:4,先利用等底三角形的面積之比等于高之比可求出第一個及第二個陰影部分的面積,再由相似比為1:2可求出面積小于2011的陰影部分的個數(shù).
解答:解:由題意得,△A
2B
1B
2∽△A
3B
2B
3,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/10.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/11.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/12.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/14.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/15.png)
,
又∵A
1B
1∥A
2B
2∥A
3B
3,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/18.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/19.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/20.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/21.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/22.png)
,
∴OA
1=A
1A
2,B
1B
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/23.png)
B
2B
3繼而可得出規(guī)律:A
1A
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/24.png)
A
2A
3=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/25.png)
A
3A
4…;B
1B
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/26.png)
B
2B
3=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/27.png)
B
3B
4…
又△A
2B
1B
2,△A
3B
2B
3的面積分別為1、4,
∴S
△A1B1A2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/28.png)
,S
△A2B2A3=2,
繼而可推出S
△A3B3A4=8,S
△A,4B4A5=32,S
△A5B5A6=128,S
△A6B6A7=512,S
△A7B7A8=2048,
故可得小于2011的陰影三角形的有:△A
1B
1A
2,△A
2B
2A
3,△A
3B
3A
4,△A
4B
4A
5,△A
5B
5A
6,△A
6B
6A
7,共6個.
故答案是:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191445474854140/SYS201311011914454748541011_DA/29.png)
;6.
點評:此題考查了相似三角形的判定與性質(zhì)及平行線的性質(zhì),解答本題的關(guān)鍵是掌握相似比等于面積比的平方,及平行線分線段成比例,難度較大,注意仔細觀察圖形,得出規(guī)律.