【答案】
分析:(1)OP
6旋轉(zhuǎn)了6×45°=270°,得到點(diǎn)P
6落在y軸的負(fù)半軸,而點(diǎn)P
n到坐標(biāo)原點(diǎn)的距離始終等于前一個(gè)點(diǎn)到原點(diǎn)距離的2倍,故其坐標(biāo)為P
6(0,-2
6);
(2)根據(jù)兩組對(duì)應(yīng)邊的比相等,且它們的夾角也相等,則這兩個(gè)三角形相似得到△P
OP
1∽△P
1OP
2∽△P
n-1OP
n.然后求出S
△P0OP1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/0.png)
×1×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/1.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/2.png)
,再求出
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/3.png)
,利用相似三角形面積的比等于它們的相似比即可得到△P
5OP
6的面積;
(3)分類(lèi)討論:令旋轉(zhuǎn)次數(shù)為n,①當(dāng)n=8k或n=8k+4時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在x軸上,此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(2
n,0);②當(dāng)n=8k+1或n=8k+3或n=8k+5或n=8k+7時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在各象限的平分線上,此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/4.png)
×2
n,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/5.png)
×2
n);③當(dāng)n=8k+2或n=8k+6時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在y軸上,此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(0,2
n).
解答:解:
(1)根據(jù)旋轉(zhuǎn)規(guī)律,點(diǎn)P
6落在y軸的負(fù)半軸,而點(diǎn)P
n到坐標(biāo)原點(diǎn)的距離始終等于前一個(gè)點(diǎn)到原點(diǎn)距離的2倍,故其坐標(biāo)為P
6(0,-2
6),即P
6(0,-64);
(2)由已知可得,△P
OP
1∽△P
1OP
2∽△P
n-1OP
n.
設(shè)P
1(x
1,y
1),則y
1=2sin45°=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/6.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/8.png)
×1×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/9.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/10.png)
,
又∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/11.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/12.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/13.png)
;
(3)由題意知,OP
旋轉(zhuǎn)8次之后回到x軸正半軸,在這8次中,點(diǎn)P
n分別落在坐標(biāo)象限的平分線上或x軸或y軸上,但各點(diǎn)絕對(duì)坐標(biāo)的橫、縱坐標(biāo)均為非負(fù)數(shù),因此,點(diǎn)P
n的坐標(biāo)可分三類(lèi)情況:令旋轉(zhuǎn)次數(shù)為n,
①當(dāng)n=8k或n=8k+4時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在x軸上,此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(2
n,0);
②當(dāng)n=8k+1或n=8k+3或n=8k+5或n=8k+7時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在各象限的平分線上,此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/14.png)
×2
n,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/15.png)
×2
n),即(2
n-1![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/16.png)
,2
n-1![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164254073392401/SYS201310221642540733924016_DA/17.png)
);
③當(dāng)n=8k+2或n=8k+6時(shí)(其中k為自然數(shù)),點(diǎn)P
n落在y軸上,
此時(shí),點(diǎn)P
n的絕對(duì)坐標(biāo)為(0,2
n).
點(diǎn)評(píng):本題考查了旋轉(zhuǎn)的性質(zhì):旋轉(zhuǎn)前后的兩個(gè)圖形全等,對(duì)應(yīng)點(diǎn)與旋轉(zhuǎn)中心的連線段的夾角等于旋轉(zhuǎn)角,對(duì)應(yīng)點(diǎn)到旋轉(zhuǎn)中心的距離相等.也考查了三角形相似的判定與性質(zhì)以及各象限和坐標(biāo)軸上的點(diǎn)的坐標(biāo)特點(diǎn).