【答案】
分析:(1)何時(shí)矩形的一邊恰好通過?ABCD的邊AB或CD的中點(diǎn),題目本身就不明確,到底是GF還是HE,經(jīng)過了AB的中點(diǎn)還是CD的中點(diǎn),所以必須分情況討論,即①當(dāng)GF邊通過AB邊的中點(diǎn)②當(dāng)EH邊通過AB邊的中點(diǎn)③當(dāng)GF邊通過CD邊的中點(diǎn)
(2)設(shè)當(dāng)矩形運(yùn)動(dòng)到t(s)(7<t<11)時(shí)與平行四邊形的重疊部分為五邊形,則BE、AH都可用含有t的式子表示出來.在矩形EFGH中易證△AHP∽△BEP根據(jù)對(duì)應(yīng)線段成比例,可求出EP的長,因此面積可表示出來.
(3)點(diǎn)Q在矩形一邊上運(yùn)動(dòng)的時(shí)間為多少s,這里的“一邊”是哪一邊,必須分情況進(jìn)行解釋,所以也有三種情況.
解答:解:(1)作AM⊥BC,∵AB=5,sin∠ABC=3/5,
∴BM=4,AM=3,
①當(dāng)GF邊通過AB邊的中點(diǎn)N時(shí),
有BF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/0.png)
BM=2,
∴t
1=3(s).
②當(dāng)EH邊通過AB邊的中點(diǎn)N時(shí),
有BE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/1.png)
BM=2,
∴BF=2+6=8,
∴t
2=8+1=9(s).
③當(dāng)GF邊通過CD邊的中點(diǎn)K時(shí),
有CF=2,
∴t
3=1+10+2=13(s),
綜上,當(dāng)t等于3s或9s或13s時(shí),矩形的一邊恰好通過平行四邊形的邊AB或CD的中點(diǎn).
(2)設(shè)當(dāng)矩形運(yùn)動(dòng)到t(s)(7<t<11)時(shí)與平行四邊形的重疊部分為五邊形,
則BE=t-7,AH=4-(t-7)=11-t,
在矩形EFGH中,有AH∥BF,
∴△AHP∽△BEP,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/images2.png)
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/2.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/3.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/4.png)
,
∴EP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/5.png)
,
∴S=18-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/6.png)
,
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/7.png)
(t-11)
2+18(7<t<11),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/images9.png)
由對(duì)稱性知當(dāng)11<t<15時(shí)重疊部分仍為五邊形,
綜上S與t的函數(shù)關(guān)系式為:S=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/8.png)
(t-11)
2+18(7<t<15且t≠11),
把s=16.5代入得:16.5=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/9.png)
(t-11)
2+18,
∴t
1=9,t
2=13,
即當(dāng)t等于9s或13s時(shí)重疊部分的面積為16.5cm
2.
(3)點(diǎn)Q從點(diǎn)C運(yùn)動(dòng)到點(diǎn)D所需的時(shí)間為:
5÷(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/10.png)
)=10(s),
此時(shí),DG=1+14-10=5
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/images13.png)
,
點(diǎn)Q從D點(diǎn)運(yùn)動(dòng)開始到與矩形相遇所需的時(shí)間為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/11.png)
,
∴矩形從與點(diǎn)Q相遇到運(yùn)動(dòng)到停止所需的時(shí)間為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/12.png)
,
從相遇到停止點(diǎn)Q運(yùn)動(dòng)的路程為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/13.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/14.png)
<6,
即點(diǎn)Q從相遇到停止一直在矩形的邊GH上運(yùn)動(dòng),
∴點(diǎn)Q在矩形的一邊上運(yùn)動(dòng)的時(shí)間為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163611071552049/SYS201310221636110715520025_DA/15.png)
.
點(diǎn)評(píng):此題主要考查了矩形的性質(zhì),此題在解答過程中,一定要注意分情況討論,另外還考查了二次函數(shù)的一些基本應(yīng)用,考查比較全面,難易程度適中.