【答案】
分析:(1)根據(jù)折疊的性質(zhì)可知:AE=OA,OD=DE,那么可在直角三角形ABE中,用勾股定理求出BE的長(zhǎng),進(jìn)而可求出CE的長(zhǎng),也就得出了E點(diǎn)的坐標(biāo).
在直角三角形CDE中,CE長(zhǎng)已經(jīng)求出,CD=OC-OD=4-OD,DE=OD,用勾股定理即可求出OD的長(zhǎng),也就求出了D點(diǎn)的坐標(biāo).
(2)很顯然四邊形PMNE是個(gè)矩形,可用時(shí)間t表示出AP,PE的長(zhǎng),然后根據(jù)相似三角形APM和AED求出PM的長(zhǎng),進(jìn)而可根據(jù)矩形的面積公式得出S,t的函數(shù)關(guān)系式,根據(jù)函數(shù)的性質(zhì)即可得出S的最大值及對(duì)應(yīng)的t的值.
(3)本題要分兩種情況進(jìn)行討論:
①M(fèi)E=MA時(shí),此時(shí)MP為三角形ADE的中位線,那么AP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/0.png)
,據(jù)此可求出t的值,過M作MF⊥OA于F,那么MF也是三角形AOD的中位線,M點(diǎn)的橫坐標(biāo)為A點(diǎn)橫坐標(biāo)的一半,縱坐標(biāo)為D點(diǎn)縱坐標(biāo)的一半.由此可求出M的坐標(biāo).
②當(dāng)MA=AE時(shí),先在直角三角形OAD中求出斜邊AD的長(zhǎng),然后根據(jù)相似三角形AMP和ADE來(lái)求出AP,MP的長(zhǎng),也就能求出t的值.根據(jù)折疊的性質(zhì),此時(shí)AF=AP,MF=MP,也就求出了M的坐標(biāo).
解答:解:(1)依題意可知,折痕AD是四邊形OAED的對(duì)稱軸,
∴在Rt△ABE中,AE=AO=5,AB=4.
BE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/1.png)
=3.
∴CE=2.
∴E點(diǎn)坐標(biāo)為(2,4).
在Rt△DCE中,DC
2+CE
2=DE
2,
又∵DE=OD.
∴(4-OD)
2+2
2=OD
2.
解得:OD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/2.png)
.
∴D點(diǎn)坐標(biāo)為(0,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/3.png)
).
(2)如圖①∵PM∥ED,
∴△APM∽△AED.
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/4.png)
,
又知AP=t,ED=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/5.png)
,AE=5,
PM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/6.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/7.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/8.png)
,
又∵PE=5-t.
而顯然四邊形PMNE為矩形.
S
矩形PMNE=PM•PE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/9.png)
×(5-t)=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/10.png)
t
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/11.png)
t;
∴S
四邊形PMNE=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/12.png)
(t-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/13.png)
)
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/14.png)
,
又∵0<
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/15.png)
<5.
∴當(dāng)t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/16.png)
時(shí),S
矩形PMNE有最大值
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/17.png)
.
(3)(i)若以AE為等腰三角形的底,則ME=MA(如圖①)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/images18.png)
在Rt△AED中,ME=MA,
∵PM⊥AE,
∴P為AE的中點(diǎn),
∴t=AP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/18.png)
AE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/19.png)
.
又∵PM∥ED,
∴M為AD的中點(diǎn).
過點(diǎn)M作MF⊥OA,垂足為F,則MF是△OAD的中位線,
∴MF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/20.png)
OD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/21.png)
,OF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/22.png)
OA=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/23.png)
,
∴當(dāng)t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/24.png)
時(shí),(0<
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/25.png)
<5),△AME為等腰三角形.
此時(shí)M點(diǎn)坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/26.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/27.png)
).
(ii)若以AE為等腰三角形的腰,則AM=AE=5(如圖②)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/images29.png)
在Rt△AOD中,AD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/28.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/29.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/30.png)
.
過點(diǎn)M作MF⊥OA,垂足為F.
∵PM∥ED,
∴△APM∽△AED.
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/31.png)
.
∴t=AP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/32.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/33.png)
=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/34.png)
,
∴PM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/35.png)
t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/36.png)
.
∴MF=MP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/37.png)
,OF=OA-AF=OA-AP=5-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/38.png)
,
∴當(dāng)t=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/39.png)
時(shí),(0<2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/40.png)
<5),此時(shí)M點(diǎn)坐標(biāo)為(5-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/41.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/42.png)
).
綜合(i)(ii)可知,t=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/43.png)
或t=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/44.png)
時(shí),以A,M,E為頂點(diǎn)的三角形為等腰三角形,
相應(yīng)M點(diǎn)的坐標(biāo)為(
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/45.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/46.png)
)或(5-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/47.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131101191307430453038/SYS201311011913074304530025_DA/48.png)
).
點(diǎn)評(píng):本題主要考查了矩形的性質(zhì)、勾股定理、圖形的翻折變換、相似三角形的判定和性質(zhì)以及二次函數(shù)的綜合應(yīng)用等知識(shí)點(diǎn),綜合性較強(qiáng).