【答案】
分析:(1)可通過(guò)構(gòu)建直角三角形然后運(yùn)用勾股定理求解.
(2)①△PMN的形狀不會(huì)變化,可通過(guò)做EG⊥BC于G,不難得出PM=EG,這樣就能在三角形BEG中求出EG的值,也就求出了PM的值,如果做PH⊥MN于H,PH是三角形PMH和PHN的公共邊,在直角三角形PHM中,有PM的值,∠PMN的度數(shù)也不難求出,那么就能求出MH和PH的值,也就求出HN和PN的值了,有了PN,PM,MN的值,就能求出三角形MPN的周長(zhǎng)了.
②本題分兩種情況進(jìn)行討論:
1、N在CD的DF段時(shí),PM=PN.這種情況同①的計(jì)算方法.
2、N在CD的CF段時(shí),又分兩種情況進(jìn)行討論
MP=MN時(shí),MC=MN=MP,這樣有了MC的值,x也就能求出來(lái)了
NP=NM時(shí),我們不難得出∠PMN=120°,又因?yàn)椤螹NC=60°因此∠PNM+∠MNC=180度.這樣點(diǎn)P與F就重合了,△PMC即這是個(gè)直角三角形,然后根據(jù)三角函數(shù)求出MC的值,然后就能求出x了.
綜合上面的分析把△PMC是等腰三角形的情況找出來(lái)就行了.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/images0.png)
解:(1)如圖1,過(guò)點(diǎn)E作EG⊥BC于點(diǎn)G.
∵E為AB的中點(diǎn),
∴BE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/0.png)
AB=2
在Rt△EBG中,∠B=60°,∴∠BEG=30度.
∴BG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/1.png)
BE=1,EG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/2.png)
即點(diǎn)E到BC的距離為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/3.png)
(2)①當(dāng)點(diǎn)N在線(xiàn)段AD上運(yùn)動(dòng)時(shí),△PMN的形狀不發(fā)生改變.
∵PM⊥EF,EG⊥EF,
∴PM∥EG,又EF∥BC,
∴四邊形EPMG為矩形,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/images5.png)
∴EP=GM,PM=EG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/4.png)
同理MN=AB=4.
如圖2,過(guò)點(diǎn)P作PH⊥MN于H,
∵M(jìn)N∥AB,
∴∠NMC=∠B=60°,又∠PMC=90°,
∴∠PMH=∠PMC-∠NMC=30°.
∴PH=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/5.png)
PM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/6.png)
∴MH=PM•cos30°=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/7.png)
則NH=MN-MH=4-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/8.png)
在Rt△PNH中,PN=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/9.png)
∴△PMN的周長(zhǎng)=PM+PN+MN=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/10.png)
②當(dāng)點(diǎn)N在線(xiàn)段DC上運(yùn)動(dòng)時(shí),△PMN的形狀發(fā)生改變,但△MNC恒為等邊三角形.
當(dāng)PM=PN時(shí),如圖3,作PR⊥MN于R,則MR=NR.
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/images13.png)
類(lèi)似①,PM=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/11.png)
,∠PMR=30°,
MR=PMcos30°=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/12.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/13.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/14.png)
,
∴MN=2MR=3.
∵△MNC是等邊三角形,
∴MC=MN=3.
此時(shí),x=EP=GM=BC-BG-MC=6-1-3=2.
當(dāng)MP=MN時(shí),
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/images18.png)
∵EG=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/15.png)
,
∴MP=MN=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/16.png)
,
∵∠B=∠C=60°,
∴△MNC是等邊三角形,
∴MC=MN=MP=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/17.png)
(如圖4),
此時(shí),x=EP=GM=6-1-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/18.png)
,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/images23.png)
當(dāng)NP=NM時(shí),如圖5,∠NPM=∠PMN=30度.
則∠PNM=120°,又∠MNC=60°,
∴∠PNM+∠MNC=180度.
因此點(diǎn)P與F重合,△PMC為直角三角形.
∴MC=PM•tan30°=1.
此時(shí),x=EP=GM=6-1-1=4.
綜上所述,當(dāng)x=2或4或(5-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161337657243363/SYS201310221613376572433023_DA/19.png)
)時(shí),△PMN為等腰三角形.
點(diǎn)評(píng):本題綜合考查了等腰梯形,等腰直角三角形的性質(zhì),中位線(xiàn)定理,勾股定理等知識(shí)點(diǎn)的應(yīng)用.