【答案】
分析:(1)求出∠A=∠B=45°,因?yàn)锳D=3,由勾股定理求出AE長(zhǎng);
(2)由∠ADE+∠AED=135°和∠BEF+∠AED=135°推出∠ADE=∠BEF,證出△ADE∽△BEF,得到
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/1.png)
,代入即可;
(3)①如圖,若EF=BF,由相似得到AE=DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/2.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/3.png)
,求出t;②如圖,若EF=BE,由相似求出AE,即可求出t;③若BF=BE,則∠FEB=∠EFB,由△ADE∽△BEF得出AE=AD=3即可求出t.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/images4.png)
解:(1)∵∠C=90°,AC=BC,
∴∠A=∠B=45°,
∵DE⊥AB,
∴∠DEA=90°,
∵AD=3,
由勾股定理得:AE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/4.png)
,
故答案為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/5.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/6.png)
.
(2)∵在△ABC中,∠C=90°,AC=BC=4.
∴∠A=∠B=45°,
AB=4
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/7.png)
,
∴∠ADE+∠AED=135°,
又∵∠DEF=45°,
∴∠BEF+∠AED=135°,
∴∠ADE=∠BEF,
∴△ADE∽△BEF,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/8.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/9.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/10.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/11.png)
,
∴y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/12.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/13.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/14.png)
x,
∴y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/15.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/16.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/17.png)
x=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/18.png)
(x-2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/19.png)
)
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/20.png)
∴當(dāng)x=2
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/21.png)
時(shí),y有最大值=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/22.png)
,
∵從運(yùn)動(dòng)的過(guò)程中可以得出點(diǎn)F運(yùn)動(dòng)的路程正好是2BF,
∴點(diǎn)F運(yùn)動(dòng)路程為2×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/23.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/24.png)
cm,
答:在點(diǎn)E運(yùn)動(dòng)過(guò)程中,y與x之間的函數(shù)關(guān)系式是y=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/25.png)
x
2+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/26.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/27.png)
x,點(diǎn)F運(yùn)動(dòng)路線的長(zhǎng)為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/28.png)
cm.
(3)這里有三種情況:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/images30.png)
①如圖,若EF=BF,則∠B=∠BEF,
又∵△ADE∽△BEF,
∴∠A=∠ADE=45°,
∴∠AED=90°,
∴AE=DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/29.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/30.png)
,
∵動(dòng)點(diǎn)E的速度為1cm/s,
∴此時(shí)x=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/31.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/32.png)
;
②如圖,若EF=BE,則∠B=∠EFB;
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/images35.png)
又∵△ADE∽△BEF,
∴∠A=∠AED=45°,
∴∠ADE=90°,
∴AE=3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/33.png)
,
∵動(dòng)點(diǎn)E的速度為1cm/s
∴此時(shí)x=3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/34.png)
;
③如圖,若BF=BE,則∠FEB=∠EFB;
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/images38.png)
又∵△ADE∽△BEF,
∴∠ADE=∠AED,
∴AE=AD=3,
∵動(dòng)點(diǎn)E的速度為1cm/s,
∴此時(shí)x=3.
綜上所述,當(dāng)△BEF為等腰三角形時(shí),x的值為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/35.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/36.png)
或3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/37.png)
或3.
答:x的值為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/38.png)
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/39.png)
或3
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022164741106323741/SYS201310221647411063237027_DA/40.png)
或3.
點(diǎn)評(píng):本題主要考查對(duì)二次函數(shù)的最值,相似三角形的性質(zhì)和判定,勾股定理,等腰三角形的性質(zhì)等知識(shí)點(diǎn)的理解和掌握,靈活運(yùn)用性質(zhì)進(jìn)行計(jì)算是解此題的關(guān)鍵,用的數(shù)學(xué)思想是分類討論思想.