题目
题型:广东省竞赛题难度:来源:
(1)当点Q在线段BC上运动时,点P出发多少时间后,∠BEP=∠BEQ?
(2)设△APE的面积为ycm2,AP=xcm,求y关于x的函数解析式,并写出函数的定义域.
(3)当4<x<8时,求函数值y的范围.
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050950-64564.png)
答案
当∠BEP=∠BEQ,∠ABD=∠DBC=45°,
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050950-77954.png)
∴△PEB≌△QEB(ASA),
∴PB=BQ,即8﹣x=2x,
解得:x=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050951-97607.png)
∴出发
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050951-45977.png)
(2)当0<x≤4时,如图2,
Q在BC上,过E作EN⊥AB,EM⊥BC,
∴AD∥BC,
∴△AED∽△QEB,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050951-77237.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050951-58443.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050952-12548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050952-74439.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050952-39851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050952-86533.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050953-23688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050953-31660.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050953-44681.png)
∴NE=2x
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050953-62624.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050954-41709.png)
∴S△APE=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050954-87033.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050954-43857.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050954-44737.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050954-92114.png)
即y=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050955-15541.png)
当4<x<8,Q在CD上,作QF⊥AB于F,交BD于H (如图3)
DQ=HQ=16﹣2x,
∵AD∥FQ,
∴△ADE∽QHE,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050955-59073.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050955-23710.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050955-17313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050956-45547.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050956-23263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050956-21879.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050956-18998.png)
作EN⊥AB,
∴NE∥FQ,
∴△ANE∽△AFQ,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050957-30240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050957-40535.png)
∴NE=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050957-37514.png)
∴S△APE=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050957-38527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050958-95100.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050958-15468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050958-36135.png)
即y=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050958-41078.png)
(3)当4<x<8时,由y=
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050959-82706.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050959-84579.png)
由4<x<8,可得4<
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050959-43627.png)
∵y>0,
∴16+y>0,
∴4(16+y)<12y<8(16+y),16+y<3y<2(16+y),
即
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050959-57145.png)
解得:8<y<32,当4<x<8时,8<y<32.
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027051000-24431.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027051000-30160.png)
核心考点
试题【如图,正方形ABCD的边长为8cm,动点P从点A出发沿AB边由A向B以1cm/秒的速度匀速移动(点P不与点A、B重合),动点Q从点B出发沿折线BC﹣CD以2cm】;主要考察你对全等三角形的应用等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050945-35699.png)
B.四边形CDFE的面积保持不变
C.△CDE面积的最大值为8
D.四边形CDFE不可能为正方形
(1)在图(2)中,线段BD与线段CE的大小关系是 _________ ;
(2)在图(3)中,△AMN与△ABC是相似三角形吗?请证明你的结论.
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050942-86545.png)
(1)当直角三角尺的两直角边分别与矩形ABEF的两边BE,EF相交于点G,H时,如图甲,通过观察或测量BG与EH的长度,你能得到什么结论并证明你的结论;
(2)当直角三角尺的两直角边分别与BE的延长线,EF的延长线相交于点G,H时(如图乙),你在图甲中得到的结论还成立吗?简要说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050936-31070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050929-93592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050929-37310.png)
(1)求证:BG=FG;
(2)若AD=DC=2,求AB的长.
![](http://img.shitiku.com.cn/uploads/allimg/20191027/20191027050924-81722.png)
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