题目
题型:湖北省同步题难度:来源:
(1)求证:AB
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144621-90040.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144622-31846.png)
(2)已知AB=15cm,BC=9cm,P是线段DE上的动点.设DP=x cm,梯形BCDP的面积为ycm2.
①求y关于x的函数关系式.
②y是否存在最大值?若有求出这个最大值,若不存在请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144622-49790.png)
答案
∴DE垂直平分AC,
∴AF=CF,∠DFA=∠DFC=90°,∠DAF=∠DCF.
∵∠DAB=∠DAF+∠CAB=90°,∠CAB+∠B=90°,
∴∠DCF=∠DAF=∠B.
在Rt△DCF和Rt△ABC中,∠DFC=∠ACB=90°,∠DCF=∠B,
∴△DCF∽△ABC.
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144622-29970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144622-23027.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144623-86223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144623-67771.png)
∴AB
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144623-41927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144623-58350.png)
(2)解:连接PB,
①∵AB=15,BC=9,∠ACB=90°,
∴AC=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144624-35410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144624-38098.png)
∴CF=AF=6.
∴y=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144624-83882.png)
②由EF∥BC,得△AEF∽△ABC.
AE=BE=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144625-49712.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144625-11958.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144625-22149.png)
由∠EAD=∠AFE=90°,∠AEF=∠DEA,得△AEF∽△DEA.
Rt△ADF中,AD=10,AF=6,
∴DF=8.
∴DE=DF+FE=8+
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144625-73953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144626-44140.png)
∵y=3x+27(0≤x≤
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144626-93050.png)
∴当x=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144626-99102.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144626-93582.png)
核心考点
试题【如图,四边形ABCD中,AD=CD,∠DAB=∠ACB=90°,过点D作DE⊥AC,垂足为F,DE与AB相交于点E.(1)求证:ABAF=CBCD;(2)已知A】;主要考察你对相似三角形性质等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144617-12861.png)
B.△ADE∽△ABC
C.
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144617-75567.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144617-25949.png)
D.S△ABC=3S△ADE
(1)当x=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144556-49698.png)
(2)当CQ=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144556-73459.png)
(3)①当CQ=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144556-61768.png)
②当CQ=
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144556-43728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103144556-46484.png)
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