题目
题型:不详难度:来源:
(1)用x表示AD和CD;
(2)用x表示S,并求S的最大值;
(3)如图②,当S取最大值时,等腰梯形ABCD的四个顶点都在⊙O上,点E和点F分别是AB和CD的中点,求⊙O的半径R的值.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022633-66157.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-19901.png)
解析
试题分析:(1)作AH⊥CD于H,BG⊥CD于G,如图①,易得四边形AHGB为矩形,则HG=AB=3x,再根据等腰梯形的性质得AD=BC,DH=CG,在Rt△ADH中,设DH=t,根据含30度的直角三角形三边的关系得AD=2t,AH=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
(2)根据梯形的面积公式计算可得到S=-2
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
(3)连结OA、OD,如图②,由(2)得到x=2时,则AB=6,CD=18,等腰梯形的高为6
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
试题解析:(1)作AH⊥CD于H,BG⊥CD于G,如图①,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-22578.png)
则四边形AHGB为矩形,
∴HG=AB=3x,
∵四边形ABCD为等腰梯形,
∴AD=BC,DH=CG,
在Rt△ADH中,设DH=t,
∵∠ADC=60°,
∴∠DAH=30°,
∴AD=2t,AH=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴BC=2t,CG=t,
∵等腰梯形ABCD的周长为48,
∴3x+2t+t+3x+t+2t=48,解得t=8-x,
∴AD=2(8-x)=18-2x,
CD=8-x+3x+8-x=16+x;
(2)S=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-81783.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-81783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
=-2
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∵S=-2
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴当x=2时,S有最大值72
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
(3)连结OA、OD,如图②,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-24069.png)
当x=2时,AB=6,CD=16+2=18,等腰梯形的高为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
则AE=3,DF=9,
∵点E和点F分别是AB和CD的中点,
∴直线EF为等腰梯形ABCD的对称轴,
∴EF垂直平分AB和CD,EF为等腰梯形ABCD的高,即EF=6
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴等腰梯形ABCD的外接圆的圆心O在EF上,
设OE=a,则OF=6
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
在Rt△AOE中,
∵OE2+AE2=OA2,
∴a2+32=R2,
在Rt△ODF中,
∵OF2+DF2=OD2,
∴(6
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴a2+32=(6
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴R2=(5
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-58441.png)
∴R=2
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022634-19901.png)
【考点】圆的综合题.
核心考点
试题【如图①,已知等腰梯形ABCD的周长为48,面积为S,AB∥CD,∠ADC=60°,设AB=3x.(1)用x表示AD和CD;(2)用x表示S,并求S的最大值;(3】;主要考察你对二次函数定义等知识点的理解。[详细]
举一反三
(1)求抛物线的解析式;
(2)在抛物线上能不能找到一点P,使∠POC=∠PCO?若能,请求出点P的坐标;若不能,请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022628-65456.png)
(1)求抛物线的解析式,并写出其顶点坐标;
(2)当S△MFQ:S△MEB=1:3时,求点M的坐标.
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022622-45044.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022613-43559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-86749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-51511.png)
(1)求此二次函数关系式;
(2)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-15927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-86749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-15927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-51511.png)
(3)若过点A作AG⊥
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-86749.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-51511.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022614-22560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022607-29664.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019022602-18637.png)
时间x(单位:年,x为正整数) | 1 | 2 | 3 | 4 | 5 | … |
单位面积租金z(单位:元/平方米) | 50 | 52 | 54 | 56 | 58 | |
(1)求出z与x的函数关系式;
(2)设第x年政府投入使用的并轨房收取的租金为W百万元,请问政府在第几年投入使用的并轨房收取的租金最多,最多为多少百万元?
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