2005年中考复习之平面直角坐标系

2005年中考复习之平面直角坐标系

知识考点：

1、理解直角坐标系的概念，坐标的意义，掌握坐标轴上的点和各象限内的点的坐标特征。

2、掌握关于x 轴、y 轴、原点对称的点以及两坐标轴夹角平分线上的点的坐标特征。

3、会利用数形结合思想，结合解直角三角形的知识解决有关求点的坐标问题。 精典例题：

【例1】若点A （1+a ，2-b ）与点B （4，－2）关于原点对称，则点C （a ，b ）到y 轴的距离为 。

分析：由关于原点对称的点的坐标关系：横纵坐标互为相反数，求出a 、b 的值，再由点到y 轴的距离是该点横坐标的绝对值得解。

略证：易得C （－5，4），距离为5。

【例2】若点P （22-m ，m ）在第四象限的角平分线上，则点（m ，1-m ）关于y 轴的对称点坐标是 。

解析：由题意有：0)2(2=+-m m ，解得21-=m ，12=m （舍） ∴点P （2，－2），点（2，2

1-）关于y 轴对称点的坐标是（－2，2

1-

）

答案：（－2，2

1-

）

评注：一、三象限夹角平分线上的点横、纵坐标相等，二、四象限夹角平分线上的点横、纵坐标互为相反数。关于x 轴（y 轴）的对称点是横（纵）坐标相同，纵（横）坐标互为相反数。

【例3】已知点M （3，－2）与点M '（x ，y ）在同一条平行于x 轴的直线上，且M '到y 轴的距离等于4，那么点M '的坐标是（ ）

A 、（4，2）或（4，－2）

B 、（4，－2）或（－4，－2）

C 、（4，－2）或（－5，－2）

D 、（4，－2）或（－1，－2） 解析：易知：2-=y ，x ＝4或－4

答案：B

评注：平行于x 轴（y 轴）直线上所有点的纵坐标（横坐标）相同。

【例4】如图，在矩形ABCD 中，AB ＝4，BC ＝3，将矩形ABCD 置于直角坐标系中，使点A 与坐标系的原点重合，AB 与x 轴正方向成300的角，求点B 、C 的坐标。

解析：作BE ⊥x 轴，CF ⊥x 轴，BG ⊥CF 。OE ＝ABcos300＝32，EB ＝2，所以B

点坐标为B （32，2）。

设AB 与CF 交于点H ，∠BCH ＝300，BG ＝

2

3，CG ＝2

33，OF ＝2

332-，FC

＝2

332+

。

答案：B （32，2），C （2

332-

，2

332+

）

问题图

探索与创新：

【问题】A 、B 两村在河边的同旁，以河边为x 轴建立直角坐标系，则A 、B 两村对应的坐标分别是A （－1，1），B （3，3）。现要在河边P 点处修建一个水泵站，分别向A 、B 两村送水，P 点选择在什么地方，才可使所用的水管最短？试求出P 点的坐标及所需水管的长度。

解析：先找出A （－1，1）关于x 轴的对称点A '（－1，－1），连结B A '，因B （3，3）知点A '、B 在一、三象限夹角的平分线上，故原点O 即为选择之P 点。又过A '、B 两点分别作x 轴、y 轴的平行线交于点C ，在Rt △BC A '中利用勾股定理可求所需水管的长度。

答案：P （0，0），需水管的长24。

评注：本题应用了对称思想以及两点之间线段最短的知识，这是求最短线时的常用方法。 跟踪训练： 一、选择题：

1、如果点P 既在x 轴下方，又在y 轴右侧，且距x 轴、y 轴的距离分别是a

2、a 3，那么点P 的坐标是（ ）

A 、（a 2，a 3-）

B 、（a 3，a 2-）

C 、（a 2-，a 3）

D 、（a 3-，a 2）

2、已知点P （42-a ，2

7a -）在第二象限的角平分线上，则a 等于（ ）

A 、3

B 、3或－1

C 、－1

D 、321±-

3、如图，将□ABCD 对角线交点与直角坐标系的原点重合，点A 与点B 的坐标分别是（－2，－1），（

2

1，－1），则点C 和点D 的坐标分别为（

A 、（2，1）和（2

1-

，1） B 、（2，－1）和（2

1-

，－1）

C 、（－2，1）和（

2

1，1） D 、（－1，－2）和（－1，2

1-4、在直角坐标系中，点A 、B 的坐标分别是（3，0），（0，4），Rt △ABO 内心的坐标是（ ）

A 、（

2

7，

2

7） B 、（

2

3，2） C 、（1，1） D 、（

2

3，1）

二、填空题：

1、若点（24a -，1-a ）在y 轴负半轴上，则a ＝ 。

2、已知点P （93-a ，a -1）是第三象限的点，且横、纵坐标均为整数，若P 、Q 关于原点对称，则Q 点坐标是 。

3、已知点P 1（1-a ，5）和点P 2（2，1-b ）关于x 轴对称，则2003)(b a +＝ 。

4、已知A （0，3），B （0，－1）两点，分别以A 、B 为圆心的两圆相交于M （－2，1-a ）和N （a b 22-，1），则a b 的值为 。

5、如图，相交于点P （5，5）的互相垂直的直线1l 和2l 与x 轴和y 轴相交于点A 和B ，则四边形OAPB 的面积是 。

第5题图

第6题图

6、如图，已知边长为2的正方形OABC 在直角坐标系中，B 、C 两点在第二象限内，OA 与x 轴正方向的夹角为600，那么B 点坐标为 。 三、解答题：

1、已知第一象限内的点P （a ，b ）到原点的距离为10，且a 、b 是方程

0632

=++-m mx x 的两根。求m 的值和P 点的坐标。

2、已知如图，矩形AOBC ，以O 为坐标原点，OB 、OA 分别在x 轴、y 轴上，点A 的坐标为（0，3），∠OAB ＝600

，以AB 为轴对折后，使C 点落在D 点处，求D 点坐标。

第3题图

第2题图

第3题图

3、如图，在直角坐标系中，过点C （3，6）分别作x 轴和y 轴的垂线CB 和CA ，垂

足为B 和A ，若点P 从O 沿OB 向点B 以1个单位长度／秒的速度运动，点Q 从B 沿BC 向点C 以2个单位长度／秒的速度运动。如果P 、Q 分别从O 、B 同时出发，试求：

（1）经过多少时间，△PBQ 的面积等于2个平方单位；

（2）线段PQ 与AB 能否垂直？若能垂直，求出此时点Q 的坐标；若不能，请说明理由。

参考答案

一、选择题：BCAC 二、填空题：

1、－2；

2、（3，1）；

3、－1；

4、9；

5、25；

6、（31-，31+）

三、解答题：

1、14=m ，P （6，8）或（8，6）

2、D （

32

3，2

3-

）

3、（1）1秒或2秒；（2）能，Q （3，5

6）

任务型阅读专项练习和答案

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