2009年威海市中考数学压轴题(含解题过程)

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冲刺2010 ——2009年中考数学压轴题汇编(含解题过程)

（2009年威海市）25．（12分）

一次函数y ax b =+的图象分别与x 轴、y 轴交于点,M N ，与反比例函数k

y x

=

的图象相交于点,A B ．过点A 分别作AC x ⊥轴，AE y ⊥轴，垂足分别为,C E ；过点B 分别作BF x ⊥轴，BD y ⊥轴，垂足分别为

F D ，，AC 与BD 交于点K ，连接CD ．

（1）若点A B ，在反比例函数k

y x

=的图象的同一分支上，如图1，试证明：

①AEDK CFBK S S =四边形四边形； ②AN BM =．

（2）若点A B ，分别在反比例函数k

y x

=的图象的不同分支上，如图2，则AN 与BM 还相等吗？试证明你的结论．

25．（本小题满分12分）

解：（1）①AC x ⊥轴，AE y ⊥轴，

∴四边形AEOC 为矩形． BF x ⊥轴，BD y ⊥轴， ∴四边形BDOF 为矩形．

AC x ⊥轴，BD y ⊥轴，

∴四边形AEDK DOCK CFBK ，，均为矩形． ·

··········· 1分 1111OC x AC y x y k === ，，， ∴11AEOC S OC AC x y k === 矩形 2222OF x FB y x y k === ，，，

O C F M

D

E N K

y x

11()

A x y ，22()

B x y ，

（第25题图1）

O C

D

K F E N y x

11()

A x y ，33()

B x y ， M

（第25题图2）

O C F M

D

E N

K

y x

A

B

图1

北京中考网—北达教育旗下 https://www.wendangku.net/doc/653904538.html, 电话 010-********

∴22BDOF S OF FB x y k === 矩形． ∴AEOC BDOF S S =矩形矩形．

AEDK AEOC DOCK S S S =-矩形矩形矩形，

C F B K B

D O F D O S S S

=-矩形矩形

矩形

，

∴AEDK CFBK S S =矩形矩形． ······································································································ 2分 ②由（1）知AEDK CFBK S S =矩形矩形．

∴AK DK BK CK = ． ∴

AK BK

CK DK

=． ···················································································································· 4分 90AKB CKD ∠=∠=°，

∴AKB CKD △∽△． ·

······································································································· 5分 ∴CDK ABK ∠=∠．

∴AB CD ∥． ·

····················································································································· 6分 AC y ∥轴，

∴四边形ACDN 是平行四边形． ∴AN CD =． ·

····················································································································· 7分 同理BM CD =．

AN BM ∴=． ····················································································································· 8分 （2）AN 与BM 仍然相等． ································································································ 9分

AEDK AEOC ODKC S S S =+矩形矩形矩形， BKCF BDOF ODKC S S S =+矩形矩形矩形，

又 AEOC BDOF S S k ==矩形矩形，

∴AEDK BKCF S S =矩形矩形． ·

··································· 10分 ∴AK DK BK CK =

． ∴

CK DK AK BK

=． K K ∠=∠，

∴CDK ABK △∽△． ∴CDK ABK ∠=∠．

∴AB CD ∥． ·

···················································································································· 11分 AC y ∥轴，

O C D K F E

N y

x A

B M

图2

北京中考网—北达教育旗下https://www.wendangku.net/doc/653904538.html,电话010-******** ∴四边形ANDC是平行四边形．

∴AN CD

=．

=．

同理BM CD

∴AN BM

=． ····················································································································12分