第三章位置与坐标(单元测试·培优卷)(解析版)_2026语数英7-9年级上《举一反三系列》多版本_2026《初中数学•举一反三系列》8上多版本_2026《初中数学•举一反三系列》8上(北师大)

                                                                           
                                                                       
第三章  位置与坐标·培优卷
 2024

Ⅰ
 !"#$% 10 &'() 30 )'*& 3 )+
1!$3 )+$24-25 ,-./·0123·45+678P(2,−2)'98 P :$   +
A! ;<
B!=;<
C!>;<
D!?;<
D
)@ABC8BDE'FGH;<I8BDEJ8KLMN!
O
∵2>0'−2<0'
∴8 P :?;<!
P"
D!
2!$3 )+$24-25 ,-./·0123·45+:QRSTDEUV' ;<IB8P(a,1)WXyY5Z[
\2]'9aB^_$   +
A!−5
B!5`−5
C!5
D!6
C
)@abAc8dDEYBWX' ;<IB8BDEJ8'8d y YBWXefg8hD
EBij^' ;<IB8hkDEl_m'GnoKL!
O
∵ ;<IB8P(a,3)dyYBWXC5Z[\2]'
∴|a|=5'a>0'
∴a=5'
P"
C!
3!$3 )+$24-25 p-./·q1rs·4V+:QRSTDEUV'tA(−2,2)'B(−2,−3)u8vSw'/
xyzm{BC$    +
A!AB⊥xY
B!AB⊥yY
C!AB∥xY
D!AB |t}8
A
)@AcDE~€'‚SfxYBSwƒ8BhDE„…CB†!
FGu8BhDE„e'kDE‡e'KLMNt∵A(−2,2)'B(−2,−3)u8BSw‚SfxY!
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
O∵A(−2,2)'B(−2,−3)'
∴AB⊥xY'
P"
A!
4!$3 )+$24-25 ,-./·q1ˆ‰·45+VŠ;‹ŒŽ‘'’~C“”j•B–~'’—˜™
QRSTDEU'š‹›“œ”\f8(−1,−2)B\'9:… DEU/'“ž”Ÿ:\C$   +
A!(1,1)
B!(2,0)
C!(2,1)
D!(−2,−1)
C
)@abAc8BDE' ¡¢£QRI8BDEQ¤¥¦§¨oKLMN!©ªQR
I8BQ¤¥¦§¨«¬KLMN!
O
FGQRI8BQ¤¥¦LM'
­“œ”®¯Q¤ 3 Z[\'®ƒQ¤ 3 Z[\Md“ž”B\'
∴(−1+3,−2+3)'
K‹›“ž”Ÿ:B8BDE_(2,1)!
P" C!
5!$3 )+$24-25 ,-./·°1±·45+’~':QRSTDEUV'²8P³~V´µŸ¶·®¸
}8N¹'1”º²d8P1(1,1)'2”»¼º²d8P2(2,0)'3”»¼º²d8P3(3,−2)'4”»¼º
²d8P4(4,0),⋯⋯'³½¾Bº²¥¦'8P2025BDEC$   +
A!(2025,1)
B!(2025,−2)
C!(2024,1)
D!(2024,−2)
A
)@AcDEB¥¦¿'ÀN¥¦C†'FGÁ'8PBhDE_n$nCmÂÃ+'kD
EB¿¥¦C1,0,−2,0'*4” ÄÅ'ÆnKLo!
O
1”º²d8P1(1,1)'
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
2”»¼º²d8P2(2,0)'
3”»¼º²d8P3(3,−2)'
4”»¼º²d8P4(4,0)'
∴hDEB¿¥¦C
n”BhDE_n$nCmÂÃ+'
kDEB¿¥¦C
1,0,−2,0'*4” ÄÅ'
∴8P2025BhDEC2025'
∵2025÷4=506⋯⋯1'
∴kDE_
1'
∴P2025(2025,1)'
P"
A !
6!$3 )+$24-25 ,-./·ÇÈÉÊ·45+:QRSTDEUV'8ABDE_(−4,5)'8BCyYƒËÁ
 8'9wÌABBÍ&^_$    +
A!1
B!4
C!5
D!9
B
)@Ac8dDEYBWX΂wÌÍÏ' ¡¢£„7Ð8CB†!FG‚wÌÍ
ÏB€LM'ÑAB⊥yY'wÌABÒÍ&^'ÓFG8ABDEKL!
O
∵8ABDE_(−4,5)'8BCyYƒËÁ 8'
∴ÑAB⊥yY'wÌABÒÍ&^'Í&^_|−4|=4!
P"
B!
7!$3 )+$24-25 p-./·Ô1·45+8 AÎBÎCÎD BDE)Õ_ −3'0 Î 0'−3 Î 4'0 Î
0'4 'ÖÒ Sw l |t8 −3'4 × y Y‚S'9Sw l Ø|t$    +
A!8 A
B!8 B
C!8 C
D!8 D
D
)@Ac8BDE'ÙÚ ÛSw l t8 −3'4 × y Y‚S'M½ZSwƒB8BkDEl
C4'ÓFG?8BDEÜݧ¨)'KLv!
O
 ∵ ÛSw l t8 −3'4 × y Y‚S'
∴½ZSwƒB8BkDElC4'
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
∵8 AÎBÎCÎD BDE)Õ_ −3'0 Î 0'−3 Î 4'0 Î 0'4 '
∴Sw l ØÞ|tB8C8 D'
P"
D!
8!$3 )+:STDEQRI'ß8A(−1,−2)ஃQ¤ 3 Z[\2]'Ó®¯Q¤ 4 Z[\2]'Md
B8A'BDEC$    +
A!(−2,3)
B!(3,1)
C!(4,1)
D!(−4,−7)
B
)ß A$-1'-2+®ƒQ¤ 3 Z[\2]Md$-1'1+'Ó®¯Q¤ 4 Z[\2]Md$3'1+KL
Mdá
OÆQ¤B€L7'
ß A$-1'-2+®ƒQ¤ 3 Z[\2]Md$-1'1+'
Ó®¯Q¤ 4 Z[\2]MdA'$3'1+'
P" B!
8â@A8DEBQ¤' ¡¢£Q¤B€Cã@B†!
9!$3 )+$24-25 ,-./·äÔå·45+:QRSTDEUV'2·ABCDBæç“DEYQ
¨!67(−2,−2)'(3,1)Cg2·BuZè8DE'9/xH8VLéCg2·è8BC$    +
A!(−2,3)
B!(3,−2)
C!(1,−2)
D!(−3,1)
B
)@ABCDE~'FGÆf2·BæDEYQ¨'êè8DEÆu뇅B x ^ì y
^ëÚíî'íè8(−2,−2)'(3,1)_jTè8':{ï2·BðñuZè8KL.
O
’~'2·ABCDBæç“DEYQ¨!(−2,−2)'(3,1)Cg2·BuZè8DE'
∴ðñuZè8DE_
(−2,1)'(3,−2)'
∴B òÚÁá
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
P"
B
10!$3 )+:QRSTDEUV'Ö8A(a,−1)8B(4,b)fxYjó'9$   +
A!a=4'b=−1
B!a=−4'b=1
C!a=−4'b=−1
D!a=4'b=1
D
)@AB7Ð8CfxYjóB8BJô'†C ¡¢£fxYjóB8BJô!
Æ“fxYjó8BDE'hDE‡¿'kDEg8kDEõ_„öÔKLM!
O
∵fxYjó8BDE'hDE‡¿'kDEg8kDEõ_„öÃ'
∴Ö8A(a,−1)8B(4,b)fxYjó'9a=4'b=1!
P"
D!
=!÷ø$% 6 &'() 18 )'*& 3 )+
11!$3 )+Ö8PBDEC(3,−2)'9ùdxYBWXC      
!
2
)@abA8BDE'FG8d x YBWXCkDEBij^KL!
O
8PdxYBWXC|−2|=2'
P_
2!
12!$3 )+’~Cúäû üý8B)þ¶Á~':~V')Õémÿ'm·®_ x Y'y YBm·
®˜™QRSTDEU!’—Ā¶ÿSāB8BDE_ 3.5'4 'Ā¶ĂăāB8BDE_ −2'−1 'Ąą
DE}8Ÿ:B\C          
!
ÇĆā
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
)FGĀ¶ÿSāB8BDEìĀ¶ĂăāB8BDE{ï}8B\KL!
O
∵Ā¶ÿSāB8BDE_ 3.5'4 'Ā¶ĂăāB8BDE_ −2'−1 '
∴DE}8 0'0 Ÿ:B\CÇĆā!
P_
ÇĆā!
8â@A:QRSTDEUV{ï8B\' ¡¢£g7Ð8C†!
13!$3 )+$24-25 p-./·qćĈ·4V+:QRSTDEUV'678P(a,a+1)'Q(−1,2)!ÖSwPQ
xYQ¨'9aB^_     !
1
)@ADE~'FGQ¨ x Y B8BkDE„e'ĉ˜·ĊoKL!
O
ÆÁ'a+1=2'
∴a=1'
P_
1!
14!$3 )+$24-25 ,-./·ċČ·45+678A(a−5,a−3)'B(b,m)'8 A : x Yƒ'AB∥yY'8 B d x
YBWXC 4'×m<0'98 B BDEC      
!
(−2,−4)
)nAcDE~B€'m{¢£QRI8BDEJ8CB†!
S»čª x Yƒ8BDEJ8MN a B^'FGAB∥yYoN b B^'FG8 B d x YBWXC 4 oN m B
^'§íLoN8 B BDE!
O
∵8 A : x Yƒ'
∴a−3=0'
∴a=3'
∴A(−2,0)'
∵AB∥yY'
∴b=−2'
∵8 B d x YBWXC 4'
∴|m|=4'
∴m=±4'
∵m<0'
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
∴m=−4'
∴8 B BDEC(−2,−4)'
P_
(−2,−4).
15!$3 )+$24-25 p-./·Ď˜>ď·4V+’~'△OABBè8BBDE_(5,0)'­△OABĐxY®¯Q
¤Md△CDE!’—CB=1'ĄąBEB2_             
!
4
)@AcDE~B¿BQ¤' đQ¤B€CB†áFG8 B BDEoNOB'
ÓFGCB=1'oNOC'ÍĒFGQ¤€MNBEKLM!
O
∵è8BBDE_(5,0)'
∴ OB=5'
∵ CB=1'
∴ OC=OB−CB=5−1=4'
∵ △OABĐ x Y®¯Q¤Md△CDE'
∴BE=OC=4á
P_
4!
16!$3 )+$24-25 ,-./·ēÿVĈ·45+678O(0,0)'A(2,2)'8 B : x YmĔYƒ'×>TAOB
BRĕef 3'98 B BDEC          
!
(0,3)
)@AcDE~!
àĖB(0,b)'FG>TRĕė«¬KL!
O
∵8 B : x YmĔYƒ'
∴LĖB(0,b)'
∵>TAOBBRĕef 3'
∴1
2×(b−0)×2=3'
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
M
b=3'
∴B(0,3)'
P_
(0,3)!
Ⅱ
>!$% 8 &'() 72 )+
17!$6 )+$24-25 ,-./·ĘČęÊ·4V+ĚN~VBěæABCDEHZè8BDE!
A(−2,1)'B(1,−2)'C(5,0)'D(4,2)'E(1,3)
)@AcQRSTSTU' ¡¢£8BĀ¶·zCB†!FGQRSTDEUBJ8
ĚNH8BDEKL!
O
FGSTDEUB7ÐLM
A(−2,1)'B(1,−2)'C(5,0)'D(4,2)'E(1,3)!
18!$6 )+$2025·ĆĜ· ĝ+ĞğV*Z&·ğCæ2_ 1 Z[\B&m·'△ABC\’~Ÿ¶'
×A(−4,5),B(−6,2)!
(1)ĠNQRSTDEUxOy'ĚN8 C BDEá
(2)Q¤△ABC'š8 C ¤²d8F(6,−4)!
①ĠNQ¤ĒB△DEF'êV8 D 8 A j©$‡ĚĠz+á
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
②Ö8P(m,n):△ABCI'êQ¤ĒBj©8_P′'ĚNP′BDE!
(1)ġ'(−2,3)
(2)①ġá②(m+8,n−7)
)@abAQRSTDEUBJ8'~BQ¤'¢£éƒ7Ð'ÃÙÚ)CB†!
$1+FG8A(−4,5),B(−6,2)BDE{ïDEU'ÆDEUBJ8LĚN8CBDEá
$2+①FG~Q¤B·zv~KLá
②FG8Q¤¥¦“Ģģ¯Ĥ”KLo!
O$1+
’~Ÿ¶'˜™QRSTDEU!
∴8 C BDE(−2,3)á
$2+
678C(−2,3)'Q¤d8F(6,−4)'
∴¯¤8Z[\'/¤7Z[\'
①’~Ÿ¶'△DEFK_Ÿoá
②P′BDE_(m+8,n−7)!
19!$8 )+$24-25 p-.ƒ·Ĉÿĥ1·4V+67QRSTDEUVÒ 8M(m−1,2m+3)!
(1)678N(5,4)'ÑMN∥xYĦ'o8MBDEìwÌMNB2á
(2)Ñ8MdyYBWX_ 1 Ħ'o8MBDE!
(1)M(−1
2,4)'51
2
(2)M(1,7)`(−1,3)
)@AcDE~'¢£WXlCħĨÃ'íDELéCĨÃ':ÆWXoDEĦ'ĩbĤ
ƒij^Bò	'½CB†!
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
$1+FGMN∥xY'Md M'N 8BkDE„e'oN m B^'Md8 M BDE'¸íMdwÌMNB2
]á
$2+FG8 M d y YBWX_ 1'Md|m−1|=1'oN m B^KLMd8 M BDE!
O$1+
∵ MN∥xY'
∴M'N8BkDE„e'
∵M(m−1,2m+3)'8N(5,4)'
∴2m+3=4'
∴ m=
1
2'
∴ m−1=−
1
2'
∴ M(−1
2,4)'        
∴wÌMNB2]=5−(−1
2)=5
1
2á
$2+∵8Md y YBWX_ 1'
∴|m−1|=1'
∴m−1=1`m−1=−1'
∴m=2`0'
∴2m+3=7`3'
∴M(1,7)`(−1,3)!
20!$8 )+$24-25 ,-./·Ô1ĪSīĬ.[\·45+’~Ÿ¶':QRSTDEUV'8A'BBD
E)Õ_A(−2,0)'B(a,0)'êVA:BBĢĭ×AB=6'8CBDE_(0,3)!
(1)oaB^ĮS△ABCá
(2)Ö8M:xYBmĔYƒ'×S△ACM=1
3S△ABC'o8MBDE!
(1)a=4'S△ABC=9
(2)MBDE_(0,0)`(−4,0)
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
)@AQRSTDEUVDE>TRĕ«¬'¢£8BDEHwÌ2BUCãn
B†!
$1+67AÎB:xYƒ×A:BĢĭ'AB=6'čªxYƒu8įWXė$u8hDEİıBij^ +'Æ
A(−2,0)'B(a,0)LMa−(−2)=6'n·ĊoNaB^áÓFG>TRĕė'éAB_IJ'8CdxYWX_
ij'«¬△ABCRĕ !
$2+ĖM(x,0)'àĀ¶NAMB2]'FGS△ACM=1
3S△ABCoNS△ACMB^'Óčª>TRĕėxNfx
B·Ċ
1
2×|x+2|×3=3'o·ĊMdxB^'§í{ïMBDE !
O$1+∵A(−2,0)'B(a,0)×A:BĢĭ'AB=6'
∴a−(−2)=6'Ka+2=6'
Ma=4!
 ∵8C(0,3)
∴OC=3
∴S△ABC=1
2×AB×OC=1
2×6×3=9á
$2+
ĖMBDE_(x,0)'9AM=|x−(−2)|=|x+2|!
∵S△ACM=1
3SABC'S△ABC=9'
∴S△ACM=3! 
△ACMéAM_IJ'ij_8CdxYBWX 3'
SΔACM=1
2×AM×3=3!
K
1
2×|x+2|×3=3'
Mx=0áx=−4!
∴MBDE_(0,0)`(−4,0)!
21!$10 )+$24-25 p-.ƒ·ēÿĴÊ·45+:QRSTDEUV'Ò 8P(2x−1,3x)!
(1)Ö8P:yYƒ'oxB^á
(2)Ö8P: ;<'×duDEYBWXİì_ 9'o8PBDE!
(1)x=1
2
(2)(3,6)
)@abAcDE~'8dDEYBWX' ;<I8BDEJ8': y YƒB8BD
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
EJ8' ¡¢£C@B†!
$1+:yYƒB8hDE_0'GnxN·ĊoKLá
$2+ ;<IB8hkDEl_m'8dxYBWX_g8kDEBij^'8dyYBWX_g8hD
EBij^'GnoN8PduDEYBWX'ÓFG8PduDEYBWXİì_ 9 ˜™·ĊoNxB^K
LMd!
O$1+
∵8P(2x−1,3x):yYƒ'
∴2x−1=0'
∴x=
1
2á
$2+
∵P(2x−1,3x): ;<'
∴8PdxYBWX_3x'dyYBWX_2x−1'
∵8PduDEYBWXİì_ 9'
∴3x+2x−1=9'
∴x=2'
∴2x−1=3,3x=6'
∴8PBDE_(3,6)!
22!$10 )+’~':QRSTDEUV'8A'BBDE)Õ_(−1,0)'(3,0)'ĵ…Ħß8A'B)Õ®ƒ
Q¤ 3 Z[\'Ó®¯Q¤ 1 Z[\')ÕMd8A'BBj©8C'D'Ķ»AC'BD'CD!
(1)o8C,DBDEĮ?æABDCBRĕS四边形ABDCá
(2):yYƒCķĸ: 8P'Ķ»PA,PB'šS△PAB=S四边形ABDCÖĸ:½¾ 8'oN8PBDE
ևĸ:'
yď±Æ!
(1)C(0,3)'D(4,3)'S四边形ABDC=12
(2)ĸ:'8PBDE_(0,6)`(0,−6)
)@Ac8BDEJô'8BQ¤'>TBRĕ' ¡¢£7Ð8CB†!
$1+FG®¯Q¤ 1 Z[\'hDEĤ 1'®ƒQ¤ 3 Z[\'kDEĤ 3'KLoN8C,DBDE'Óo
NCD2'KLoRĕá
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
$2+Æ$1+M?æABDCBRĕ_ 12'Óčª>TRĕėKLo!
O$1+
 ∵8A,BBDE)Õ_ −1'0 ' 3'0 '
ĵ…Ħß8A,B®ƒQ¤ 3 Z[\'Ó®¯Q¤ 1 Z[\'Md8A,BBj©8)ÕCC,D'
∴C 0'3 ,D 4'3
?æABDCBRĕ=(3+1)×3=12á
$2+
ĖS△PAB=S四边形ABDCĦ8PdABBWX_ℎ'
9
1
2×(3+1)ℎ=12'
Mℎ=6'
∴8PBDE_ 0'6 ` 0'−6 !
23!$12 )+$24-25 ,-./·ĈČĹÊ·4V+:QRSTDEUV'ĺN’/ïĻ
8 P d x Y'y Y
BWXBļ^ó_8 P B“2W”'8 Q d x YÎy YBWX„eĦ'ó8 Q _“Ľľ8”!
(1)8A(−3,5)B“2W”_______á
(2)Ö8B(4−2a,−2)C“Ľľ8”'o a B^á
(3)Ö8C(−2,3b−2)B2W_ 4'×8 C :=;<I'8 D BDE_(9−2b,−5)'yď
8 D C“Ľľ8”!
(1)5
(2)a=1或a=3
(3)ġ
)@abAcQRSTDEU'8dDEYBWX'ĿfŀŁ±łŃń'†CbŁŅ
ńņïĻB“2W”“Ľľ8”!
$1+FG“2W”BïĻKLá
$2+FG“Ľľ8”BïĻKLá
$3+Æ“2W”BïĻoN b B^'ŇĒFG“Ľľ8”BïĻoKL!
O$1+
FGÁ'M8A(−3,5)d x YBWX_ 5'd y YBWX_ 3'
∴8 A B“2W”_ 5!
P_
5!
$2+
8B(4−2a,−2)C“Ľľ8”'
∴|4−2a|=|−2|'
∴4−2a=2`4−2a=−2'
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
M
a=1`a=3á
$3+
8C(−2,3b−2)B2W_ 4'×8C:=;<I'
∴ 3b−2=4'Mb=2'
∴9−2b=5'
∴8DBDE_(5,−5)'
∵8DdxYÎyYBWXlC 5'
∴DC“Ľľ8”!
24!$12 )+QRSTDEUV'A 8_(−3,0)'D 8_(0,4)'ßwÌADQ¤ňwÌBC'ĶAB'CD!
(1)’~ 1'Ö B 8_(−2,−2)!
①S»ĚN~V„eìQ¨BwÌì C 8DEá      
②o?æABCDBRĕá
(2)’~ 2'ÖOAQ)∠DAB'oʼn
ODQ)∠ADC!
(1)①AD=BC'AB=CD'AB∥CD'AD∥BC'(1,2)á②10
(2)ġ
)@abAcDE~¿—Q¤'Q¤B€'DE~'Q¨wB€'TQ)wB
ïĻee' 7„7ÐCB†!
$1+①ÆQ¤B€LMAD=BC'AB=CD'AB∥CD'AD∥BCáFG8 A ì8 B BDELMQ¤·ė_®
¯Q¤ 1 Z[\2]'®/Q¤ 2 Z[\2]'ÓÆ8 D BDEÙÚQ¤·ėLM8 C BDEá②t8
A'C )Õv y YBQ¨wEH'FG't8 BÎD )Õv x YBQ¨wHG'EF'FGS四边形ABCD=S四边形EFGH
−S△ADH−S△CDG−S△ABE−S△BCExėoKLá
$2+ÆTQ)wBïĻMd∠OAD=1
2∠BADáÆQ¨wB€Md∠BAD+∠ADC=180°'9LMd∠OAD+1
2∠
ADC=90°'Æ>TITìï±LM∠OAD+∠ODA=90°'9∠ODA=1
2∠ADC'KLʼnďODQ)∠ADC!
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
O$1+
①ÆQ¤B€LMAD=BC'AB=CD'AB∥CD'AD∥BCá
∵A 8_(−3,0)'D 8_(0,4)'ßwÌADQ¤ňwÌBC'B 8_(−2,−2)'
∴Q¤·ė_®¯Q¤ 1 Z[\2]'®/Q¤ 2 Z[\2]'
∴8 C BDE_(0+1,4−2)'K(1,2)á
②’~Ÿ¶'t8 A'C )Õv y YBQ¨wEH'FG't8 BÎD )Õv x YBQ¨wHG'EF'
∴E(−3,−2)'F(1,−2)'G(1,4)'H(−3,4)'
∴S四边形ABCD=S四边形EFGH−S△ADH−S△CDG−S△ABE−S△BCE 
=6×4−1
2 ×1×2−1
2 ×3×4−1
2 ×1×2−1
2 ×3×4
=24−1−6−1−6
=10á
$2+ʼnď
∵OAQ)∠DAB'
∴∠OAD=1
2∠BADá
∵AB∥CD'
∴∠BAD+∠ADC=180°'
∴1
2∠BAD+1
2∠ADC=90°'
∴∠OAD+1
2∠ADC=90°'
∵∠AOD=90°'∠AOD+∠OAD+∠ODA=180°'
∴∠OAD+∠ODA=90°'
∴∠ODA=1
2∠ADC'
∴ODQ)∠ADC!
教辅资源，关注公众号★全科AA+