2017-2021年山东中考数学真题分类汇编之图形的变化

《2017-2021年山东中考数学真题分类汇编之图形的变化》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

2017-2021
ὃᦪ⚪ᑖḄᓄ.⌱⚪ᐳ16⚪1.2020•☘᝞ᡠḄ!"#$ᐸ&'(3.2021•☘*+(ᡃ-.ὁḄ0123.4ᑡ67*+ᫀ9:;<Ḅ(

15.2021•☘᝞$=>?ABᐜABC஺ᢥ⌮G┐IᔣKL90°,Nᔣ4OP47ᓫR$Sᑮ>?ᑣCAḄ;VC4Ḅᙶ᪗(A.1,-6
B.-1,6
C.1,-2
D.--2
6.2021•Z[᝞ᡠḄ!"#$;ᐸ\']^_`Ḅ(A.a'b&'cdB.\7'efcdC.g'ba'cdD.g'b&'cd7.2021•hᓭjklmn⍩pᢈ3rstVᵨvwxᱥz{.᝞$|w}ᱯ⁐ᙢᵨjkl;ᙽᵪz{x}G$ᙠᙢ☢9135%Ḅ4ᜐ{SᵪNᜐ&943°,jklᚖ440,“Bᜐ${Sᵪaᔊᜐ&935°,UM,N1Ḅ9ὃᦪtan43°^0.9,sin43°g0.7,cos35°QO.8,tan35°0.7,¡¢£ᶇ᦮ᦪ

2C.286mD.312,7?8.2021•Z[|ᖪ§¨ᜓᦋᗐBᨵ®Ḅ¯ᐰឋ²$³ᚁᵫ37°¶30°,r·B
ὃᦪsin37°¾¿$cos37°¾Àtan37°®¸95¹$º᦮»Ḅ®¼½¸55A.6¹B.3¹C.2¹D.1¹9.2021•Á᯿ÃÄOÅÆḄÇÈÉ᤮ÅḼÌÍ7ÎÈ$ᐸ\'᝞ᡠ$ᑣÏÃÇÈÉᐳᨵÎÈḄ7ᦪ9B.12C.14D.1810.2021•☘᝞$ᙠ6+ᱏ488$AD//BC,AB=10,/B=60°,=+ᱏᢚÒ$ÓCBÔᙠADÉḄCGᜐ$ᢚÕ9EF,ÌNBFE=45°,ᑣBFḄ¸9

3A.5B.3ÖC.5MD.×511.2021•ØÙ᝞$ÚÄOᐝ>᯿ᙠᨵÜÝÞḄO☢ßÉ$Ìᐭáᐝ>âãáᐝ>Ḅᜳ960°,ᑣO☢ßḄᚖ>âÄOᙢ☢ḄᜳaḄᦪ(
ᐝ12.2020•hᓭ᝞$å4BCஹåçEDèé9êëᕒḄíè$êëᕒ'>PBâᙢ☢8EḄᜳNP8E=43°,'>PEâᙢ☢8EḄᜳ/PEB=20°,CA,ç9'>âîïðñḄòC$AF//BE,AC±BE,FDLBE.ÌACᑮBCḄA8=16”$ᑣíèOEḄ¸(ὃᦪsin43°"0.7,tan43°*0.9,sin20°«=0.3,tan20°^0.4
A.2.6/zzB.2.8"?C.3AmD.4.5m13.2019•⃩᝞$ᙠ_IA8C஺$C஺(;>ACஹBQḄòC$ôC஺xá>OMஹONᑖõòBCஹCD
EஹF,ö/EOF=90°,OCஹEF
G.÷ã4ᑡ¡øজåCOEgZX஺஺/ûঝOGEsFGCঞ6CEOçḄ☢ý9_IABC஺☢ýḄþûট஺
+Bf2=OGOC.ᐸḄ4A.জঝঞটB.জঝঞC.জঝটD.ঞট

414.2021•᝞ᙠRtZiAOBNAOB=90°,OA=3,OB=4,஺ᙊ2ḄᙊOBC,!C"CC08ABD,#$OA%Ḅ&.(PC+POᨬ*+OPḄ,24215.2019•-ᓭ/ᦪ1234567ឋẆ:ᙠᜧ<ᓭ=A>?ᔊABCᙠᓅDE37°FᔣHIᔣᓅJ1053KᑮMNṹᜮB,>?ᔊABCᙠNṹᜮBḄᓅDE53°Fᔣ.RSTUAᓭ=AᔊABCVWḄXYZ[ὃᦪ]^tan37஺33,tan53஺FᓉA.225mB.215mC.300mD.315nz16.2021•᝞ᙠF`A8COPAB%U&a4ஹBcᔠ,fghACஹ8஺i0,!Pᑖk"ACஹ8஺ḄᚖhᑖkACஹBDEஹF,A஺ஹ8cMஹN.Aᑡno^জpqr£tupq“£tঝPM+PN=ACঞP*+PF2=PO2ট/\P0Fs/\BNFঠ஺ᙠMஹNxḄyh%.ᐸḄ

5A.জঝঞটB.জঝঞঠC.জঝঞটঠD.ঞটঠ~⚪(ᐳ3*⚪)17.(2019•)᝞ᙠgᙶ᪗*F`Ḅ$,ᙳ1ᓫ,pA3஺Ḅ⚔ᙶ᪗ᑖkA(-2,-1),8(-2,-3),O(0,0),pA1B1O1Ḅ⚔ᙶ᪗ᑖkAl(1,-1),B\(1,-5),0\(5,1),pA3஺pA1B101஺Ḅ`ᑣ#Ḅᙶ᪗.18.(2018•᯿)᝞U¡¢Ḅ£¤(¥¦ᓫ^cm),᪷]ᡠ«ᦪ]ST¬¡¢Ḅ⊤☢¯19.(2019•°±)᝞ᙠ²`A8C£)AB=4,BC=JWE#CD$%&,·4BCE¹BEᢚ»¼?C½ᑮ²`ᑁFḄÀyÁAF,Âtan/BAF=ÇᑣCE2

6£.ÈÉ⚪(ᐳ4*⚪)20.(2018•°±)ÊËpA8CAB=AC,/84C=90°,DஹEᑖkABஹACḄ,·pADEÌAᢥÎ+┐FᔣÐÑUga(0°,8஺ACE,஺஺F.(1)Ò/D4FḄᦪt(2)ÒÓ^AE2=EF'ED(3)ÒÓ^஺஺Ḅᑗh.22.(2020•E⃩)᝞1,ᙠÛῪ£g`48c/A=120°,AB=AC,஺ஹEᑖkᙠ$ABஹ4c%AD^AE,yÁBE,ஹMஹNஹPᑖkOEஹBEஹ8cḄ.(1)ÝÞßà.1háNMஹNPḄᦪâᐵ,NMNPḄᜧ)#.(2)ä:Ó<åÌAÎ+┐FᔣÐÑᑮ᝞2ᡠ«ḄÀyÁæ#ஹBDஹCE,ᑨè

7Ḅ`éêë<ᳮᵫt(3)î5ïðåpADEÌAᙠ×☢ᑁñᵫÐÑÂAO=1,48=3,RÒòpMNP☢¯Ḅᨬᜧó.23.(2020•-ᓭ)ᙠÛῪpABCAC=BC,pADEg£g`ZDAE=90°,ZADE=L/ACB,yÁ80,BE,F8஺ḄyÁCF.2(1)(NCAB=45°+.জ᝞1,(⚔஺ᙠ$AC%+RÁᑏòNE4BNCBAḄᦪâᐵ.háBEháCFḄᦪâᐵtঝ᝞2,(⚔஺ᙠ$AB%+(1)háBEháCFḄᦪâᐵᔲöᯠᡂùúÂᡂùRûüÓ<ÂaᡂùRë<ᳮᵫt1ýþ!ÿ
⚪Ḅ
ᜧὃῪ48CḄCM,BEḄ!M"ᑭᵨ%&'ᐰᡈ*+ᨵᐵ./ᩭ⚪12஺EḄ!G,454G,CG,6CAG7!C⌮9┐;<90°,"ᑭᵨ;<ឋ?ஹ%&'ᐰᡈ*+ᨵᐵ./ᩭ⚪.(2)BNC4B=30°9
᝞D3,B⚔!஺ᙠAC9
ᑏHIBEJHICKḄᦪMᐵN
OPᳮᵫ.

82017-2021TUVὃᦪWX⚪ᑖZ[\]D'Ḅ^ᓄὃ`ᫀJb⚪᪆.⌱e⚪fᐳ16h⚪i1.2020•☘li᝞DᡠnḄopq
ᐸstDufiூὃ!௃xᓫzᔠqḄ%tD.ூ|⚪௃ᢗ~JtD1op.ூᑖ᪆௃ᑮ☢ᡠᑮḄD'ᓽ
ᡠᨵḄᑮḄ⊤ᙠstD.ூ`௃☢u'
'ḄᜐᨵᩩᔣḄH
HḄᨵᩩᔣḄH.ᦑ⌱A.ூ!௃⚪ὃ%tDḄ./
stDuᱥqḄ☢ᑮḄtD.2.2019•¡¢£iᑡD'
¥u¦§¨
©uª§¨ḄufiA.C.ூὃ!௃¦§¨D'1ª§¨D'.ூ|⚪௃«¬ஹ;

9Cஹ¶u¦§¨D'
uª§¨D'
ᦑ⌱⚗┯¹;஺ஹu¦§¨D'
¶uª§¨D'
ᦑ⌱⚗┯¹.ᦑ⌱B.ூ!௃⚪ὃ¦§¨D'Jª§¨D'Ḅᭆ±.ª§¨D'Ḅᐵ½u¾§¨ª
D'¿ᑖᢚÁÂÃᔠ
¦§¨D'u⌕¾§¨¦
;<180Å¿ᑖÃᔠ.ூὃ!௃ª§¨D'.ூ|⚪௃«¬ஹ;

10D.ூὃ!௃xᓫopqḄ%tD.ூ|⚪௃ᢗ~JtD1op.ூᑖ᪆௃ÊÈ☢ËopqᡠᑮḄD'ᓽ.ூ`௃ËopqḄÈtDÌᦑ⌱A.ூ!௃⚪ὃxᓫopqḄ%tD
ᳮtDḄÎ
ÏÐ%tDḄÊÑu»¼`ᫀḄÒÓ.5.(2021•☘l)᝞D
ÚHIᐜ7Ü!஺ᢥ⌮9┐Þᔣ;<90°,"ᔣ«¬4ᓫâ
ᑮHIAb,ᑣ!AḄ§!4Ḅᙶ᪗u()A.(1,-6)B.(-1,6)C.(1,-2)D.(-1,-2)ூὃ!௃ᙶ᪗JD'^ᓄ-;<1ᙶ᪗JD'^ᓄ-«¬.ூ|⚪௃«☢&ᙶ᪗N1«¬ஹ;

11ᓫâÂḄ!Ḅᙶ᪗ᓽ.ூ`௃A!7஺!⌮9┐;<90°,ᑮ!A"(-1,2),A"ᔣ«¬4ᓫâ
ᑮ4(-1,-2),ூ!௃⚪ὃᙶ᪗JD'^ᓄ
íï᪷®⚪ÊHIAB;<ஹ«¬ÂḄD'u⚪Ḅᐵ½.6.(2021•ðñ)᝞DᡠnḄopq
§ᐸ%tDòó»¼Ḅu()A.ÈtD¯stD*ôB.%tD¶*ôC.õtD¯ÈtD*ôD.õtD¯stD*ôூὃ!௃xᓫzᔠqḄ%tD.ூ|⚪௃ᢗ~JtD1ö±.ூᑖ᪆௃ᑖ÷øùpqḄ%tD
úû`ᫀ.ூ`௃᝞Dᡠn

12ᦑøopqḄõtD¯ÈtD*ô.ᦑ⌱C.ூ!௃⚪ὃ%tDḄ./
»¼6Ð%tDḄÊÑu⚪ᐵ½.7.2021•üᓭiþÿ
⍩ᢈᵨᱥ.᝞ᱯ⁐ᙢᵨ!"#ᙽ%&ᵪ()*ᙠ,ᙢ☢./0135,”ḄAᜐ3%&ᵪ4567Nᜐ89043°,!ᚖ<=>40,"@BᜐA3%&ᵪB567Mᜐ89035°,ᑣM,NDEḄ,F0GIGJὃᦪMNtan43°O0.9,sin43°P0.7,cos35°Q0.8,tan35°^0.7,RSTᶇ᦮ᦪIA.188wB.269mC.286/nD.312WூὃY௃[<9\9]Ḅᵨ-_989`⚪.ூb⚪௃[<9\9]cᐸᵨefᳮhi.ூᑖ᪆௃᪷M⚪m3no<9\9]pAONஹ4BOM,rs[tno<9\9]u3OBஹONḄv/(wᓽyuz{ᫀ.ூ[{௃[Nᵫ⚪m3NNN=43°,ZM=35°,AO=135m,BO=AO-AB=95m,ᙠRt/XAON~taiW=9=lan43°-NO..NO=——%150஻?tan43°ᙠ~,

13tanM=^i=tan35°,MOMO=——P135.1m,tan35..MN=MO+NO=135.7+150g286”.ᦑ⌱NC.ூY௃⚪ὃ[<9\9]Ḅᵨ-_989`⚪.[⚪ḄᐵRᔠ]ᑭᵨ\9¡ᦪ[<9\9].8.G2021•£¤Iᖪ¦§ᜓᦋᗐ«ᨵ®Ḅ¯ᐰឋh²ᚁ9ᵫ37°´@30°,µ¶«®v05·¸᦮¹Ḅ®º»vGIGJὃᦪMNsin37°O3,cos37°Q¿tan37°55A.6·B.3·C.2·D.1·ூὃY௃[<9\9]Ḅᵨ-ᚁ/ᚁ9`⚪.ூb⚪௃[<9\9]cᐸᵨeᵨmÄ.ூᑖ᪆௃᪷MÅÆḄÇÈuzBO,᪷M<9\9]ḄឋÊËÌ3ᑮ{ᫀ.ூ[{௃[NᙠRtZ\BA£I~48=5·NB4D=37°,ᑣ8£>=AB,sinNBAOQ5x3=3G·I5ᙠRtZ\BC£>~ZC=30°,..BC=2BD=6G·Iᑣ¸᦮¹Ḅ®º»vN6-5=1G·Iᦑ⌱ND.ூY௃⚪ὃḄ[<9\9]Ḅᵨ#ᚁ/ᚁ9`⚪×Øᚁ9ḄᭆÚஹÛÜ┦9\9¡ᦪḄÇÈ[⚪Ḅᐵ.

149.2021•Þ᯿I#àáâãäḄåæç᤮ãḼêëoìæᐸ\í᝞ᡠᑣtàåæçᐳᨵìæḄoᦪ0GIB.12C.14D.18ூὃY௃ᵫ\íᑨñòóô.ூb⚪௃òó<õ.ூᑖ᪆௃ö8í÷øᨵ\ᑡìæúí~y¶B5ìæᨵ6o45ᨵ2o᪷M\íḄûüy[{ý⚪.ூ[{௃[Nö8íy¶ýåæᐳ᤮ãḼ\ᑡìæ.úíy¶B5ìæᨵ6o45ᨵ2owBíy¶B5ᨵ4o45þúíḄB5ìæÿ
ᐳ12
ᦑ⌱B.ூ௃⚪Ḅᜧ
⌕ὃḄᙠ!"#$%Ḅ&ᵨ.10.2021•☘+,᝞
ᙠ./01ᱏA8CQ%
AD//BC,AB=10,NB=60°,41ᱏᢚ6
7B8ᙠAD/9ḄGᜐ
ᢚ;
bc./0AHFGd0
ef\aAH=GF,

15ᓽhijF⚪.ூik௃iᵫᢚ6BF=GF,NBFE=NGFE,■:NBFE=45",:.NBFG=90°,ᙠRt^AB”%
AH=sin60°l48=mn1஺=5«,'AD//BC,...NGAH=/AH8=90°,:.NGAH=NAHB=NBFG=90°,..../0A”qGd0
:.FG=AH=5M.BF=GF=5-/3-ᦑ⌱C.ூ௃⚪⌕ὃrCᢚḄឋL
HIJḄឋL
d0ḄᑨtRឋL
ᱯvwḄwxᦪm
^z{J᪀⌼~ww0i⚪Ḅᐵ.11.2021•,᝞
HᐝJ᯿ᙠᨵtwḄH☢9
ᐭᐝJRaᐝJḄᜳw<60°,ᑣH☢ḄᚖJRHᙢ☢ḄᜳwaḄᦪ?,
ᐝூὃ௃i~ww0Ḅ&ᵨ-ᚁᚁwF⚪.ூM⚪௃i~ww0ᐸ&ᵨGWXGUᳮWX.ூᑖ᪆௃^CO1■H☢
ᚖ

16ᙢ☢ᡠᙠ~JHI
fh\ᜳwaḄᦪ.ூik௃i᝞
^H☢
ᚖ?,?µὃᦪsin43°¶0.7,tan43°·0.9,sin20°«=0.3,tan20°*=0.4
A.2.6mB.2.8/wC.3.4mD.4.5mூὃ௃i~ww0Ḅ&ᵨGஹw¸¨.ூM⚪௃i~ww0ᐸ&ᵨG&ᵨ¢.ூᑖ᪆௃✌ᐜc»./0AC஺qd0
`aAC,XqᓽhijF⚪.

17ூik௃iAC±EB,C.DF//AC,'CAF//EB,../0ACDFHI./0
VZACD=90°,..../0ACQFd0
.DF=AC,ᙠRtaACB%
VZACB=90°,
AC=AB.sin43°·1.6X0.7=1.12'm(,.DF=AC=IA2Cm(,ᙠRtz^OEF%
•NFDE=9Q°,tanZ£=-5L,DE12=2.8?Û,,0.4ᦑ⌱B.ூ௃⚪ὃi~ww0Ḅ&ᵨ
i⚪Ḅᐵᳮi⚪¢
ᮣ$ᵨᡠÝijF⚪
Þ_%ὃßὃ⚪à.13.?2019•â⃩,᝞
ᙠäå0A8CD%
஺SwJACஹBXḄ
]஺^JOMஹONᑖèBCஹCDEஹF,éNE஺q=90°,OCஹEF3ஹG.êaëᑡíîজ§COEg/^஺஺qGঝ§OGfs§õGCGঞ./0CEOFḄ☢÷<äå0ABCD☢÷ḄøG@DF2+BE2=OG'OC.ᐸ%äùḄ?,4A.জঝঞটB.জঝঞC.জঝটD.ঞটூὃ௃ûüw0ḄᑨtRឋLGᐰmw0ḄᑨtRឋLGäå0ḄᑨtRឋL.

18ூM⚪௃d0þ0äå0.ூᑖ᪆௃জᵫäå0c»OC=OZ),NODF=NOCE=45°,/COM=NDOF,ÿ
ঝᵫᐰ
OE=OR
NOEG=/FCG=45°,ᑭᵨ⚔
AOGEsaFGCঞᐜSACOE=SஹDOF,$QOF=S[SABCDA0CDটOEGs/\OCE,
OGOC=O4,BE1+DF2^EF2,ᵫEF>OE,
.ூ௃"জABC஺%OC=OD,AC1BD,NODF=NOCE=45°,■:NMON=90°,..NCOM=/DOF,.♦.COE&OOF(ASA),ᦑজ(ঝ;COECOF,OE=OF,■:NMON=90°,..ZOEG=45°=ZFCG,'.ZOGE=ZFGC,:.AOGEsAFGC,ᦑঝ(:SACOE=SஹDOF,..SCEOF=SA0CD)ABCDᦑঞ(ট:COEOOᐘOE=OF,+"NEOF=90°,/XEOF%Ὺ-,

19..ZOEG=45°=ZOCE,NEOG=NCOE,:.AOEGSAOCE,..OE.OC=OG.OE,..OG-OC=OE1,,:CE=DF,BC=CD,..BE=CF,+RtACEF/CF2+CE2=EF2,..B^+DF1=EF1,".EF2>OE1,:.BE1+DF2>OG-OC,ᦑট┯1ᦑ⌱"B.ூ34௃5⚪78Ḅ:ᔠ⚪<⌕ὃ?@ḄឋBᐰḄᑨDEឋBஹGḄᑨDEឋBஹHIDᳮḄ:ᔠKᵨ.⚪MNO"ᐰḄPGḄPᡂRS.14.2021•UVWX᝞ZᙠRtZ\AOB/NAO8=90°,04=3,08=4,\30]ᙊ_2]`aḄᙊE08b83C,c3Cdb4B83஺3P%04eḄf3.gPC+P஺ᨬiM0PḄj]kX

20A.AB.3c.1D.3242ூὃ3௃tu-ᨬwxyz⚪{|yᑖy~ᡂRS.ூ⚪௃y~ஹஹbyE{|y{ஹEuEᙊᨵᐵḄPᵨO.ூᑖ᪆௃jCOb஺஺83E,EP,bAO83P,ᑣPC+P0Ḅᨬiᑭᵨ{|yᑖy~ᡂRSᑖCD,POḄjᓽ.ூ௃"᝞ZjC஺b83£,E஺bA஺83P,MPC+POḄᨬi.•:CD1.OB,..ZDCB=90a,+NAOB=90°,ZDCB=ZAOB,..CD//AO-BC_CD"BO'AOVOC=2,OB=4,..BC=2,2_=CD
czX=3A432\'CD//AO,EOPOᓽ2=¤
PO=3A=ECDC4342ᦑ⌱"B.ூ34௃⚪<⌕ὃ?@tu——ᨬw¦§z⚪¨Mὃ?@{|yᑖy~ᡂRS,©ªtuឋB«{|yᑖy~ᡂRSDᳮ%⚪Ḅᐵ¬.15.k2019•®ᓭX°ᦪ²³´µ¶·¸ឋẆºᙠᜧVᓭ¼A½
ᔊ¿ÀCᙠᓅÂÃ37°ᔣÆÇᔣᓅÈ105᪷ÊᑮÌÍṹᜮB,½
ᔊ¿ÀCᙠÍṹᜮ8ḄᓅÂÃ53°

21ᔣ.ÒÓ¿ᓭ¼AEᔊ¿ÀCÔÕḄ¦§Ö]kXk×ὃᦪØ"tan37஺ÙÚ,ூὃ3௃-ḄPᵨ-ᔣz⚪.ூ⚪௃-ÛᐸPᵨ.ூᑖ᪆௃᝞Zd8E.ÝEC=xm,BE=ym.᪀ßàáx,yᓽâz⚪.ூ௃"᝞ZdCEJ_B48E.ÝEC=wi,BE=ym.即4X---ᙠRtZ\ECB/tan53°=EC-3yEBEC-ᙠRtZ\AEC/tan370=AE
x=180,y=135,AC=VEC2+AE2^V1802+2402=300ᦑ⌱"C.ூ34௃5⚪ὃ?-ḄPᵨ-ᔣã⚪Ḅᐵ¬%²äåæçᵨèéy᪀⌼-âz⚪²äᑭᵨ×ᦪ᪀ßàáâz⚪78/ὃçὃ⚪ë.16.2021•UVWX᝞ZᙠABC஺/3ì%ABeÓf3k)EAஹ8íᔠX,

22yACஹb83஺c3PᑖdACஹ8஺ḄᚖyᑖbACஹBDEஹF,bA஺ஹ3c83MஹN.¿ᑡ"জðñ£"òAMEঝPM+PN=AC;ঞPE"2+PF2=po2ট4P0FS/\BNFঠ3஺ᙠMஹNô3Ḅye.ᐸ/(Ḅ%kXA.জঝঞটB.জঝঞঠC.জঝঞটঠD.ঞটঠூὃ3௃GḄᑨDEឋBᐰḄᑨDḄឋB.ூ⚪௃õöPᵨO.ூᑖ᪆௃÷ØḄឋB\ÛHIDᳮஹõḄᑨDøᓽᑨù4PM«ABPA\9XAPEஹABPìú%Ὺ-PEOF%õûüdᑨù.ூ௃"•••ABCO%/BAC=ND4C=45°.:ᙠ?1ñ£"«/ZPAE=ZMAE•AE=AE.ZAEP=ZAEM.".APE^AAMECASA1,ᦑজ:.PE=EM=LPM,2ᳮFP=FN=LNP.2•.,ABCDAC_LB஺PE_LAC,PFLBD,..NPEO=/EOF=NPF0=9G°,H./XAPEAE=PE

23...PEOF..PF=OE,..PE+PF^OA,FP=FN=LNP,OA=1AC,222..PM+PN=AC,ᦑঝ•.•PEOF..PE=OF,ᙠOPFO>+Pᓲ=PO2,..PE1+PF2=PO2,ᦑঞ.•••BNFῪ!POF"#$Ὺᦑট┯';)*OM,ON,'.0Aᚖ,ᑖ./PM.0Bᚖ,ᑖ./PN,..OM=OP,ON=OP,..OM=OP=ON,.•.0஺Ḅ᜜*ᙊḄᙊ4,.ZMPN=90°,5MN6AM,0,Nᐳ.ᦑঠ.ᦑ⌱:B.ூ0<௃>⚪ὃAḄឋCஹḄᑨ$ஹFG$ᳮHIJIAPMK8PNLMAPEஹ8PFNῪPEOFᐵᏉ.5.QR⚪(ᐳ3T⚪)17.(2019•VW)᝞Yᙠᙶ᪗\]^TḄ_ᙳa1^ᓫc_dAB஺Ḅ⚔0ᙶ᪗ᑖfaA(-2,-1),B(-2,-3),O(0,0),AAIBIOIḄ⚔0ᙶ᪗ᑖ

24faAlh1,-1k,Bih1,-5k,Oih5,1k,ABOoAiBi01L0racs4ḄcsYᑣr0Ḅᙶ᪗ah-5,-1k.ூὃ0௃csuᣚᙶ᪗oYឋC.ூw⚪௃YḄxs.ூᑖ᪆௃ᑖfz_84ஹ010ஹ44,}~x0P,ᯠᑏP0ᙶ᪗ᓽ.ூ௃᝞YP0ᙶ᪗ah-5,-1k.ᦑᫀah-5,-1k.ூ0<௃>⚪ὃAcsuᣚ᝞^Y"xsY!⚔0Ḅ).x#0x,᪵Ḅ^YcsY,^0cs4.csYḄឋCᨵ^Yxs0Ḅ).N#0,ᡈᐳ..18.h2018•¢᯿k᝞Y#^¤¥¦Ḅ§YhY¨©ᓫccmk,᪷Yᡠ¯ᦪ±²^¤¥¦Ḅ⊤☢µ4nc/«2

25ூὃ0௃ᵫ§Yᑨº¤¥¦ᙊ┵Ḅ±².ூw⚪௃¤¥Y.ூᑖ᪆௃ᵫ¼§YK½§Y$¾¦┵¦¿ᳫ¦Áᵫ§Y$ᐹ¦Ä$ᙊ┵ḄÅ._KÆ☢Ç6È!$ᐸ⊤☢µ.ூ௃ᵫ¼§YK½§Yaᑨº┵¦ᵫ§Yᙊᑨº^¤¥¦Êᙊ┵᪷§YHÊᙊ┵ḄÅ._a
h2&2+]2=Ë7஻,Æ☢Ç6a1cm,ᦑ⊤☢µ=Í”+11/=11Ð1X3+TTXl2—4TTcm2,ᦑᫀa4ircm2,ூ0<௃ὃAÑÒ§YÓÔÕdKᮣ×ØᵨÚÛᐵÜᵫ¼§YK½§Y$¾¦┵¦¿ᳫ¦.19.2019•ÝÞßk᝞YᙠA8CD4B=4,8C=JQ,EaC£à#0á48CEâ8EᢚäåæCçᑮᑁ0rḄcê)*AF,ëtan/B”=4ᑣCE=25-⍩ூὃ0௃íᢚuᣚhᢚäî⚪kḄᑨ$oឋC.ூw⚪௃Ὺoï,ðஹñòoóYḄxsMᐸᵨ.ூᑖ᪆௃ôHtan/BAF=õö÷ø.᪀⌼ûüHᦪᑭᵨFG$ᳮ2þFMஹBM,ÿ
FN,ᑭᵨᢚḄឋ
EC.

26ூ௃MN஻A஺ABஹC஺ᑖ!"஻ஹN,ᑣMN_LA8,MN1CD,ᵫᢚ%EC=EF,BC=BF=^ZC=ZBFE=90°,&/W=x,ᑣAM=2%,BM=4-2x,2AMᙠ,ᵫ-./ᳮ%%2+(4-2x)2=(5/5)2,%X]=\,X2=4L>2235.FM=\,AM=BM=2,.FN=y/s-1᧕GABMFS^FNE,...6᝞ᓽ⌭=2,EFFNEFV5-1%EF=-~J^.=EC.2ᦑᫀDE⌭.ூF௃ὃHIḄឋஹJḄKᐵMஹNOPḄឋQRḄឋSTUᔠ〉ḄXYZា\Ḅᑭᵨ⚪^,Ḅ_Tᩩabcd⚪Ḅᐵe..⚪(ᐳ4h⚪)20.(2018•jk)_TlABC,AB=AC,/8AC=90°,DஹEᑖ!bABஹ4cḄ,mnAᢥpq┐sᔣuvwxya(0°

27ூὃ௃Ḅᑨ/ឋSῪJ,Z/ᳮuvḄឋ.ூ⚪௃⚪.ூᑖ᪆௃(1)✌ᐜuvḄឋ,Ḅ/~A஺'=AE',ᯠᑭᵨS4S~lB஺'A^/XCE'A,ᨬᐰSḄឋ~ᓽ(2){|஺஺'ᐜ~lA஺஺'DSKᯠ~lA8஺'DJ|ᩭ~l8஺'sXAFE',ᨬḄឋ
ᓽ.ூ௃(1)~AB=AC,DஹEᑖ!bA8ஹACḄ,.AD=BD=AE=EC.ᵫuvḄឋTZDAD'=NEAE'=a,AD'=AD,AE'=AE..AD'=AE',:./XBD'A^/XCE'A,.BD'=CE'.(2){|Q஺'.'ZDAD'=60°,AD=AD',.AADD'bSK.AAADD'0=60°,DD'=DA=DB..ZDBD'=NDD'B=30°,AZBD'4=90°.VZDzAE'=90°,AABAE'=30°,

28:.NBAE'=NABDH-.4BFD'=NAFE',../\BFD's,•BFBD'BD',----=---:—=---;—•AFAE‘AD'Rt/\ABD',tan/BA஺'—=«,AD.•°=«.AFூF௃±⚪²⌕ὃHḄbᐰSḄᑨ/ឋஹḄឋᑨ/ஹuvḄឋ´µlBF஺'^/\AFE'b⚪Ḅᐵe.21.(2018•₹·)᝞}lABCᑁ|"஺஺AB=AC,ZBAC=36°,AAQ஻BC,NABCḄ¹ᑖZ"஺AC"E,஺0"F.(1)
ND4Ḅyᦪ(2)
~A^=EF,ED(3)
~AOb஺0ḄᑗZ.ூὃ௃Ḅᑨ/ឋᙊᕜ/ᳮᑗZḄᑨ/.ூ⚪௃ᙊḄᨵᐵᭆÁRឋᙊᨵᐵḄÂᐵM}ḄÃÄÅÆÇᳮÅÆ.ூᑖ᪆௃(1)
NABCஹNAC8Ḅyᦪ᪷ᙊᕜ/ᳮ
NAF8=NC=72°,᪷¹ZḄឋ
/£»,᪷Ḅ᜜ឋ
ᫀᓽ(2)
᪷Ḅឋ%ᓽ(3){|A஺
/஺4஺=90°ᓽ.ூ௃(1)'JAD//BC,:./D=NCBD,'.AB=AC,N54c=36°,AZABC=ZACB=^X(1800-NBAC)=72°,2

29:.NAFB=NACB=72°,'.ZD+ZDAF^ZAFB,..ZDAF^ZAFB-ZD=72°-36°=36°(2)~NCBD=36°,/FAC=NCBD,..ZFAC=36°=ND,ZAED^ZAEF,../\AEF^>/\DEA,•AE=EDI"EFAE";.A/=EFXED(3)~{|஺4ஹOF,NABF=36°,AZAOF=2ZABF=72°,\'OA=OF,..ZOAF=ZOFA=^X(1800-ZAOF)=54°,2ᵫ(1)T/D4=36°,..ZDAO=36°+54°=90°,ᓽOA^AD,OADÏÐb஺0ḄᑗZ.ூF௃±⚪ὃHÑᑗZḄᑨ/ᙊᕜ/ᳮᑁ/ᳮḄ᜜ឋSῪḄឋSTUÅÒᔠÃᵨ/ᳮÇᳮb6⚪Ḅᐵe.22.(2020•Ó⃩)᝞}1,ᙠSῪABC,NA=120°,A8=AC,஺ஹEᑖ!ᙠKABஹACÕAD=AE,{|BE,MஹNஹPᑖ!DOEஹBEஹ8cḄ,.

30(1)×ØÙÚ.}1,ZÛNMஹNPḄᦪÜᐵMbNM=NP,NMNPḄᜧ-N#60°.(2)ÝÞ~ßlAOEnApq┐sᔣuvᑮ᝞}2ᡠáḄÂ{|MPஹBDஹCE,ᑨâlMNPḄãäåᳮᵫ(3)æçèéßlAQEnAᙠ¹☢ᑁëᵫuvìA£>=1,AB=3,î
☢ïḄᨬᜧ.ூὃ௃ñòóᣚÒᔠ⚪.ூ⚪௃}ḄᐰSSῪJ¹õஹuvOPÃÄÅÆÇᳮÅÆ.ூᑖ᪆௃(1)ᐜ~ᵫAB=AC,AD=AE,OBD=CE,ᵫḄ,Z/ᳮ%NMNPḄᦪÜᐵMᵫ¹Zឋ%NMNḄᜧh(2)ᐜ~lABOöl4CE%8D=CE,ᵫḄ,Z/ᳮ%NM=NP,ᵫ¹Zឋ%/MNP=60°,᪷SKḄᑨ//ᳮ%÷ø(3)ᵫBOW4B+AD,%MN<2,ᵫSKḄ☢ïûü%lMNPḄ☢ïᐵ"MNḄýᦪᐵMüᵫýᦪឋ
%ᨬᜧþ.ூ௃(1),.AB=AC,AD=AE,..BD=CE,•.•MஹNஹPᑖ!DOEஹBEஹBCḄ,..MN=LBD,PN=LCE,MN//AB,PN//AC,22:.MN=PN,NENM=NEBA,ZENP=ZAEB,../MNE+NENP=ZABE+ZAEB,VZABE+ZAEB=1800-NBA"60°,NMNP=60°,ᦑᫀNM=NP60°

31(2)MNP.ᳮᵫ᝞ᵫZBAD^ZCAE,AB=AC,AD=AE,..^ABD^/\ACE(SAS),..BD=CE,NABD=NACE,•.•"MஹNஹPᑖ%஺EஹBEஹBCḄ(".:.MN=LBD,PN=%E,MN//BD,PN//CE,22:.MN=PN,NENM=NEBD,ZBPN=ZBCE,..NENP=NNBP+NNPB=NNBP+NECB,'.4EBD=NABD+NABE=ZACE+ZABE,..NMNP=ZMNE+ZENP^ZACE+ZABE+ZEBC+ZEBC+ZECB^\SO°-NBAC=60°,...MNP(3)᪷*⚪,BD^AB+AD,ᓽ8OW4,:.MN&2,:•4MNPḄ☢/=᧡1•ᑏMX3*5224:.PMNPḄ☢/Ḅᨬᜧ8ூ":௃<⚪Ḅ5=>ᔠ⚪@⌕ὃCDḄᑨFḄ(GHFᳮᐰḄឋKLᑨFḄឋKᐵNOPᐰQRᵨ(GHFᳮTUVLWVὶYZᩭ.23.(2020•\ᓭ)ᙠῪABC(AC=BC,AOEaND4E=90°,ZADE=^ZACB,bcBD,BE,"d8஺Ḅ("bcCF.2(1)eNC4B=45°f.জ᝞h1,e⚔"஺ᙠACjfkacᑏlNE48L/C8AḄᦪnᐵYNEAB=NCBA.HoBELHoCFḄᦪnᐵYCF=1BE2ঝ᝞h2,e⚔"Z)ᙠABjf(1)(HoBELHoCFḄᦪnᐵYᔲsᯠᡂvwxᡂvkyzOPx{ᡂvk|Pᳮᵫ

32}~l⚪Ḅᜧὃ5ῪABCjḄCM,8EḄ("MᑭᵨᐰᡈᨵᐵVᩭ⚪OEḄ("G,bcAG,CG,CAG"C⌮f┐90°,ᑭᵨឋKஹᐰᡈᨵᐵVᩭ⚪.(2)eNC4B=30°f᝞h3,e⚔"஺ᙠACjfᑏlHoBELHoCFḄᦪnᐵY|Pᳮᵫ.ூὃ"௃>ᔠ⚪.ூ⚪௃¡¢>ᔠ⚪£ᵨ,.ூᑖ᪆௃(1)জ᝞h1(¥DE¦4B§■✌ᐜOPBD=BE,ᑭᵨa«(HḄឋK⚪ᓽ.ঝ¬5᝞h2-1(ABḄ("M,BEḄ("N,bcCM,MN.OPCMFBMN(SAS)®.¬᝞h2-2(¯EḄ("G,bcAG,CG,CAG"C⌮f┐90°ᑮCBT,bc¯7,GT,BG.OP±BEGT²³±±஺GB7²³±®.(2)®BE=2^/3CF.᝞h3(ABḄ("T,bcCT,FT.OPBAEs4C"®.ூ௃(1)জ᝞h1(¥஺E¦AB§T.

33Dh1CA=CB,ZCAB=45°,..ZCAB=ZABC=45°,NACB=90°,VZADE=^ZACB=45°-ZDAE=90°-2..ZADE=ZAED=45°,..AD=AE,\9ZDAT=ZEAT=45°,..AT±DE,DT=ET,•ABᚖa²ᑖDE,:.BD=BE,9.ZBCD=90°,DF=FB,..CF=1-BD12:.CF=1BE.2VZCBA=45°,Z£AB=45°,..ZEAB=ZABC.ᦑᫀ/EAB=NABC,CF=LE.2ঝ®{µ.¬5᝞h2-1(A3Ḅ("M,BEḄ("N,bcCM,MN.

34D□/¸F~^B/τEh2-1VZACB=90Q,CA=CB,AM=BM,..CM.LABCM=BM=AM,f¥AD=AE=y.FM=x,DM=a,ᑣ஺d=F3=Q+X,.•y+a=a+2x,..y=2xᓽAO=2dM,fEN=BN,:.AE=2MN,MN//AE,:.MN=FM,NBMN=NEAB=90°,:/CMF=/BMN=90°,:ACMFABMN(SAS),•CF=BN,•:BE=2BN,CF=LBE.2¬᝞h2-2(஺EḄ("G,bcAG,CG,CAG"C⌮f┐90°ᑮCBT,bc஺T,GT,BG.

35h2-2".AD=AE,/E4D=90°,EG=DG,..AGVDE,/EAG=/ft4G=45°,AG=DG=EG,'.ZCAB=45°,..ZCAG=90Q,J.ACVAG,..AC//DE,".ZACB=ZCBT=90°,..AC//BT//DE,'.AG=BT,..DG=BT=EG,...±8EGT²³±±DGBT²³±,LGTº²ᑖ:"F80Ḅ(".•.BZ
LGT¦§"F,GF=FT,•••GCTῪa..CF=FG=FT,CF=LBE.2(2)®BE=2y/3CF.ᳮᵫ᝞h3(ABḄ("T,bcCT,FT.

36DE@3'.CA=CB,..ZCAB=ZCBA=30°,NACB=120°,".AT=TB,.CT±AB,..AT=yf3CT,AB=2«CT,,:DF=FB,AT=TB,..TF//AD,AD=2FT,../FTB=/CAB=30°,NCTB=NDAE=9Q°,:.NCTF=NBAE=60°,VZADE=1ZACB=6O0,2..AE=-j3AD=2-j3FT,ᙢ=½=2«,CTFT:.BAEs/\CTF,.•.Á=ᙢ=2«,CFCT..BE=243CF.ூ":௃<⚪§>ᔠ⚪ὃCDῪḄឋKᐰḄᑨFQឋK,²³±ḄᑨFQឋKḄᑨFQឋKV⚪ḄᐵN}ÃÄÅÆᵨÇÈH᪀⌼ᐰᡈ⚪§(ὃËÌ⚪.

37ὃ"ᓱᱏ1.ᙶ᪗LhឋK1ஹ"ᑮᙶ᪗ÌḄÑÒLÓ="Ḅᙶ᪗ᨵÔ%Ḅ⊤Öᙠ×=Ø☢জᑮXÌḄÑÒLÙᙶ᪗ᨵᐵᑮÚÌḄÑÒLÛᙶ᪗ᨵᐵঝÑÒÜÝÞᦪßᙶ᪗ÞᦪᙠᵫÑÒàᙶ᪗f◤⌕ÅjាeḄãä.2ஹᨵh(5å"Ḅᙶ᪗à☢/fUV"ᔣᙶ᪗ÌᚖHᯠçàlᐵḄHoèÓé⚪Ḅê<جQëì.3ஹxᙶ᪗YᑁḄ±Ýëᑣ±îÆᵨ²³§ᙶ᪗ÌḄÇÈHᵨ“ᒘஹ⊡”¬ó⚪.2.²³HḄឋK1ஹ²³HឋKFᳮFᳮ1×ᩩ²³HõöᩩaHᡠøùG.úᓫ|ᡂ×aH²³ùG.Fᳮ2×ᩩ²³HõᙢᩩaHᡠøùüᑁº⊡.•úᓫ|ᡂ×aH²³ùüᑁº⊡.Fᳮ3×ᩩ²³HõöᩩaHᡠøᑁ┯.úᓫ|ᡂ×aH²³ᑁ┯.2ஹ×ᩩ²³HþÿḄ
ᜐᜐ.3,ᐰḄᑨIᑨᳮ1SSS--ᩩᑖḄᐰ.
2ᑨᳮ2SAS--ᐸᜳᑖḄᐰ.
3ᑨᳮ3ASA--ᐸᜳᑖḄᐰ.4ᑨᳮ4A4S--ᐸḄḄᐰ.5ᑨᳮ5HL--Ḅᐰ.ᢣ"#ᐰḄ5$ᑨ%⌱ᵨ($%)*+⚪-Ḅ./ᩩ0%1./%ᑣ345Ḅᜳᡈ781./%ᑣ9:;3<%=⌕?Ḅᜳ%1./%ᑣ3@<%ᡈ3AḄ@<B.

384.ᐰḄᑨឋEF1HᐰḄᑨ?IᔠᐰḄឋEKLMNOḄP⌕Qᐹ.ᙠᑨᐰT%ᐵV?⌱WាYḄᑨᩩ0.F2HᙠᵨᐰḄᑨT%⌕[\]Ḅ^ᐳO^ᐳ%9⌕T`a〉YcdM᪀⌼.5.ῪF1HᩩḄhiῪ.F2HῪ?$ᱯkḄ%ᐹᨵᡠᨵḄឋE%nᐹᜓῪOḄᡠᨵឋE.ᓽ#┦r?45°,wMஹyᑖMஹwḄz%Mᔠ%ῪwḄz{᜜}ᙊḄz{ᑁᑗᙊḄF{ῪḄᙳ{45°,zᚖ+%ᡠᙳ{Ὺ%ᑣῪH8F3H1ῪᑁᑗᙊḄr=l,ᑣ᜜}ᙊḄR=&+1,ᡠR=lV2+1.6.MᳮF1HMᳮ#ḄMy+7%=+7Ḅ.F2H:᝞%#஺ஹEᑖ?A3ஹACḄ.DE//BC,DE=%C.7.¤ḄᑨឋEF1Hᐵ+¤%¥y¦ḄᑁḄ§ᓄw©ªᐸᱯkឋ#ᑁ?Ḅy¦%«¬Ẇ®ᐸᱯᨵḄឋE#?¯°ஹᑁr?ஹM.±Ty¦ḄឋE¤²rᐹᨵ.ᙠᜐᳮ³´µ⚪%1¶ᮣ¸¹ᵨ¤ḄAºឋE%ᑣ»¼ᣩᙢ¿*ஹMNᨵᐵḄµ⚪.F2HÀ☢ḄIÂ+K⚪²?ᨵᵨḄ#জÄ048ஹÄOBCr?Ὺ8ঝNOAB=N

39

40

41

42

43

44

45

46