书书书图书在版编目(CIP)数据数学.九年级.下册杨俊英编著.-石家庄:河北教育出版社2012.7(2019.10重印)义务教育教科书ISBN978-7-5434-9538-8.数.杨.中学数学课-初中-教材.G634.601中国版本图书馆CIP数据核字(2012)第168434号主编杨俊英副主编王洁敏缴志清程海奎编者(按姓氏笔画排序)王佐李会芳苏桂海徐建乐简友书名义务教育教科书数学九年级下册责任编辑王东芳吴丽霞责任印制王淑英装帧设计呼玉迈内文插图老迈视觉设计工作室出版(石家庄市联盟路705号)发行河北省新华书店制版保定市佳美制版中心印刷河北新视野彩印有限公司开本787mm1092mm116印张7.5字数127千字版次2014年10月第1版印次2019年10月第6次印刷印数1129001-1419000书号ISBN978-7-5434-9538-8定价7.40元冀发改价格2019761号冀价审2020002073版权所有请勿擅用本书制作各类出版物违者必究如有印刷质量问题请与本社出版部联系调换电话:18603114066购书电话:0311-88643600书书书!#$%&(!#$%&!&()%+&-.#012#34&5678#9:9#ABC#DE$F&GHI)JKLMNOPQR#SMITUVWXYZ&K#R$!#$%!&_ab%cdefg#hijk#l%mn$&()!eopqrstumvl%nv%fwexy#Iz&|o#h$+&!#9q#I+%=l%#$-.!e$%&#%=#op$01#2$!#nv#fwxy#k#I+#%=$A#I)Q|$!#$%&()&st“#+%=#q$I#&st”“B%“B#.z#e#%H#l%fwxy$+-&+ql$l#eS#+“qxr#5$.01%23&.RzH#$I#&-%=estqT#zeqTe#fwxy$45#67&#I#!S+#$#nvu$#%#+&#t#qT()fwxy#+)+#$%(#-&.e+0|12#$-.12#0q%#34&l%A|o#50#l%=67#(8$3456789!#$:%书书书!#$%!&()+-.!#$!#$%&(!#$!)#$%&(!%#$&!+$-.0!#$(!+102!#$%!3456#!)!789:!A!)&$!A$BC.-!#&$&!DEFG!$HIJ0A!&#&$(!A$KL!(!&$%!AMNOP$(!%!789:!%(!=!%!#3%!456789!%#&!$!J0QR.STQR!)&!$!STQR$UV!)&!WMW!UVX$YZ!+&!$&!LV_UV!+!&!$(!LabcdeQR$UV!+!Afgh!ijklmn!(!789:!%!=!)!#%!:(=!#&$!ko!#&$!pB!#(!WMW!qr$GLst!%&$&!)uv.#w$xyzB!)!789:!(A!BCDE!%书书书!#$%&!#$!%&()+-.%!($0%!(1!%&2)345-.6#789:5A!#$%&(!)+$!)!-.0123415#(67!89!:!=3ABCD(EFGH$)I3BMNEFGH%)3O&)3+P)3L!#(EFGHQR#$ST!R#$UVP#!#(WXY$!3Z#2(Y%!$!_abR4#(EF!)cdef$%2(ghGH!()#+!#$%&()!ijklR4cT$%2(ghGH%!#1%!#$%!#!m%$nop!qr%#$!%$!%$s!tu#(EFGH!BC$%DE$FGH%!IJKLMNOPQRSN!LSNTOPQRMN!vwxy%$#)#O%#$!#)#+%$!&#)#L%$!R#$!QR#$!)&(!(%#(!&%)+!(%+!z&Y3Z&+YWX3!|%$#(&(EFGH!#&(EFGH!&#&()&(EFGH!%UV&(WX$%&+!$#Y&(%&-(槡%)-槡%&+#%$!Sz()&+!#Y&%)+#&+%$!Sz)&O!&#Y)&%$&%.)+$&+%$!Sz)&L!#!#$%&!$!)H!zY3Z(#(WXY)!za#(EFGH%&!#!-&!#!(!-#!)$!)#!#!QR!5z&Y3Z&+YWX(3415Y(1!&!&2($+!Q$+0(“!)sL!($s15$!$#1$!#$%$%)!&#()#+#)%&!-).!+!01!23)%!+)#!)$!1!+!4�:&()!&%&!&)%!AB1&C&D&!E!(!)01FGH1-&I!3FGH1-&J!KLM&#$%$#NOPQ!2$!RS)CT(#1!B(C)%D)!UB(CVWDXY0Z&(!#$1&-).!XY0Z&(Z#_!#$1&-)I!XY0Z&(Z#NOPQ!#&$1&-)J!XY0Z&(Z#NOPQ!#$%C)!&#(a&#+C!#.(a&b-)#+C#1cd()#+!#$%&()$!(&2)345-.#UVWXY.Z(&25-.6#!&B!m3B=s!3-.(m3B0!q3(1(m3Bs!3-2!#&P(MNEFGH!QR#$ST!R#$3(EFGH8zBGgh2(GH!QR#!#(WXY$!3Z#(Y%!$#m#&s!$%noABCD(ghGH$#m%$!%$%#$s!#noABCD(EFGH$%!#$%&R#!8%$#%$!#%$!&#%#$!QR#&!)34&(!(%#(!&(%&+!(%+!z(Y3Z!+!.+!&+noYWX(!&no(BCD(EFGH$Y$!R#&%QR#!6()(&!Y)!)34&(!&%&(5(槡%&5槡%)+#!M4(K!|!&()(%&)()!R#()%&()(&%&6)%.+#!3Z(&(%.+!m$%+s!%#$!&(!m$%.+s!%$!&(!m$%&+s!%$!&(!$!UV=3(XY$!Q=3(3Zq(no&!)!7!q=3(EFGHnoCD($()#+!#$%&()&!#!QR!&#%&(!-Y#+!#-%7+!z-Y3Z3!mWX$no+!&+!+s!#&-noBCD(EFGH$Y$1$!RS#$%C$!&#()#+C%!e)4#fg1!hi%4$#j78=#!-)Zkl8&#)!$m#!)$n1!B1&C&%&#oC$%D&!KLM.pmp9:4ᘮ!#2$!-XY0Z&(!&(%$(!&%&(%!KLMB1&C&%C$%#&4(#5678!-&(!(%#(!%&(!#C&.H1!#&%0!#$%$%$!-qrstq!9:0#NOPQ!#$&(4#u!#$&(4#uv!#&$&(4#uw!#$%&!_!a(&2bcd!2efc(5ghFH25ij!25c(k)lmno#UVpqrs(925c(6#)(!3(4!Q“(!8z:3(A!R#&$3s!B!#$QR#&$!Y#(Y1!#1YWX!)+$%!&#!#1P#(ghGH!()#1ABCD(EFGH!R#&+!#1(!QR#&!+#1v!6#(!Y!Y#!Sz#$#1-.#!3Z#(3(WX!#!P#01!Sz#1(!3415678049:!()#+!#$%&()(!R#&!QR#&!#&Y#(WX!6&!(#&!$#Q$cT#WX(2!%cT3Z#(!$%ABCD(ghGH$#(3$Y(#&!Y&!Sz%$!#!89:4567A9:461B341!R#&QR#&!Y#+(!tM4=36(!$!QR!&Y#(!Y&!#%!&#%&(!5#(WX!$#!#!QR!()Y#(X!&)(62+!&2#!&%8(!5)2(7g!&#&!QR!&6#+(!|#&%#!(&%(!&#ABCD(EFGH$89!)!#$%&1$!nd&B1#C&#x&!y&#a&z&#v!(Cv1!(
C|i$#!#!&#!-#!&C)%!&)Ca!1#v4&)#ow1(!3&)%)(!&)#_!&!RS&!&C#)%!&1%)(!&1%&!&14#uv!2$!RS&!&C#)%!(C#v!v1C!a&)#(!&()#v!$#!#!.k.!B78+#_C#!#!1&#!&1$%!“Hq$%B(G!#$%C7&9+!eL(.$+$()#+!#$%&()!t2r%5(&2ubc!t2$r%v)rs25c(!t21r%)s25c(!wx215%fc%5s(yz|)345XY.6#QR#$!UV#:3O!Q6#($R(%!QR#!&#!z#YX3!#&!R#$!R#&!Y&!#(!$#tY&!#(3$!89!#()&!ABCD(ghGH!+!&!A#(!&%!R#&aKB9%63OC3SD(q(2!UV%QR#&!#O!&!no#&!)!#$%&5B%&%!R#B9%QR#!&#&!#!#!)34&#&P34&#!:!&!no#&!&(#&!(#!#&%#%#(!E:!#&%#!#%#!34&#&)34&#!&%!F&!(2#(2!1CDE!83(qS4(K!)!UV%QR#)!6(qno#&!+YLM&+N&!($O!6+(no&!(!)!5B%&()(P2&!B9%:!&!()A#(!R#)!&%!(+%(&!)+%)!&()(P2%(5)5()%(5)5(+5)+%(5)5(&5)%&5%&!4Q3!RUVM4(MA!UV%QR#7!&(!5%3!R&(MA!nS%5(3&(MA!T=3(3Z()#+!#$%&()!&(M(A!Sz3ZM4=LJn(!3(WXM43!#633)(!Y)!&#z3Y3Z&3)(2YWX3!3YS5!R#7!R#9!R#8QR#8!32(&(!35(&!noY2!5!R678V3)%32%35!Y3(WXY3)!Sz3&(M&!(!&(no5!)!2!M4(MA(3BVBnoY)!2!5!$#RBW($#&)%!2%&!(5%$!5&(P2!$#!#!QR!)&(!&%)(!(LZY3!53(7g!书书书数学九年级下册组已知犗的半径为，点犘与圆心犗之间的距离为过点犘作犗的两条切线，求这两条切线的夹角和切线长如图，犗为犃犅犆的内切圆，切点分别为犇，犈，犉，犃犈，犅犉，犆犇求犗的半径（第题）（第题）已知：如图，四边形犃犅犆犇的四边犃犅，犅犆，犆犇，犇犃分别与犗相切于点犈，犉，犌，犎求证：犃犅犆犇犃犇犅犆如图，犘犃，犘犅是犗的两条切线，犃，犅是切点，直线犗犘交犗于点犇，犈，交犃犅于点犆（）请写出图中所有具有垂直关系的直线（）请写出图中所有的全等三角形（第题）（第题）已知：如图，犃犅为犗的直径，犈为犗外一点，犈犅，犈犆分别切犗于点犅，犆求证：犃犆犗犈()#+!#$%&()$!2$!&#%7(!-C#.#H1!#-%!B1-C&%C$%D&!#&1#!#&m-uv!#&b#Z_G$#!RS&!&C#)%!(v#1!&(w#1!12-(.!232v#1!&(2%2!#)%!#$%&!F&5bF!TbF!k)bF&TbF5!7&2)c5!$!aK(MXRY!88RYZ!4!h!&(4MNOPABCDEFAE=#!F!=a84!F=a8434KQM!OP!=3M!OP4Q3!O3(3Za4(RL!O3(WXa4(9:!b#Y=ac4(Z!#&YWX!&#YZ!#7(2YZ!$!V9388+&#n!8za84!Q9=38n$()#+!#$%&()&!ad4(Ze7$Q93dn!ad4$f6n3Z!8za4!W!g4!8z4Q3(La4!)!4Q3(La”4!UV%QR#)!#!5%a”4&()L#!R#)!R#)&U%$#QR#)&!qh(X&(!)!#_&!(!()!)&!R678V!i=ZA#(!Sz&%(%()%)&!Y&(!)AX!Sz&(%()%()&%)&%#(!i4&()Y#(La”4!$!jk3(Lac4(Z7g!uac4(2PO3WX2(ghGH!4Q3(Lac4!lRmn!v5fU#!QR#)!&(Y#(LaM4!Q#(WXY$!5=aM4(2PZ!R#)!R#)!#$%&%QR#)!&#!6#)(!Y)!)34&#)!:!#)%&(!#%$!#)%$!)%槡&$!(%)%槡&$!=aM4(2Y槡&$!ZY$!$!WM4!QM!(M=o!p6!QM=!(Mo!Wq4!Qrs+P+=qr($O=!tB=4a43$u89!=a4(Z2(vY$!5=a4(g!1$!-#!jZDHdW!UDdW#sZ!$#!0Z#WoC7!0dZCHdW!dW#!#()#+!#$%&()(!&!$%C#&#I0Z!W#Wo%W+%Zm!e”-q&3(La42ZZK2$!-&.WoC$)+#!b#&#)%zCG!&7+o#-.QH!ABq“&0Z%Wm&!“#y!#$%&!F!UVWX)!YWZ)wx!yHz3&3(EFGH!)|+!GHz3(#Po&2oz:a43(GH!)yH3&3EFGH(67!zn()!3Z(%WX$2(ghGHT3(EFGH!yzg4()!$!3&3(EFGH!3&3noBMNEFGH!Qn($CD3Z(%WX$2(ghGHklMNEFGH$efac!3(EFGH3Z(%WX$(GH3(EFGH(=g3Z(%WX$(GH!(#Po!(#%3(6(WX!(o%6WX(O|qWX(3(!2o%63OS(3(q(2!()#+!#$%&()!)!&!a43!f6n38z3(La4!4Q3(La”4PLac4(”U!Q3(La4Pa4!1$!-34&(!(%#(!(%&!&(%!B1(C&%(oC$%D&!1&4(#5678!34&(#W&%7!)ZW&(%&!&C1(!$%9:C!#nd&4&=#5678$%Co!&4(uv!&#&!-34&(!(%#(!&(%)!(%$!#&-T(&.!3#$%C&!#$1#41(!#4)&=#5678#$!e#)(
#)(&$!#(XG!#4)&uv!“XW0Z#J&4Iv&#O!)!-&(!&%&(%!B&C)%#&4&(uv1&!4(uw1)!9#!)#!7#7!-)Zkl8!-4.puw1&#!$!#8!$!4puv1(!&-#kl!9#9!-34&(!(%#(!1#-&.!B1#C&%#&oC$%#&4&(!&9:w1)!2!3()%&!)4#5678!2#!#$%&8!Hd&6-G(.!bC!-!(&m(#v!v19:&!e#$%C槡&+!3&%7+!&(#_!8#!#!b!zxjHd#)%!)%m0Z6#.!2j&%&.)+!$#)%!$#$!RS&!&C#)%!(C#.H1!1)-&#o.!3)(%&!#$&()C#v!#$)%&(!)%$+!#$%!2$!&!)29:v#1&!(!#$%C)+!#oC$&+!&)2#%oG$#!RS&!-34&(!&(%#(!&(#9w(1)!2C&.#H1!)2%)(!B1)C&%)oC$%)!&#$&(C)#v!#$&52%&(!#第二十九章直线与圆的位置关系233.已知:如图AB为O上的两点AOB=120D为劣弧AB的中点.(第3题)(1)求证:四边形AOBD为菱形.(2)延长线段BO至点P交O于另一点C且BP=3OB连接AP.求证:AP为O的切线.4.已知:如图AB为O的直径C为O上一点AD和过点C的切线互相垂直垂足为D.(第4题)(1)求证:AC平分DAB.(2)过点O作线段AC的垂线OE垂足为E.若CD=4AC=45求垂线段OE的长.C组1.已知:如图AB为O的直径BCAB垂足为B连接OC交O于点ED为O上一点ADOCDFAB垂足为G.(第1题)(1)求证:E为BD的中点.(2)求证:CD为O的切线.!#!#$%&#&$I#&(#1#4kl-1!6CnsZ#w1!1&-.p.!1(#klC#!7$!(#%&(!B#(.H1C&%&C$%#&.4#uv1)!#$1#kl!#$&(m#5678!20!#&$RS2C4(#w1!)2#o!%!#$%&!rX0X!ZX!R&$!$!QR&$ST!QdI(a”44K!CvCb)!4K-OK0Y!nY!CbB8!$#(gY!8(gcT!T%!8ABCD(gGH$#(gY9!8(gcT9!T9%!98(g3$88!7(Y8!j(J2Y.!$#7(Y!T%!M7(Y9!T9%!#!9A.(g3$(cBvI$+K(g9%858-7!9%8.5$7.58!(cAh(!Q=h.P2(gGH8zcT%:.5.5:!C4LB=4E=#!:Hg!Hg!g!(_+!+-!&!+K(=g!z:%.5.5$!%-).5$).5)!%&.!%-$.57!Ag!$!nou+Ky(gc(Hg&HgPg!“g&pvg&gBvI!IKsh!B0I!b2A!cTI(g!$#0(gcT!890(g!ugW(:!(!#B)I!TD(ge$!$!uabg(:!(%$#%-).5&.5$(#%.5$#-$(&#%-.57!2”4”!(2v+!(2Y.+!KY+!f.cT(gc!.(g3$!u(:!(!1$!-qr1!2341!#$%&#$%.)!#$%-$.5$)#&$%$.)#$%.5)#)$%.-.5)#7$%-$#.5$5$!#WoC.+!C+!”.#15!.#341&!H67-8$9:!C.!n67#8C9:!7.#15!2$!4.O#1O&.-)-&-$&%.5.5&%-.-.5#!341%.5.5&!cd.#O4%9uAB=#.#O!(_+!+-!(!rX0X5now4!X5!ZX5no!)!#$4abcd!UVg%.!8(R!$#bc%.-&-$&-%.-#$#-#k%QR&$!)Hk(!&#&%QR&!J_&!g%.(R!R&$!R&g%.(R!ab%$#9%.(RW!(nt3$%.(RvWR4$Q!(Wq$#%.(RB-3$QB!-($!#$%&$!)QR&ST(H!Uz%.(R!g%-.(R!R&!R&!)QR&ST(H!Uz%.(R!g%-.(R!Wvg%.%-.!%.%-.(R!g%:.za%$#R(”CP(-#:(ABCD(GH$#RvWR4$Q!(Wq$&#_R!89(.(!g%:.(RCMEC#!(W(W!(W(g0N4O#!(_+!+-)!#$%:.%&()c”CW.(-#%:.:#($%!(a.$d!.5“)a.#d!.5)%!(!a.%d!%:.:$(%!(a.$d!.5)a.#d!.5“)%!(!a.%d!%!Y”!FY.%!F6:!#.(Y.%:(F.Y%!F6!#(Y%!g%:.Y%:.!$!vR!ug%-#.R(”C&W&z:-#!u%-$&.(”C&WP!(R!1$!CqrDE#FG%Hpm1kl!#$%7.)!#$%-.)!#&$%&.)!#$%-$).!9:CDE%&.4%-&.#FG%Hp%1klmI.#JyKLM#NO!2-xH)Zkl8DP#Q!2$!DE%.m%.!RCP2Sx1!eDE%:.T1-#!)$!:#O!2CDE#FG!#$%&)!#$#%#Z!#$#%#$&4abcd!)I&(”!z(R!.-&-$&-%.-#$#-.-$&)7-%.-&#-#$#-.-)-&-$-%.5#-#$#-R&)!4+:!g%.-&#!%.5#(Rg%.(RI(!(EFvI!$#%.-&#(R8z%.(R”CJe=E27$#%.5#(R8z%.(R”CJe=E27$8zV!g%:.-=#(R8z%:.(RQaJ%m=#s!CJ=E27(m=$s!CJ-=-=E27!(_+!+-!g%-.(R!no6CD(J8zabg(R$#%-.5$#(#%-.-#(&#%-.-&#!)R&7(H!Uzg%.-&#(R!$#t)Hg%.-$P%.-&#-&(R!#89g%.-$P%.-&#-&(R8znog%.(R6CD(J!&#fg%.-$P%.-&#-&(R(W!R&7$#89g%-.(R!no6CD(J!8zg%-.-#57P%-.5)#-(R!#!#$%&#ug%-.-#57P%-.5)#-(R(W!|89Qo(!#$%:!.-=5%&()c”CW.(-#%:.-=(5:#($(.%=!(a.$=d!.5“)a.#=d!.5)%=!(!a.%=d!%:.-=(5:$(.%=!(a.$=d!.5)a.#=d!.5“)%=!(!a.%=d!%)!$#5g%-$.5)#-(-#!#9g%-$.(RCJ=E27!CaJ&=E27!fJ(R(gc!%$#-$!VgB-!m.%-)s!g%-!-%-!#J(R(gcY%-$.5#-&!ec%c”CW.(-#%-.-#5&%$.5&#-&%-.5$#5)(_+!+-$!1$!=UV0%.#QTW(1%#.-7$59#QXzY&Z7d5o_!U.9d5o_!z&Z7d5o_!U.9d5o_!z&Z.9d5o_!U7d5o_!x_#0!Cqr1Q#FG%Hp%1klm1#y#az$O!#$%.7.!%.7#.-$!%.7#.-$5!#$%-&.!%-&#.5$!%-&#.5$-!#&$%-7.!%-7#.-&$!%-7#.-&$58!2$!341%-8.#Q&d5o_!Uqd5o_!Q#15!Cqrbnd341Q#5678!#$%&.4%-&.!#$%&#.5$4%&#.-$!#&$%-#.5$54%-#.5$-!#$%#.-$5$4%#.5$-$!)!#$(4abcdb=g%:.5.5(4%!#$%&:.:.:.#:.):).:#A:):.:#:%:(RP#!efg%:.5.5.(-#%:.5.5:#(%:.5.5:$(g%:.5.5(R-:!:-:#!m.$-:s!.(m.#-:s!.(m.%-:s!%-!-%:-:!:-:#!m.$-:s!.(m.#-:s!.(m.%-:s!%-!-%:-:!(_+!+-&!Y”!Fg%:.5.5Y-$!-#!$#bc%.-&-$-%.5.-$-$-$-R&9#)H!k!&!g%.5.-$(R!QR&9!_abqr!o(c!$#%-.5B.5C(Y-&!)#!#%:.5.-76&-$!&#P!-7#!%$#:!%-.5B.5C%-.-B#5B58C8!B%-&!B58C8%)!B%-$!C%-$&!Sz(cY%-.-$.-$&!#&-$!&#P!-7#(p!(c!:-7%&!:5-7%-7.0!:%&!%-7.0!Sz(cY%&.-7.-7!$!5ab(WP!|u(”C!$#%&.-7.5$(!#%-.-7.-$!%.-.5(R!|89m.%-P.%-$s!=W(g!#$%&1$!qrDE#Hpm1kl!2CP#FG!#$%.-.58)#$%-).5&.-!RSDE%-.5.5!#$sDE#Hp!1kl!#$DsDE!#&$sDE.&#.$!$!#.!$n1#ckld.$#.#$!efQ!C$4#yz78!&!g!hqr1MC%:#.-=$5#5!2C1#y#az$O!#$%-.-).5#)#$%.5&.)#&$%).-.5$)#$%-&.5#.-!2$!ijqrst!DE#5!#$DE%.5.5#1C-#!&$!#$DE%:.-.5T1&#-$!$m#!-#$!RSDE%.-B.5$7#1-klp.!KLMB#O!(_+!+-(!_%5!OPqrX!UVu(%5ZqX6#)H!vM(8zog(c!UVv(M&$!&#!-#!(-$!$#!CDo6M(g%:.5.5$!UVM&!$#!$!#!(!&#!5MSo(gc!%S5gY%:.5.5(nogc!%$!:55%!:55%&.0!#!#$%&:%!%-&!&!&#!-&!#!($!#!5&!(MSo(gc!5&$!9#!8#!(&!$#MSo(gc!1$!RS341%:.-.5#QT1&#-$!-$!#&!-#$!#$d341#5!#$”sDE#Hpm1kl!0&#$!$!#&!$!(#-!$01bLM#3415!2$!RS341%:.5.5#nLj.m1#9O!q&.-&-$-$&-)-#-)9Sx#d341#5!&#.$!$!#.!$!(#.&!&$01#1