De thi HSG Toan 9 nam 2014

Goi H Id truc tdm tarn giiic ABC, M r,,ti N tuong u,ng la truns diOrn ag vi CH.. a Chring rninh ring A'MB'N ld hinh vu6ng..[r]

so Grio nuct vA DAo r,{0 I}EN'TITE Tt{I HOC SINII (;I0I L()I'9'I'RUN(; IIOC CO S() t,{AN,{ IIOC 2(}t3 - 20t4 DE TIri'i gian: MOn: f},iX phrit (kh6ng k0 lrhf t etO) 150 Ciu I (5 di0rn) lu) I{Lrt r:t.rn biCu thric: A = l-t- { \ h) Cho l.r = + viiJ Ti'h 5- ll:x -1 Vx-JIs-Jx*Jtu t/r'r/x -18 rtvi x)Jts :l tri cira bieLr thirc: +) 2? -Y -r+r g iri 1\ Chu (4 di6m) - 2.r -l- 2023 -0 Giai he phuong trinh : Ciu (3 di6rn) Tirn nghi0rn nguy6n duong cua phuo-ng trinh: r, + , - I 3(x - l/) = Cf,u (3 di0rn) CtLo hinh thang ABID (AB // CD), giao (i€rn hai dubng tlreo la O Dubr-rg rh3 g qua O song song vdi AB cit aO va BC IAn luqr rai M vd N a) Chung minh: Y*Y AB CD =r b) BiCt Soou = m2i Scoo = n2 Tinh Sorc, theo m vi n (v6i Sn,,u, Sc,rr, Sno.,, lAn luqt la dign tich tam gi6c AOB, di0n tich tam gidc COD, dion tich t['gi6c ABCD) CAu (5 di6m) Cho tam gi6c nhon ABC n6i ti6p dudng tron tdrn O vit IdB = 450 K6 c6c AA' vd BB' Goi H Id truc tdm tarn giiic ABC, M r,,ti N tuong u,ng la truns diOrn ag vi CH a) Chring rninh ring A'MB'N ld hinh vu6ng b) Chring rninh ring A'B', MN, OH d6ng quy du"crng cao HET SO GI,qO DIIC VA DAO TAO TiEN'I]I1E HTJoNG DAN crrAnn xV rut cHoN rroc srNn (;ror Lop e rnuNc Hec co sO ruAm Hec 2013 lll0n: TO,.ilV -_ Ciu la NEiiruiig- Di6rn bi€r-r thu'c: t (2,5d) r-\ I / A:{ Vx- JtS -Vx+JtS 1rlt, /x' -lti \ /' / rtr.i r ) vTs Taco: A: :(Jx-vTs -J* *JT) (- - Ji,r) o' : (* -vffi *x + J8 -z,F - rs;( a' : (2" -2J.' -ls)( 6t- 8) 0.5 rl=; 0.5 0.25 A2:2(x'-x'+ts) 0.5 flt2 -Jo -tr Nhung theo gi6 thitit rh6y o tT : (J.-Jr8 - J, Jtr)J* - J- Cho ru 0.25 .9 0.5 2x:5 + 13 Tinh gi6 tri c0a bi6u thf;: B:x5 2x = - -\ +A:-6 1b 5* JiJ * 5x4 4x3 - 2; - nx * :)2x- - ,fiJ (2* -5)' = 13 = (2.sd) 2023 0.5 1y2-5x+3=0 B :x5 - 5x4 + 0.5 4; -2; - t2x +2023 B_ (r'- ) 5xa - ;hi i;iii- :hri chri Rrit gon t -zatl +3x')+(r, B; (; -5x+:) + ulal trinh B:2014 h -5-r2 +3x )+ (r*, -15x+9)+ 2014 *.(? -5x+r) + (; -5x +3) +2014 : 0.5 0.5 0.5 (4d) I l i I - :.r')' - I T("r'' - 9.1,' - +(t ) + -i.r.)' ) ,r+J.l'_r -) ' r I 5(.r *3r, - :r,)' - I 7(x' * 9,),' ) - u( t * i.,,)' =0 (/) -r+3r,* *=5 r-iv ki6nxiicdinh:r+3_i, Di6u x= r\.thi Ndu (/)o{ f s4r;.r'' = I | I I o I <= .)- -\ -J /)', *I :hePT( 0.5 l0 I ) vo nghi6m FN6u x *+3y Chia2 vd phuong triph (l ) cho (;r +:y)(, - 3).) c6 : -3y)' {,'t, lrc* -3Y I x+3v -l I x+3y+ ^l/ 0.5 t Edtt-n ' ?,' - - '/ th\ x+3y 4-1 (*)ots/-tt-!=o \./ =[,-i)[,.+)=o<+/= f-_ 35 t +Voi Thay A /=-J n thi x+3v x-3y - -21v 0.25 0.25 vio (**) Ta co : i -1 -l8r'-l=5.6-r'= -8y 4'' 36 0.25 a -J- rI \ - -,1 o Vti'i r = -I. > 1tlrolr nrirr I)K\l)) z{+ l=., -l(; Vo'i,r,' = + vo'i r = j = r 015 l i l ii l ( thoa rnirn DKXD) () 12 l-i I I t-]l' x+3y= l= thi I' -r = 2.1, - 1-5ay, vao (*,F) 'l-a l 0.25 I CO: I 1' -) _i' [,,, = = lt i I + -2t0 - Vti-5 I I 0.25 1-a;*t L)'= r0 I i I I V0i y =:* VCIi $.tt, = r - , i.,tr y=*-*J8s=.r-,-iJ" vay hq phuong rrinh c6 nghie," , ( thoa man DKXD) ( thoa mdn DKXD) 0.2s [?,+j, [*,-+), 10 l( s' to' ) Tim nghidm nguydn ducrng cria phuong trinh: 0.5 (3d) *' + y' -13(x -t)=0 *'+y' -13(x -/)=o <+ (x + y)' + (* - y)' -2.t3(x - y) +132 = 69 <+ (x + y)' + (13 -x -t y)' =169 =122 + 52 Vi x,yeZ* =0< x+y<13,0<13-x+y <13 | **!=lZ v{[ ,+y=5 =( fl3-x*y=5 [13-x+y=12 [x=10V<(x=3 -Jt -/1 lv=2 _ lv=2 IJ IJ 1.0 0.5 0.5 0.5 0.5 cho hinh thang ABCD (AB ll cD), giao di€m hai duong cheo ta o Duong thing qua o song song v6i AB cit ao vd BC lAn lucrt tai M vA N -4- i I I (, lJss, I aGl,(, - !,6j,1'z- a*;) ( s 0.25 (3d) jl1A' ,d'l\/ - _ _ + .4t] b) ili0t S.,.,,,, : n'rl; ct) S(.r,, = Sn,,,, S,.,.,,, S,\r(.r) liin lrrot 'l'inlr ltl Ii ,,\lr(,r) thcrl nr r li n (r'tii tanr giac AOI] di0rr lfch cliCn tfch tam giiic COD di.^n rfch ru.giiic AIICID) I'linh vC ,\ l] - ^\ ,'\ ->(-FL/t)\\ Nr l"' l/' l) 4a) -'\\ -\ r _ >\ ( IVIO AM MO MD CD AD' AB AD o - - MO MO AM + MD Suvra 0.5 laC()-=-.-= - CD AB AD TNONO lu(rngtu,ta + AD =_:l ' AD CD AB /l\ \I'l 0.5 t)t \-, 0.25 MO+IVO /r\.,A,.\^., (r)\,'a(l)su-! MO+IVOq_-) CD - 0.5 S.ro, =oB snoo oD'Scoo-oA oc' 0.25 oB oA 0.25 Suou oD -:-_ Srou_Sooo oc s.nro sro, = Sloo = m2 n2 S ooo = m.n Tuong tU Suo = m.n V4y Souc, = m, + n2 + 2ntn (m * n)2 = \-no ram glac nnon AIJU nQr ti€p dudng tron t6rn o vd ^ ACB 5a 0.25 MN _+ 'CDAB SU\,ra 4b) MIV AB = 45o Ke cric duong cao AA,, BB, Goi H ld t.rc t6rn tam giitc ABC, M vaN tucrng irng ld trung di6m AB vd CH a) Chfng minh r6ng A'MB,N ld hinh vu6ng b) Chrine minh rins A'B' MN OH r{Ano n,,, Hinh vE -5- 0.2s 0.25 (sd) l - (' l i l I I I I I I I I I TacoMA,=MB-: !as I il) I I 0.,5 I I NA' : NB' : lcu I I (2) 0.5 Niat klrac ,i FiB= 45" ncn LCAA' va ,\BA,l/ = LCA'H = LAA'B(c.g.c) CH = AB = NA'= MA' (3) = = ^ MA'N = AA'C = 900 n€n I 5h (l), (2), (3) va (4) suy A'MB'N la hinh vu6ns (*) Do O vd A' thuQc dudng trung truc cfia c4nh AC N0n 0l' L AC O,4'll B' (5) = Tucrng tu, ta co OB'll A' (6) Tt (5) (6) suy A'FIB'O ld hinh binh (**) Tir (*),(**) suy A'B', OH, MN clt tqi trung di6m m6i duong henh -6- I 0,5 (4) H H I vu0.g cirr Me ffd=m,_fffi,_@) Ti.r I I 0.5 0.s 1.0 0.5 0.5 0.5