C5 thd n6i rang (*) thd hiOn mdt mOnh dd mh ta goi 1b dinh li ttong tt dinh ll Pythagore trong hinh hoc kh1ng gian d6i v6i tti di€n "vu6ttg":
"Trong m1t trt di€n vudng, binh phuong di€n t{ch mdt huy€n bdng tdng binh phaong di€n tlch ba mdt g6c vLr6rtg". C6 nhidu ciich chring minh
hO thfc (*); ching han, c6 thd dirng phuong
ph6p thd tich. Tinh V =V ptsc bang 4 c6ch kh6c nhau: 3V - aS, = bS2 = 65', = /lS, trong d6 a, b, c
vd i ldn luot th d0 dei cia PA, PB, PC vI chidu
cao PH ha xudng mdt ABC; sau d5 su dung h0
llll
thfc quen thuoc " =
-* " * " (mh ta de
NADC
ddng chring minh duoc) th) suy ra duoc (*;.
(r
I tu = tu.or*=1.ur,,
=.1 2
I
IHK =OPsinq = rlsinq
(a (t \24l -+-coscr I 4l -+-coscr I l, \- - 1 /I (2) +dz sin2 a PHz + PK2 < ,12 111 +coscx)2 +sin2 cr) 2 = 2tlzf.oro 9*rin' 9.nr' 9) ( 2 2 2)
= 2d2 cos2l[.or'9+sin2 )\ ) - !)=rr'cos2 )l ') $ ) 13.y
Sit dung dinh li Pythagore ta c6
(AB + CD)2 <2(AB2 + CD\ = g@Ff + ct()
= 8(o A2 -o ti21 + 1o c2 -o t$l
= 8(212-oH'*oK')
= 8 (2r2 -(o P2 - p tl)-e r' - p rio)
= 8(212 *2c12 + qruz + er?y < s( 2r' \. -2ct2 +2ct2.or' 9) 2r'(rheo (3)) = rc( -"1.'"1,' ,'' *d'(t-.or'9'l)2)) = rc( ,2-rt'rin'9) . \ 2) 25
, - ---(t t r)
Ap dung BDT l '+ + tlx+y+21 > 9 doi
\x Y z)
vdi ba sd duong x, !, z rdi thay x, l, Z ldn lucrt
b&i 52 +S,? (i = 1, 2,3) vd srl dung (*) thi duoc:
i=+ -2 Q)
- S'+Si 45'
'2 's3
Thav ' -]-: .S2 +.sJ l-,', .S2 +s3 v6ii= l.2.3vdo
(2) thi thu duoc BDT (1) .aJ, ti.n.
Ding thrlc & (1) xiy ra khi vi chi khi
S2 +Sf = 52 +Sz2 =S2 +S.2 e Sr - 52 = S.j e
a = b = c, BC = CA = AB ttic lir khi vi chi khi
PABC li rnot tit di€n vudttg cdn d P.
Nh:in x6t. l) Bii todn nhy thuoc loai d0, s6 ban tham
gia girii khri d6ng vd ddu cho ldi giai dring. Tuy nhi6n,
vdn cbn mot so ban quen kh6ng xdt khi vi chi khi n)o th) dingthrico(l)xriyra.
2)Cdcban sau dAy c6 ldi girii gon ging hon cii:
Hi Ndi: Pham H6ng San, I lA1, THPT chuyOn DHSP
Hd Noi; Hi Tny: Trinh Ngoc Tieh, 11 Torin, THPT
Nguy6n llue, Hi Dong; Vinh Phrtcz L€ Ti€it Dlrc,
Nguyin l,'dn Ngoc, Phing Dinh Philc, llguy€n Kint
Tlrudt, 114'1, THF'| chuydn Vinh Phric, Nguy€n Htlu Hidn,12A4, THFrf Lc Xoay, VTnh Tudng; Thanh H6a:
Trinh Vdn Vtong, I lT, TH[rf Lam Son, Pham Khdc
Tlrunh, I lA, THPI- Tri0u Son, Mai Vdn Ctdtrg, I2T,
THPT Mai Anh Tudn, Nga Son, Thanh H6a; Nghd An:
Tb Hdng Sor, I lAl, THF'f Phan Boi Chau, TP Vinh,
Ddng Cbng Vinh, l1l, THfrf Nghla Din; Hi finh:
Nguydn Thi Hanh Dung, ll Toiin, THPT chuyCn Hi
Tinh; Quring Binh: Phant Ti€h D6ng, I I Torin, THPT
chuy0n Qu:ing Binh; Di Nnng: Ngu'yin Nhr Qu6'c Trung,10A2, THPT chuyen LC Quf Don, TP DA Ning; Beh Tre: Hu)nh Tdh Dat, 12 To6n, THPT Ben Tre;
Clr Mau: Chdu Ngoc Huy, l2Al, THPT Ddm Doi,
Cir Mau.
NGUYEN DANG PHAT
Bei Lil339. Mdt con lac ld .ro, gim ld xo cd
dd cLTng k = 50 (i'llm) vd vdt nctng lui = 500 (d,
dao ddng didu hda vdi bidn dd A,, doc tlrco truc Ox trdn nfit phting ndm ngang, H€ dang clao
ctdng rhi mdt vcir r, = '?' (g) bdtt vdo M theo
3"'
phtong niim ngang viti vdti tdc v,,=l (mls). Gici
thi€i va cham ld hodn tocin ddn hdi vd xliy ra
vdo thd'i didtn ld xo c6 chii)u ddi nlu) nha't. Sau
khi va charn vdt M dao ddng diiu hda ldm cho ld xo crl chiiu ddi cLt:c tlei t,d c(c tidu ldn luo't ld
1,,,,,.r= I 00( c nt) vd. 1,6,,= B0( cm ). C ho g= I Q (p1 s2 1.
I ) Tim vdn to'c cila cdc vdt ngay sau va cham.
2) Xac dinh bi€tt dQ dao dbng trwic va cltqm.