AH ~ AA'~ 4
V — — V ^GBCD - ^ *ABCD-
Chung minh tuong tu cho cac hinh chop khac theo d^u bai => dpcm. 27. a) De cho gon, ta dat SA = a, SB = b, SC = c. K e S H J , mp A B C . Coi A
la dinh, SBC la day thi the tich tir dien la: V = - abc. (1) 6 •
J^u coi S la dinh, A B C la day thi the tich tir dien la: =ls,3c-SH(2) • (1) va (2) cho: | abc = S^BC-SH (3) lat khac, ta c6: 1 SH' • , 4 1 ^ S H = abc a' b' c yla'b'+b'c'+c'a' 93
Thay vao (3) cho: SABC= ^ +6'c' + c ' a ' . (4) Theo BDT Bunhiac6pxki, ta c6: ab + bc + ca^ Thay vao (4). < i(aV + bV+cV) S^Bc = - V^VT^VTTV > - . ( a b + b c + c a ) 2 2 V3 Vs SABC ^ - ab + - be + - ca ABC 2 2 2 « V S SABC ^ SsAB + SsBc + SsAc (dpciTi)
b) De tha'y: V = - SA.SB.SC (xem (1) eua eau a) => V = ^ ax(k - x).
6 o Ta nhan tha'y x + (k - x) = k khong doi nen tich x(k - x) Idn nha't khi va Ta nhan tha'y x + (k - x) = k khong doi nen tich x(k - x) Idn nha't khi va
chikhix = k - x => x = - k SB = SC= ^ k .
2 2
28. a) Ke SH 1 day. Theo tinh chat hinh chop d^u thi H la giao hai ducmg
1 a^I2
cheo day: AH = 2^^^ ^~2~' ^""^ '^^^ Pitago vao tarn giac
vuong SHA cho SH =
1 n^Jl
Vay V = -dt ABCD.SH=
3 6 b) Goi J la trung diem BC thi SJ b) Goi J la trung diem BC thi SJ
1 BC (ASBC deu) va HJ 1 BC theo dinh li ba ducmg vu6ng goe
=> BJ 1 (SIJ) => (SBC) 1 (SU). . Do do, ke HK 1 SJ thi HK ± (SBC) va HK la khoang each tur H de'n (SBC).
Trong tarn ^iac vu6ng SHK cho: HK.SJ = SH.HJ
HK = SH HJ
SJ
_ 2 "2^ _ qV6
Vi day la hinh chop deu nen d6 tha'y khoang each tir tarn den cac mat ben bang nhau.
29. a) OH ± (ABC) => OH 1 BC
OA ± OB
OA 1 (OBC) =^0A1BC OA ± OC OA ± OC
=> BC 1 (HOA) =^ BC 1 AH
=> AH la mot ducmg cao A ABC (1)
OA ± (BOC) =t> OA 1 BC
OBI OA] OB L {CAO) => OB LAC' OB L OCJ ^ OH 1 (ABC) => OH 1 AC
=> AC 1 BH => BH la mot ducmg cao cua A ABC (2) Tir (1) va (2) cho thay H la true tam A ABC
b) H la true tam ntn AH 1 BC
O A I O B OA I O C OA I O C
BC 1 mp HOA => BC ± OH (3)
OBI OA) OB 1 (CAO) ^ OB ± AC iOB 1 OCJ H la true tam, nen BH 1 AC iOB 1 OCJ H la true tam, nen BH 1 AC
AC 1 OH (4)
r (3) va (4) cho OH 1 (ABC).
E) Dap so: SA^BC = | V^V+^VTT^ (5)
AC 1 (BOh) AC 1 (BOH) . 2 ^rl . _ AI ^ AI' a) Ta CO tg Z? = — ma BI BI' b'-Of
OI = OB. PC BC be 2„2 2„2 b'c Thay vao (6): AI' a'b'+b'c'+c'a' BP b' Hinh 67 b'tgB = 2dt AABC(theo (6)
Chiing minh tuong tu: a'tgA = 2dt AABC, c'tgC = 2dt AABC => dpcm. 30. Dapso: V= ^abc. 30. Dapso: V= ^abc.
31. a) Goi a = (x, y) vori 0 < a < Iv.
- Neu m = 0 M = A va k = n' => MN" = AN' = = AB' + BN' = d' + n' = d' + k. = AB' + BN' = d' + n' = d' + k. - Neu n = 0 N s B va k = m' => MN' = d' + k. -Neu m, n => MN' = MN^ =AM + AB + BNf = m^ + n^+ d^-'2AMAB-2 AM .BN + (m + AB + BNf = + 2AB.BN = k + d'-2mn . cos( AM , BN) = d^ + k. Neu mn = 0. MN'' = d' + k -2mn.cos n6u mn^OwhiAM, BN)= a (1) MN' = d' + k' + 2mn.cosa ne'u mn?tOva(^M, BN) = 2w-a (2)
Gia tri Ion nha't, nho nha't ciia MN: - Neu a = Iv thi cos a = 0 va - Neu a = Iv thi cos a = 0 va MN' = d' + k Vm, n thoa man m' + n' = k => gia tri 16n nha't cua MN bang gia tri nho nha't ciia
MN= -Jd^+k
- Neu 0 < a <lv thi cos a > 0 va 2nin < m' + n' = k ( dang thiic xay
ra khi m = n = J— (3) n&n c/' +k-kcosa < d' + k - 2mncosa <
' min
(Vdidk(l)&(3))
< d^_+Js <d' + k + 2mncos« < +k + kcosa .
MN' max
(V6i dk (2) & (0))
b) Do X 1 y va AB la duomg vudng goc chung ciia chung nen AM 1
mp(BAN) va A BAN vudng a B.
Vay V = - dt A BAN.MA = - . - BA.BN.MA = - mnd < 3 3 2 6 ^^ 1 , /n' +«' 1 , , T^,- , , ^^ 1 , /n' +«' 1 , , T^,- , ,
S - d. = —kd. Dau = xay ra khi m = n = J— .
6 2 12 V2
1 fk
Dap s6': gia tri 16n nha't cua V = — kd vai m = n = J - .
12 V2
32. a) Goi d la cac du&ng thang vu6ng goc mp ABC tai A. Vi AABC can
dinh A nen A MBC can dinh M. Ke MI 1 BC thi I la trung diem BC
=> AI J. BC. MI, AI vijra la ducmg cao, dudng trung tuyen ciia AMBC va AABC =:> G, H la trong tarn, true tarn, AMBC thuoc MI. Hieo tinh chat va AABC =:> G, H la trong tarn, true tarn, AMBC thuoc MI. Hieo tinh chat
trong tam AMBC: Gl_ MI 3 I IG =- IM 3 G la anh cua M trong
phep vi tu tam I, ti s6' ^ => M chay tren d thi G chay tren ducmg thang d' la anh ciia ducmg thang d trong phep vi tu tam I, ti s6' - : d' 1 (ABC) d' la anh ciia ducmg thang d trong phep vi tu tam I, ti s6' - : d' 1 (ABC) tai trong tam AABC, dV/d.
Quy tich true tam H: (HS tu lam) ^ b) Do d 1 mp ABC nfin mp ^ b) Do d 1 mp ABC nfin mp
f MAI 1 mp ABC, ke HK 1 AI
thi HK 1 mp ABC nen HK la
ducmg cao hinh chop HOBC, c6
V = | d t (ABOC).HK.
Nhimg A HOC CO dinh nen V Idn nha't khi HK dai nha't HK = ^ OI (OI la dudng kinh dudng tron ngoai tiep A HOI (theo cau tren).
33. a) Dap so: S^p = a'(l + V3 )
V =
6
b) Goi M la trung diem SA, theo gia thiet cac mat ben la cac tam giac d^u SA 1 B M , SA ± D M => B M D la goc phang ciia nhi dien
\SAB)XSAD) .
Ta C O M B = M D = , DB = a V2 . Ap dung dinh li cosin vao A BMD: DB^ = MB^ + MD^ - 2MB.MDC0S B M D
cos B M D = MB^ +MD' -DB'
2MB.MD
34. Goi F, E \in lugt la trung di^m cua cac canh AB, CD. K la hinh chieu ciia F tren SE, de tha'y DC 1 (SEF) FK 1 DC nen FK 1 (SDC) => (ABK) la mat phang qua AB va vuong goc (SDC)
va D'C = (ABK) n (SDC). Thiet dien ABCD' la hinh thang can. De dang chiing minh SEF la tam giac deu (SF = EF = SE = 2a) D'C = a, FK = a V 3 , ta c6 AB = 2a. (h. 70).
3
1
Vay dien tich thiet dien ABCD': dtABC'D' = - (AB + D'C).
b) De dang chiing minh duac SK ± (ABCD').
V3a^
Goi O la tam day, khi do S O la duong cao ciia hinh chop d^u S . A B C D
va S O cung la ducmg cao ciia tam giac d^u S E F canh bang 2a, nen
I S O = a V3 . vay VS^BCD = - 2a.2a.a V3 = - . V3 a'.
3 3 ,j
Dodo V^^^^,^. = F , , , , , - V^^^^,, = - . V ^ a ^ -VJa' 5V3
a l 35. Xet duomg 1 bat k i di qua O
va (1; P) = « , lay M bat ky tren 1, ha M H ± mp(P) => M O H = a
Du5ng thing d 1 mp (P) tai Oc6'dinh,d l O H .
vay (. luon tao vdri d m6t goc 90" - a khong ddi va di qua di^m O c6' dinh e d =>