C thudc mat non tron xoay
A BMH vuong tai B', ha M K
AB tai K => A BMB' =AMBK vi CO MB chung va B, = M , => MK = BB' khong doi (vi B va mp P cho trudrc) => khoang each tijf M den AB khong d6i. VSy M thuoc mat tru c6 true la ducmg AB va ban kinh R = BB'.
42. HD: Mu6'n xac dinh goc cua hai duofng thang cheo nhau a va b th6ng thucmg ta lay m6t di^m A e a, tilt A ke b' // b thi goc giiia hai dudng thang a va b' chinh la goc giua hai duomg thang cheo nhau a va b.
Giai:
Dung ducmg sinh A'B => 00'//A'B. vay goc giiia AA', 00" chinh la A A ^ .
Khong kho khan lam ta tinh duoc AB = 2Rsin 15° => t g A A ^ =
}'
2sinl5° => A A ' B = 27°27'. 43. Gia SIX c6 hinh vu6ng ABB'A' n6i
tiep hinh tru (h78). Tiit B dung ducmg sinh BC, vay BC 1 (O'), CB' la hinh chie'u ctia BB' trSn day, ma A'B' 1 BB' (dinh li ba dudng vu6ng goc) => tam giac A'B'C vuong tai B'.
B
Hinh 78
Dat canh hinh vu6ng AB = BB' = A'B'= AA' = x
Ta c6: 4R' - = - h^ (R la ban kinh day, h la chi^u cao hinh tru)
4R^ +h'
196 + 4 = 10(/M).
44. Tie'p dien ciia hinh tru la mat phdng chi chiia m6t ducmg sinh ciia hinh tru. Ta da bidt cac ducmg sinh ciia hinh tru song song vdi nhau, boi vay hai
ti€p di6n cua hinh tru mac nhien chiia hai dudng thang song song. Vay
giao tuya'n ciia chiing neu c6 thi song song v6i ducmg sinh va song song
vdi true ciia hinh tru.
45. D6 dang xac dinh goc phang ^ ciia nhi dien tao bdi mat ben va
mat day, do la goc SIH = « ( I la trung di^m ciia BC).
Hinh tru nfii tie'p hinh chop tii giac deu c6 true la dudng cao SH ciia hinh chop, day dudi thude day hinh chop, day tren
I ndi ti^p hinh vudng thudc mp song song vdi day dudi, tie'p
xiic vdi canh hinh vudng tai ' trung di^m cua nd, di^m nay
cung thudc trung doan cua mat ben hinh chop, nhu I ' 6 SI chang han.
Theo d^u bai: HH' - HT, va IJ = HI - HJ. Tam giac vudng I'JI cho: U = IJ cotga
<» H I - H J = HJcotga; <=>-a = HJ(1 + cotgor)
HJ = 2(1 +cot g « ) a
V = TtHJ^ = m
%{\ coi gay
, , cosa + sina V2sin(45°+a)
Bien d6i: 1+ cotga = = '-
sma
Thay vao V dugc dpcm.
46. Hinh chieu cua hinh cau tren day dudi hinh lang tru la m6t duofng tron noi tiep day hinh lang tru. Dudng tron nay c6 tarn la hinh
chie'u cua tarn hinh cin irtn day du6i (trung \d\o hai dudng
cheo day), c6 ban kinh bang ban kinh hinh c^u (h 80)
Goi ban kinh hinh cin la r, ta c6:
sin a C \0 B' 3 3V Hinh 80 K e D I 1 A B t h i D I = 2r(h.81) DI l.r
Tarn giac vu6ng DAI cho: DA = = sin a sin a
Ducmg cao lang tru cung bang ducmg kinh hinh cSu la 2r (h.82) V„ = dt (ABCD). A'A = DA' sin a .2.r 8.r
sma
Hinh 81 Hinh 82
48
, Hinh 82 cho tu: di6n d^u ABCD canh a vh cung tao dugc hinh lap
phucmg chiia tii dien d6u nay (xem bai 2, chuong IV: Kh6'i da dien). Hinh c^u ti6'p xiic vol cac canh cua tur dien d^u nay se n6i tiep hinh lap phucmg, tiic la tiep xiic vdi cac mat ciia hinh lap phuong tai tarn cua cac mat va do do ti6p xiic cac canh cCia tur difin d^u tai trung di^m cac canh
(hinh 194). Nhu vay ducmg kinh hinh ciu bang canh hinh lap phuong c6
dirdng cheo bang a (canh tu: diSn d^u) => canh hinh lap phuong bang . Do do ban kinh hinh cSu bang
2 4 (SAB) ± (ABCD) (do SA
±(ABCD)) AB = (SAB) n (ABCD) ma BC 1 AB; BC c (ABCD) => BC 1 A B
Mat khac ABXSC
AB' J. BC (chiing minh trfin) => A B ± (SBC) => AB' 1 BC Tuong tu AD' X D'C, AC 1 SC =i>
49,
ABC = AB'C = AC'C = AD'C
= ADC = 90°. Hinh 83
vay A, B, C, D, B', C, D' cimg nam trSn mot mat ciu.
a) va b) Hoc sinh tu lam.
c) Do A ABC vu6ng tai A va O la trung diem BC. Qua A, dung mp (Q) I d tai I (I e d). A I la hinh chieu cua AO tren (Q). AO c6 do
dai khdng d6i, (Q) c6 dinh, d c6' dinh ntn A I c6 d6 dai kh6ng doi. Nhu vay, mp (Q) cat mat c^u (O) noi d cau a theo ducfng tron C<^ tam I , ban
kinh lA, thu6c (Q). Dao lai, nSu la'y bat ki di^m Me ( ^ thi lA =IM =>
OA = OM => M e mat cixx (O).
d) Khi g6c vu6ng x ^ quay quanh O thi M chay titn d => ducmg thang
A chay trfin mp (Q) chiia (d) va (Q) I P , mp trung true R ciia SO c6 dinh => I e A' (la giao tuye'n ciia (R), (Q), A' //d va each d m6t khoang bang - SO).
Dao lai, lay m6t di^m bat k i I G A' => I €(R) va I each d^u S, O. N,. ^ cSu (I) ban kinh SI = OI cat (d) 6 A va B. Tilt I ke mot ducmg thing song song SO cat (d) d M . Do OI = l A = IB va M I J . (P) nen MO = MA ^ MB => M la tam ducmg tron ngoai tiep A BOA => AOB = Iv. Vay 11;,
tam mat c^u ngoai tia'p tii dien SOAB vdi AOB = Iv.
Ke't luan: Quy ti'ch tam I ciia mat c^u ngoai tiep tur dien SOAB la dudiig thing A', giao cua hai mp (R) va (Q).
50. Goi H la trung didm ciia BC, AABC can nen A H I B C ma mp (ABC) ± mp (BCD) =^ A H 1 (ABC)
A BDC vu6ng tai D => HB = HC = HD Nhimg diem nam tren A H each deu cac diem B, C, D.
Dung mat phang (a) la mat phang trung true ciia canh AB. Goi 0 = (a) n A H . Thi ta c6 OA = OB = OC = OD, nen O la tam mat c^u ngoai tie'p t i i dien ABCD.
Goi R la ban kinh mat ciu nay thi R = OA = OB = OC nen R chinh la ban kinh ducmg tron ngoai tie'p A ABC.
Ap dung dinh l i Pitago vao tam giac vuong ABH ta c6: