1) (a) la mat phang chua SD va vuong goc vol (SAC). Xac djnh va tinh di^n tich thiet dif n cua (a) voi hinh chop S.ABCD .
2) Goi M la trung diem cua S A , N la diem thuoc canh A D sao cho A N = x . Mat phang (p) di qua M N va vuong goc vol ( S A D ) . Xac djnh va tinh dỉn hch thiet di^n cua hinh chop cat bai (p).
Jiuang đn giai
^) Goi E la trung diem cua canh ABva O la giao diem cua AC va DE thi •^DCE la hinh vuong c6 tam la Ọ
Ta CO SA 1 (ABCD) => SA 1 O D , them niia O D 1 AC =^ O D 1 (SAC).
Tir do ta c6 OD 1 (SAC) => (SDO) 1 (SAC). •*
Phiwug phiip giai Todn Hinh hgc theo chuyen da - Nguyen Phii Khdnh, Nguyen Tat Thu
Vay (SDO) chinh la mat phang ( a ) .
Thiet dien ciia hinh chop v6i mat phang (a) la tam giac SDE . Ta CO SO = VOÂ + A S ^ = ,.| \ + = a j | . 2) Taco Vay B C = D E = aN/2 , do D E 1 ( S A C ) => D E 1 A O ^ S^^^ = |sỌDE = ^ . a ^ . a V z = . A B I ( S A D ) M e ( p ) n ( S A B ) A B c : ( S A B ) (p) n ( S A B ) = M Q / / A B , Q e S B . A B / / ( p ) N e ( p ) n ( A B C D ) A B c ( A B C D ) A B / / ( p ) ^ (p) n ( A B C D ) = N P / / A B , P e B C .
Thiet dien la i\x giac M N P Q . N P / / A B M Q / / A B ' M N C ( S A D ) A B I ( S A D ) ' Tuong tu, Do Lai CO N P / / M Q (l) A B I M N (2)
Tu (l)/(2) suy ra (tu giac MNPQ la hinh thang vuong tai M va N .
Do do SMNPQ = | ( N P + M Q ) M N . M N = V A M2 +A N2 = a 2 — + x = 4 4x^ , MQ = - A B - a 2 2 N P D N ,^^__^J_±A-_2[.-.) A B D A DA c ^Ini \? + 4 ? ( 3 a - 2 x ) V ? + 4 ? V^y SMNPQ = - ( 2 ( a - X) + a) = ^ . 106
Cty TNHH MTV DWH Khang Viet
§ 2. G O C
I Goc giua luii duang thdng cheo nhaụ ' '
E)e tinh goc giiia hai duong thang cheo nhau a va b ta c6 cac each sau
Cdch l:T\m goc giira hai duong thang a, b bang each chon mot diem O
thich hop (O thuong nam tren mot trong hai duong thang). Tu O dung cac duong thang a b ' Ian lugt song song (c6 the trung neu O nam tren mot trong hai duong thang) voi a va b. Goc giua hai duong thang á, b' chinh la goc giij-a hai duong thSng a va b. u •]>;(
Chu ^'De tinh goc nay ta thuong su dung dinh l i cosin trong tam giac:
cosA = • b ^ - c ^ - a ^ 2 be
Cdch2:Tur[ hai vec to chi phuong u ^ U j ciia hai duong thSng a, b
Khi do goc giua hai duong thang a, b xac dinh boi cos(a,b) U1.U2
Chu y: DG tinh U j . U j , U j , U2 ta chon ba vec to a, b, c khong dong
phang ma c6 the tinh duoc do dai va goc giOa ehiing, sau do bieu thj cac vec to U j , U 2 qua cac vec to a, b, c roi thue hien cac tinh toan.
2. Goc giua duang thdng vai mat phang
De xac djnh goc giiia duong thclng a va mat phang (a) ta thuc hien theo cac budc sau:
• Tim giao diem O = a n (a) • Dung hinh chieu A ' ciia mot diem A e a xuong (a)
• Goc A O A ' = (p chinh la goc giua duong thang a va ( a ) .
• De dung hinh chieu Á cua diem A tren (a) ta ehgn mot duong thang b 1 (a) khi do A A V / b .
• De tinh goc (p ta su dung he thuc luQ'ng trong tam giac vuong AOAÁ.
ll^goai ra neu khong xac djnh goc cp thi ta c6 the tinh goc giira duong thang a
Phucmg fthtip gidi Todn Hhth hoc theo chuyen de - Nguyen Phii Khdnh, Nguyen Tat Thu
va mat phang (a) theo cong thiic sincp = n la vec ta c6 gia vuong goc vol (a) .
3. Goc giua hai mat phang
De tinh goc giua hai mat phing (a) va (p) ta c6 the thuc hien theo mot trong cac each sau:
Cdch l : T i m hai duong thang a, b Ian lugt vuong goc vai hai mat phang (a) va (p).
Khi do goc gii>a hai duong thang a,b chinh la goc giiia hai mat phang (a) va (p).
fa 1 ( a )
ụn
trong do u la VTCP ciia a con
(a)
/<y)
b l ( p ) .((a),(p))^(a,b).
Cach 2;Tim hai vec to nj,n2 c6 gia Ian lugt vuong goc vai (a) va (p) khi
do goc giiia hai mat phang (a) va (p) xac dinh boi cos(p =
Cdch S.-Su dung cong thuc hinh chieu S' = Scos(p, tu do de tinh coscp thi ta can tinh S va S'.
Cdch 4: Xac dinh cu the goc giua hai mat phSng roi su dung hf thuc lugng trong tarn giac de tinh.
Ta thuang xac djnh goc giiia hai mat phMng theo mot trong hai each sau: a) • Tim giao tuyen A = (a) n (p)
• Chgn mat phang (y) 1 A / ( p ) • Tim cac giao tuyen a (y) n (a), b = (y) n (p), / . M
k h i d 6 : ( ( ^ 0 ^ ) = (Xb). \
b) • Tim giao tuyen A = (a) n (p) ^ ^
• L a y M E { p ) . ^ 7 \ " V y V
Dung hinh chieu H cua M tren (a) 108
Cty TNHH MTV DWH Khang Vift
• Dvng H N 1 A => M N 1 A .
phuang phap nay c6 nghia la tim hai duong thang nam trong hai mat phang (a)'(P) va vuong goc voi giao tuyen A tai mgt diem tren giao tuyen.
Chu ^."Cho hinh chop S.A]A2... c6 duong cao SH . De xac dinh goc giua
matphSng (SAjAj) ( i,j e {l,2,...,n}; i j) voi mat day ta lam nhu sau: • Tu H ve HK 1 A j A j , K e ÂÂ , .,, .
• Khi do SKH la goc giua hai mat phang (SAjAj) va mat daỵ
Vidu 2.2.1. Cho t i i dien ABCD. Ggi M, N Ian lugt la trung diem ciia BC va
AD, biet AB = CD = a,MN = - — . Tinh goc giua hai duong thang AB va CD.
Xgigidị Cdch J.-Ggi I la trung diem ciia AC .
^^^'^C/CD"^(^H"^)-
Dat M I N = a
Xet tarn giac I M N c6
. . . AB a CD a . ..^ aVs I M = = — , I N = = - , M N = — I M = = — , I N = = - , M N = — 2 2 2 2 2 Theo djnh l i cosin, ta c6 cos a - I M ^ + I N ^ - M N ^ U 2IM.IN => M I N = 120° suy ra ( A B , C D ) = 60°. 'h2: C O S ( A B , C D ) = C O S ( I M , I N ) = I M . I N I M I N
MN = IN - IM MN^ = (IN - IM)^ = IM^ + IN^ - 2IN.IM;
IM^+IN^-MN^ â IN.IM = - = ; COS(AB,CD)= C O S ( I M , I N ) IN.IM = - = ; COS(AB,CD)= C O S ( I M , I N ) 8 I M . I N IM I N V^iy (AB,CD)=60''. ' 1 109
Phumig phapgiai foan Hinh hoc theo chmjen dc - Nguyen Pliii Khdnh, Nguyen Td't Thu
Vidu 2.2.2. Cho hlnh chop S.ABCD c6 day ABCD la hinh thang vuong tai A
va B, AB = BC = a, AD = 3ạ Hinh chieu cua S len mat phang day trung voi
trung diem canh AD. Mat phang (SCD) tao voi day mot goc 60". Goi M la trung diem doan CD. Tinh c6 sin cua goc giua hai duang thang AM va SC.
£gl gidị
Goi H la trung diem cua AD, ta c6 SH 1 (ABCD). Ta CO ABCH la hinh vuong canh ạ
Tit H ve UK 1 CD , K € CD .
Suy ra CD±(SHK), nen SKH la goc giOa mat phang (SCD) voi mat day, do do SKH = 60''.
Ta c6: