Phương pháp giải toán hình học theo chuyên đề part 2

Phutnig
phdp
gidi
Todn Htnh hoc
theo
chuyen
de -
Nguyen
Phu Khdnh,
Nguyen
Tat Thu
Jiuang
ddn
gidi
Goi I
la trung diem ciia AD
.
S
AD
Ta
CO
CI
= lA =
ID
= —
suy
ra
AACD
vuong tai C
=> CD
1 AC (1)

SA1(ABCD)=>SA1CD
(2).
Tu
(1) va (2) suy ra
CD
1SD
=>
ASCD
vuong.
B
C
Goi d^;d2 Ian
lugt la khoang each
tit
B,H den mp(SCD)
Ta
c6:
ASAB
-
ASHA
SA
SB SH SA^ 2 SH d, 2 ^ 2^
a 2 3 1
SH
SA SB SB^ 3' SB dj 3
The
tich khoi tu dien
SBCD:
VSBCD
=

-SA.iAB.BC
=
3 2 6
Ta
c6:
SC
=
N/SA^
+
AC^
= 2a,
CD
=
Vci^TlD^
=
Via
SscD =
^SCCD
=
yfla^
.
3.
>/2a3
Ta
c6: V<
SBCD
- g'll-SsCD-^"1 - 2
Vay
khoang each
tie

H den
mp(SCD)
la
dj =
3
Bai
2.3.6.
Cho hinh lang try diing
ABCAjBiCi
c6 AB = a, AC = 2a,
AAj
= 2aV5 va
BAC
=
120*'.
Gpi
M la trung diem cua canh
CCj.
Chiing
minh
hai
duong thang MB va MAj vuong goc voi nhau.
Tinh
khoang each tu diem
A den m^t
phang
(AJBM)
.
Jiuang
ddn

gidi
Taco: BC =
VAB^
+
AC^
-2AB.AC.eosl20° =
ayf?
BM = VBC^TCM2
=2>/3a;
AjM
=
^AjCj^
+
C^M^
=3a
BAj
=
7AB^+AA7
=
\/2Ta;
BM^
+
A^M^
=
BA^^
Suy
ra MB 1
MAj.
Ke CH1AB CH1
(ABAj).

Ta
CO
CH
= AC.sin60° = >/3a .
142
Cty
TNHH MTV DWH Khang Viet
fhe tich kho'i tu di^n
M.ABA
:
V„ ,=icHlAB.AA,=:fi.
Qgi K la hinh chieu cua A len mp{AiBM)
ta CO:
VM.ABA,
=^AK.1MB.MAI
=>AK =
3 2
MB.MA
Vay
khoang each tu A den
(A^BM)
la
AK
=
Bdi
2.3.7.
Cho hinh chop
S.ABC
c6 goc giiia hai mat phang
(SBC)

va
(ABC)
bang 60°, eae tam giac
ABC
va
SBC
la tam giac deu canh a.
Tinh
theo
a khoang
each tu B deh mp(SAC) .
Jiuang
ddn
gidi
Gpi M,
N Ian lugt la trung diem cua
SA, BC
.
Ke
BHlCNtai
H suy ra BH la / / \ v :
khoang each tu Btoi mp
(SAC)
Ta
CO
SNA = 60° la goc giiia hai m|it
phing(SBC),(ABC).
Tam
giac
SAN

deu c^nh SA = AN =
Taco
CM
=
VCA2-AM2
=
>/l3a
;MN
= VCM2-CN2=^ C
Ta c6:
BH.CM
=
MN.BC
=:> BH =
MN.BC
sViSa
CM
13
^^i
2.3.8.
Cho lang try diing
ABCAiB^Cj
c6 tat ca cac canh deu bang a, M la
'^ng
diem cua AAj. Chung minh
BM
1
B^C
va tinh khoang each giua hai
•^"^ing

thang BM va B^C.
Jiuang
ddn
gidi
, '
Gpi
E la diem tren duong thSng AAj sao cho Aj la trung diem cua
ME,
taco
BM/ZBjE.
JiC =
V2a,BiE
= BM = VBA2+AM2 CE = VCA2+AE2 = ^
y/l3a
•i.>
•
Phumtg
phdpgiai Todtt
Hinh
hgc theo chuyen de- Nguyen Phu
Khanh,
Nguyen Tai Thu
B^C^ +
BjE^
= CE^ => BjC 1 BjE => BjC ± BM
*
Taco
BM//(BjCE)^d(BM,BiC) = d(M,(BiCE))
Gpi
H la

trung
diem cua AjCj ta c6
BHl(ACqAi)
The
tich
cua
khoi
chop
Bj .CME :
V
=
-B,H.S.
B].CME
- 2
•'l^-^CME
1
Vsa 1 VSa^
=
—. .—a.a =
3 2 2 12
Goi I la
hinh
chieu ciia M len mp(BjEC) ta c6 :
3Vp
r-k^c
TsOa
V,
=
-MI.S
'B,CE

>MI
= -
Bi.CME
^BjCE
10
V^y
khoang
each
giCra hai duong thang BjC,BM la MI =
10
Bai
2.3.9.
Cho lang try dung ABC.A'B'C c6 day ABC la
tarn
giac
vuong,
AB
= BC = a, canh ben AA' = aV2 . Gpi M la
trung
diem ciia canh BC.
Tinh
theo a the
tich
cua
khoi
lang try ABC.A'B'C va khoang
each
giiia
hai duong
thang

AM, B'C.
Jiuang
dan gidi
Tu
gia
thiet
suy ra tam
giac
ABC vuong can tai B. B" .
The
tich
kho'i
lang try la:
VABCA'SC = AA'.SABC =
(<Jvtt).
GQi
E la
trung
diem cua BB'. Khi do B'C / /(AME)
Suy ra
d(AM,B'C) = d(B'C,(AME)) ^
=
d(C,(AME)) = d(B,(AME))
Gpi
h la khoang
each
tu B den mat phang (AME).
Do tu di?n BAME c6 BA, BM, BE doi mot vuong goc nen:
1
1

1
—- +
1
1
7 . aV?
- = ^=>h =
h^
BA^ BM^ BE^ h^ 7
V|y
khoang
each
giira
hai duong
thing
AM va B'C la
a^/7
Bai
2.3.10.
Cho
hinh
chop
tu
giac
deu S.ABCD c6 day la
hinh
vuong canh A
Gpi
E la diem doi xung ciia D qua
trung
diem cua SA. M la

trung
diem ciJ^
144
^g,
N la
trung
diem cua BC. Chiing
mirJi
MN vuong goc vai BD va
tinh
(theo
3)
khoang
each
giiia
hai duong
thing
MN va AC.
Jiuang
ddn gidi ? ^
Gpi
P la
trung
diem ciia SA. •-: ; ;, ,
I
Ta
CO
MP
la duong
trung

binh
cua tam
giac
EAD
=>MP//AD=>MP//NC.
H
Va MN = |AD = NC.
I
Suy ra MNCP la
hinh
binh
hanh
=> MN //CP =^ MN //(SAC).
Ta de ehung
minh
dugc
BD1(SAC)=^BD1MN
Vi
MN//(SAC) nen: d(MN,AC) = d(N,(SAC)) = -d(B,(SAC)) = -BD= ^
Vay d(MN,AC) =
^f2a
Bai
2.3,11.
Cho
hinh
chop
SABC c6 tam
giac
ABC vuong can tai B, AB = BC = 2a,
(SAB) va (SAC) eiing vuong goc voi (ABC). Goi M la

trung
diem AB, mat
phang qua MS
song song
voi BC cat AC tai N. Biet goc
giiia
(SBC) va (ABC)
bang 60°.
Tinh
the
tich
khoi
chop
S.BCNM va khoang
each
giiia
hai duong
thang
ABvaSN.
Jiuang
dan gidi ^
Do hai mat phang (SAB) va
(SAC)cat
nhau
theo giao tuyen SA va cung vuong goc
voi (ABC) nen SA 1 (ABC), hay SA la
duong
jcao cua
khoi
chop

S.BCNM.
Ta CO:
SBCNM
= SABC - SAMN
= 2a2 MA.MN =
2a2-ia2=^
'
Do
BCIAB
(SAB)lBC.
BC 1 SA
Nen
SBA chinh la goe
giira
hai mat phang (SBC) va
(ABC),
the thi theo
i
gia
thiet
ta c6
SBA
= 60°.
145
Trong
tam
giac
vuong SAB
ta
c6

SA
=
AB tan 60^
= 2a>/3
.
Vay Vs.BCNM
=^SA.SBCNM
=i.2a>/3.^
=
V3a3(dvtt)
Goi
P
la
trung
diem cua
BC
thi AB
/
/NP, AB (2
(SPN)
nen
AB
/
/(SPN)
do
do
d (AB,
SN) =
d (AB; (SPN))
=

d
(A;
(SPN))
Tir
A ha
AElNP,EePN
thi \> PN1
(SAE)
;ha AHISE
thi
[PNISA
V ^ •
AH
1
(SPN)
=^
d(A;(SPN)) = AH.
Taco
AE
=
NP
=
a;SA-2aN/3=>—^
=
-^
+
-i-
= ^^=>AH = aJ
AH^
AS^ AE^ 12a^ V

'12
13
Vay d(A;(SPN)) =
a^
Bai
2.3.12.
Cho
lang
try
ABCD.AiBiCiDi
c6 day
ABCD
la
hinh
chii
nhat,
AB = a,
AD
= a\/3
.
Hinh
chie'u vuong
goc
ciia diem
Ai
tren
mat
phang
(ABCD)
trung

voi giao diem
AC va
BD.
Goc
giiia
hai mat phang (ADDiAi)
va
(ABCD) bang 60°.
Tinh
the
tich
khoi
lang
tru
da
cho va
khoang
each
tir
diem Bi
den mSt phSng (AiBD) theo
a.
Jiit6m.gddngi.di
^,
Gpi
O
=
AC n
BD,
I

la
trung
diem canh
AD.
Tac6ADl(AOI)
•
Alio = ((ADDiAi),(ABCD))
=
60°
Vi
OI = - , nen ta suy ra Ajl = 201 = a
^AiO
=
OI.tan60°
= ^.
Do do
VABCD.A,B,CID,
= AJCSABCD = ^ ^
Gpi
Bj
la
hinh
chie'u ciia
Bj
xuong mgt phSng (ABCD)
Do BjC
/
/AJD =^
BjC
/

/(AJBD)
^
d(Bi,(AiBD)) = d(C(AiBD)) = CH
Trong
do CH
la duong
cao
ciia tam
giac
vuong BCD
Ta
co:
CH =
,
=
—
V^y
d
Bi,(AiBD) =
—
VCD^+CB^
2 2
146
Cty
TNHH
MTV DWH
Khang
Viet
0di 2.3.13. Cho
hinh

chop
S.ABC
c6
day ABC la tam
giac
vuong
tai
B, BA = 3a,
= 4a; mat phang (SBC) vuong
goc
voi mat phang (ABC). Bie't SB =
2a73
va
ggC=
30"
.
Tinh
the
tich
khoi
chop
S.ABC
va
khoang
each
tir diem
B
deh mat
phang (SAC) theo a.
s "

'•
'
Jiic&ng
dan
gidi
Goi
H
la
hinh
chie'u
cua
S
xuo'ng BC.
Vi
(SBC)l(ABC)
Aen
• SHl(ABC).Tac6 SH =
aV3.
Do
do
Vs.ABCD =
^SH.S^Bc
=
2a373
.
Ta
CO
tam
giac
SAC

vuong
tai
S I"
Vi
SA = ar/2T,SC = 2a, AC =
5a
Til
••tfyt.li'
va
S^AC
= a^>/21 nen ta
c6
duoc
d(B,(SAC)) =
-j^.
^ASAC
v7
Bai 2.3.14.
Cho
hinh
chop
S.ABCD
c6
day ABCD
la
hinh
vuong canh
a. Goi
M
va

N
Ian luot la
trung
diem cua
cac
canh AB
va
AD;
H
la giao diem cua
CN
va DM.
Bie't
SH vuong
goc
voi mat ph5ng (ABCD)
va SH
=
aS.
Tinh
the
tich
khoi
chop
S.CDNM
va
khoang
each
giiia
hai duong

thiing
DM
va SC
theo
a.
Jiuang
ddn
gidi
Taco:
VgcDfji^
=-SH.S^[^DC
SMNDC
= SABCD -
SAAMN
- S/^MBC
2
a^ a^ 5a^
=
a
= .
8
4 8
Nen
^S.CDNM
8
24
(dvtt).
Laithay:
DM.CN = i(2DA-DC).i(2DC-DA)
=

DA2-DC^
=0.
2V—
/ 2'
V^y
CN
1
DM
h:r
do
SC
1
DM bai vay:
d(SC;DM) = d(H;SC)=^^'^"SC-^"-^^
SH.CH
SC
SC
VsH^+CH^
147
Phumig
phiifigidi
Tiu'ui Ilinh hoc
theo
chuyen
de -
Nguyen
Phti
Khiinh,
Nguyen
TA't

Thu
DM
'^is
Laico:
CH = ^^^^
DM
Hay ta c6
khoang
each
can tinh la: ^ay— •
Bdi
2.3.15.
Cho hinh lap phuong
ABCD.A'B'C'D'
c6
canh
b^ng a . Goi M,N
Ian lugt la trung diem cua AB va B'C. Tinh
khoang
each
giCra
hai duang
thing AN va DM.
JIuang
ddn
gidi
1.1
Gpi E la trung diein
ciia
BC .

Dethay
AADM
= ABAE
nen AMD = AEB, ma
AEB + BAE = 90'
,0
D
:^ AMD + BAE = 90"
=>
DM 1 AE.
Lai
CO
EN 1 (ABCD) => EN 1 DM
dodo
(AEN)lDM
tai I.
f
/
7
\
E
\
\
\
\
;
\
\
\
\

\
•
\
\
N
D'
Xet
phep
chieu
vuong goc len
(ANE),
ta c6 AN chinh la hinh
chieu
cua no
nen d(DM,AN) =
d(l,AN)
Goi K la hinh
chieu
cua I tren AN thi
d(l,AN)
= IK.
Taco
AAKI-AAEN,suyra
i|^-^r^IK
= ^^^ (l)
EN AN
AN^
= AE^ + EN^ = AB^ + BE^ + EN^ =
1
AN

AN
= —.
4 2
1
1
AI^
AD^ AM^ a^
Thay
vao (l) ta
duac
IK -
14
5 aS
+ — = — => AI =
15
Vay d(DM,AN) =
2aS
15
148
Cty rNIUI MTV DWII
Khang
Viet
§ 4. THH TIC:H
KHOI
DA
DlfiN
, .
De tinh the tich
riui
mot khoi dn

dii'n
(lang try va hinh
chop)
ta thuang
thuc
jiien
theo
cac
cacii
snii
Qdch
//Tinh trirc tiep ,
^^^•''"^'^
Su dung cac
cong
thlie: /iO :.
<
, •
, The tich khoi
chop:
V =
-h.S^^,
trong do h la
chieu
cao, Sj la dien tich day.
3
Dacbiet:
Ne'u hinh
chop
S.ABC

c6
SA,SB,SC
doi mot vuong gck thi:
Vs.ABC=^SA.SB.SC.
•^•-^'^^
• The tich khoi lang tru: V = h.Sj, trong do h la
chieu
cao cua lang try, la
dien tich day.
Dacbiet:
+) Hinh hop
chi>
nhat
ba
canh
a,b,e : V = abe
+) Hinh hop lap phuong
canh
a: V = a''
/lift,*
' '
Cdch
2. Tinh gian tiep.
• Ne'u hinh H
duoc
taeh
thanh hai hinh roi
nhau
Hj, H2 thi V,]^ = Vj^
~^H2

•
Tren
cac duong
thang
SA, SB, SC
ciia
hinh
chop
S.ABC
ta lay Ian lugt cac
. , „, ^, ,
SA'.SB'.SC',,
diem A ,B ,C . Ta co:
VS.A'WC
=
g^SBSC
^'^"^
"
Chijy:Khi
xet ti so the tich
ciia
hai khoi
chop
thi ta thuong tim
each
chuyen
ve
hai khoi
chop
c6

ehung
mat
phang
day.
Vidu
2.4.1.Cho
hinh
chop
tu
giac
deu
S.ABCD
. Tinh the tich khoi
chop
biet
1)
Canh
ben
bang
a\/5 va mat ben tao vol day mot goc 60"
2) Duong cao cua hinh
chop
tao voi day mot goc 45" va
khoang
each
giira
hai duong
thang
AB va SC b^ng 2a.
Goi O la tarn cua day, ta c6 SO 1

(ABCD)
suy ra : Vg
yi^g,;^
=
-^SCSy^gco
J[ffi
gidi.
L
(ABCD;
1)
Goi M la trung diem CD, ta c6: CD 1 (SMO)
Do do goc SMO la goc
giua
mat ben voi mat day, nen SMO = 60°
I
Dat AB = 2x => MO = x,OC = xV2
Trong cac tam
giac
vuong
SOC,SOM
ta c6: A=:;Ar^ ii;
SO^ = SC^ - OC^ - 5a2 - 2x^ SO = OM. tan 60" =xS , ,,
•
.,
149
Phuang phdp gidi Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tai Thu
Nen ta c6 phuong trinh: 5a^ - 2x^
= 3x^
=^
x =

a
Vay Vs.^3CD
4'<^-(2x)^
=^x^ =^a3.
3 3
2) Goi K la hinh chieu ciia O len AM,
ta
CO
OK
±
(SCD)
nen
OSK
la goc giiia
duong cao
SO
vol mat ben nen
OSK
=
45°
.
Goi N la trung diem AB.
Do AB//(SCD)
=^
d(AB,SC)
=
d(AB,(SCD))
= d(N,(SCD)) = NH = 2a
Trong do HN
//OK

=>
OK
=
^NH
= a
1
2
Cac tarn giac
SKO,SOM
la cac tam giac vuong can nen ta c6
SO =
OKV2
=a>/2, OM = SO = a72
VayVs.^3CD=^aV^(2aV^f=^.
Vi du 2.4.2. Cho hinh chop S.ABC c6 day ABC la tam giac vuong
AB =
a, AC
=
aVs
,
SA
1
(ABC).
Tinh the tich cua khoi chop S.ABC
cac truong hop sau
1)
Mat phang (SBC) tao vai day mot goc 60°
2) A each mat phang (SBC) mot khoang bang
4
tai A,

trong
Xgigidi.
,2
a'Vs
Ta
CO
BC =
2a, S^pc = - AB.AC
=
^
va V<
S.ABC
= isA.S
AABC
SA
1)
Goi K la hinh chieu cua A len BC,
taco BCl(SAK).
Suy ra SKA
=
((SBC),
(ABC))
=
60°.
Taco:
AK='^^^
=
^
BC 6
nen SA =

AK.
tan 60° = |. Vay
Vg
^BC
= ^2
2) Gpi H la hinh chieu cua A len
SK,
ta c6 AH L (SBC)
150
Cty TNHH MTV DWH Khang Viet
Trong tam giac SAK, ta c6:
1
1
AH^
SA^ AK'
.,3
Vay
Vs,ABC
=
•
SA^ AH-^ AK^
a-^76
12
2
fit
vcnoVT'
X^i
du
2.4.3. Cho hinh chop S.ABCD c6 day ABCD la hinh thang vuong tai
va

B,
AB = BC = a, AD = 4a. Tam giac SAD la tam giac deu va nam trong
0iat
phang vuong goc voi day. Mat phang (SCD) tao vai day mot goc 60° .
Tinh the tich cua kho'i chop S.ABCD theo a .
Xffigidi.
Goi
H
la trung diem doan
AD,
ta CO SH1 AD ^ SH1 (ABCD)
Goi
K
la hinh chieu,cua
H
len
CD,
ta CO CD 1 (SKH).
Suy ra
SKH
la goc giii-a mat phang
(SCD) voi mat day, do do SKH = 60°
Goi
E
la hinh chieu aia
C
len AD,
suy ra ABCD la hinh vuong canh a.
Ta c6:
CD

=
VCE2+ED2
=aylw
Do ACED ~
AHKD
nen ta c6:
HK
KD
CE DE
JVTO
=*HK
=
KD.CE
DE 3a
Suy ra SH =
HKtan60°
-
Vay the tich khoi chop la: VS.ABCD =
^SH.SABCD
^ ^^-^
•
du.
2.4.4. Cho hinh chop S.ABCD c6 day ABCD la hinh vuong canh a,
SA
vuong goc voi day. Mat phang (SBD) tao voi day mot goc 60°
.
Goi M,
1^ Ian lugt la hinh chieu cua A len
SB,
SD. Mat phang (AMN) cat SC tai P.

Tmh the tich khoi chop S.AMPN
.
JCsngidi.
Goi O la tam ciia day, ta c6 BD1 (SOA)
Phtfcmg phdpgiiii Toan Hinh hoc
theo
chuyen de-
Nguyen
Phu Khanh,
Nguyen
Tat Thu
suy ra goc SOA la goc giiia hai mat 5
phing (SBD) va mat day nen SOA = 60°. /Vs.
Trong
tam
vuong
SAO ta c6: / j \Vr\,.
SA:.AO.tan60"
==^.73
= ^. / ]^»\
BC ± AB /
A
' '
^
^ \
BC
±
AM
=>
AM

1
(SBC)
ri>
AM
1
SC
/
~'X^=zz-\
V-'
Tuong
tu: AN 1 (SCD) => AN 1SC, '-
'"'o'^^
tu
do suy ra; SC1 (AMN) ^ C
Nen AP la
duong
cao cua hinh
chop
S.AMPN
Suy ra:
Vg^^^p^
= -
AP.S^^^,,^,
Ap
dung
he thiic
lugng
trong tam
giac
vuong

SAC ta c6:
SP ^
SP.SC
^ SA^ ^ 3 _ gp^ 3g^^ 3aVT4
SC" SC^
SA^+AC^
7^ 7 14
^
r-AT,
SA.AB
as!\5
Trong
tam
giac
vuong
SAB ta co: AM = =
SB 5
Acnr^
.cn^^ MP SP
SP.BC
SaTsS
Do
ASBC
~ ASPM
=>
= —
=>
MP = =
BC SB SB 35
a2V2T

Suy ra:
S^MPN
= '^^.^AMV =
AM.MP
=
35
1
3aVl4
3^721 3a^V6
3 14 35 70
Vi du
2.4.5.
Cho hinh
chop
SABCD
c6 day
ABCD
la hinh
vuong
tam O, SA
vuong
goc voi (ABCD), AB = a,SA = aV2. Gpi H,K Ian lupt la hinh
chieu
vuong
goc cua A tren SB, SD.
Chung
minh: SC 1
(AHK)
va tinh the tich cua
khoi

chop
OHAK
theo
a . .
JCffigidi
. Ta CO
:
BC 1 (SAB) =^ BCl AH ma AH 1 SB
^ AH 1 (SBC) AH 1 SC .
Tuong
tu AK1 SC SCI(AHK).
SH
SH.SB
SA^ 2a^ 2 SK
SK.SD
SA^ 2a^ 2
Do
SB SB2 SB^ 3a2 3 SD SD^ SD^ 3a^
SH
SK „^ / /no > 2 „^ 2 /r
2^f2a
— = — => HK / /BD va HK = - BD = -av2 = .
SB SD 3 3 3
Cty
TNIIIl
MTV DWII Khaug Viet
Gpi G la
giao
diem
cua SO vh KH

thi
G la trung
diem
ciia
KH, ma
AH = AK = ^a ^ AG 1 HK .
De thay G la trpng tam cua
ASAC:
nen AG = - AM = IsC = —
3
3 2 3
(M la trung
diem
cua SC).
Vay
SAHK=^AG.HK
^ 1 2a
272a
^ 2V2a^
2' 3 • 3 ~ 9 -f-s u! <
.H-,:;:,
Gpi I la trung
diem
ciia
AM, ta c6 OI / /CM => OI1
(AHK)
' '
'
'
. CM SC a c 1 la

2723^
723^
va OI = — = — = Suy ra
VQ.AHK
= 3
OI-S^HK
= 3' 2= •
Cdch 2: Gpi E la hinh
chieu
ciia
A tren SO thi AE 1
(OHK)
nen AE la
duong
cao cua hinh
chop
A.OHK
Ta c6:
1
1
- +
•
1 2 5
-
+
—=
•
AE^ AS^ ' AO^ 2a^ ' a^ 23^
AE = a.,-
'SHK

SgoH
_
BH.BO
1
BH.BS
1 BA^ 1
'SBD
BS.BD
2' BS^
2BS2
6^^^°"
1 2
Tuong
tu SpoK ~ ^ ^OHK = SgBo - (SgHK +
SBOH
SDOK
'
Ma SsBD -
-SO.BD
= i
VAS^TAO^"
.BD = -,23^ +
—a72
=
SBD
2 2 2V 2 2 ,
: =*S
OHK
U^c 1/2 a^Vs a^72
^

•
V3y
V.oHK
= 3
AE.S0HK
= 3^5-— =
3^75
Pi du
2.4.6.
Cho hinh
chop
S.ABC
c6
C3c
canh
day AB =
53,
BC =
63,
AC = 7a.
Cac m3t ben t3o voi dsy mot goc
bang
nhau
V3
bSng
60°. Tinh the tich khoi
chop
S.ABC
V3
tinh

khosng
each
tir A deh mat
phSng
(SBC).
Bie't hinh
chieu
cua dinh S
thupc
mien
trong t3m
gidc
ABC. ____
153
Phuattg
phapgiai
Todn Hinh hoc
theo
chuyen
de-
Nguyen
Phii Khanh,
Nguyen
Tat Thu
GQ'I
I la hinh chieu vuong goc ciia S tren
(ABC), A',B',C'
Ian lugt la
hinli
chieu cua I tren BC,CA,AB . Tu gia thie't suy ra SAl = SB^I = SCI = 60° . Cac

tam
giac
vuong
SIA',SIB',SIC'
bang nhau nen lA' = IB' = IC => I la tam du6n»
tron
noi tiep tam
giac
ABC .
Goi
p la nua chu vi tam
giac
ABC
5a + 6a + 7a
=^p =
-
= 9a
SAABC
=
VP(P-BC)(P-AC)(P-AB)
=
79a(9a
-6a)(9a -7a)(9a
-5a) = sSa^
Goi
r la ban
kinh
duang
tron
noi tiep

tam
giac
ABC,
ta c6 :
^ r _
SABC
^
6V6a^
^ 2^a
p
9a 3
276 a
SABC
=
Pr
•
=^IA'
= r =
Taco:
SI-IA'tan60° =^^^3
=272a
3
Suy ra
Vg^gc
=
ISLS^BC
=
^2V2a.6^/6a
=
sVSa^

.
Vi
du
2.4.7.
Cho hinh chop
S.ABCD
c6 day ABCD la hinh
thoi
canh a, SA =
SB = SC = a. Tinh SD
theo
a de khoi chop
S.ABCD
c6 the tich Ion nha't.
Xffigidi.
Goi
H la hinh chieu ciia S len mat day, ta suy ra H la tam duang
tron
ngoai
tiep tam
giac
ABC nen H thuQC BD.
Mat
khac
•g^^^^^ACl(SBD)^0
= BDnAC la hinh chieu cua A
len
mat phang
(SBD),
ma AS = AB = AD = a => O la tam duang

tron
ngoai tiep
tam
giac
SBD
=i>
ASBD
vuong tai S. Dat SD = x
Ta c6:
SH.BD
=
SB.SD
=>
SH = va
S^BCD
= |
ACBD
Nen
V.
S.ABCD JACBD =
iAB.SD.OA
MaOA^=AB^-:52l
=
a^-^
3a2-x2
154
Cty
TNHH
MTV DWH
Khang

Vigt
Dodo:
Vs^ABCD
=
—.a.x.Vsa^
-
x^
y\
dung bdt Co si ta c6:
Suy ra:
V<
S.ABCD
^ g -^^ 2
x2+3a^-x^ Sa^
1
3a'
Dang thiic xay ra
<=>
x = 3a - x
<=>
x =
Vay
ABCD
^^^^
<=>
SD
=
't
v '
Vi

dii
2.4.8.
Cho hinh chop
S.ABCD
c6 day ABCD la hinh thang vuong tai
A
va D, tam
giac
SAD deu c6 canh bang 2a, BC = 3a. Cac mat ben tao vai
day cac goc bang nhau. Tinh the tich cua khoi chop
S.ABCD
.
J!gi
gidi.
GQI
I la hinh chieu vuong goc cua S
tren(ABCD), tuong tu nhu <i du tren
ta Cling CO I la tam dudng
tron
npi tiep
hinh
thang ABCD.
Vi
tu
giac
ABCD ngoai tiep
nen AB + DC = AD + BC = 5a
Dien
tich hinh thang ABCD la
S

= i( AB + DC) AD =
5a.2a
= 5a^
2 2 ^
Goi,
p la nua chu vi va r la ban
kinh
duong
tron
noi tiep cua hinh thang ABCD thi
AB+DC+AD+BC
/"
/'I
/'I
/'I
J''
1
I'
1
V
10a ^ . _ S
= 5a va S = pr
=>
r =
—
=
2 ^ p
5a^
5a
\

Tam
giac
SAD
deu va c6 canh 2a nen
SK
= ^ = aV3^SI = VsK2-IK2=V3a2-a2=aV2
5V2a3
= a^IK = r = a
i'
|:
Vay
V
=
|SI.SABCD4"^-^''=
3
^idy
2.4.9.
Cho hinh chop
S.ABC
c6 SA = SB = SC = a va ASB =
CSA = y. Tinh the tich khoi chop
S.ABC
theo
a, a,
(3,
y.
=
a, BSC = P,
155
Phuonig

phapgini
Todn
Hinh
hoc theo
chuyeit
tic
- Nguyen
Phil
Khanh,
Nguyen Tat Thu
JCgrigidi.
V
Ap
dung dinh h' ham
so'co
sin cho tarn
giac
SAB ta co:
AB^
=SA^ +56^ -2SA.SB.cosa
=
2a^ (1
-cosa)
= 4a^ cos^ ^
=>AB
=
2acos—
2
B
V

Tuong tu: BC = 2a cos ^, CA = 2a cos ^ A
Goi
H la hinh chieu ciia S len mat phang day
(ABC),
ta
CO
H la tarn duong
tron
ngoai tiep
Aor^
XT- ATT
AB.BC.CA
_ „
tarn
giac
ABC. Nen AH = , -5 = b
4S
^^^^^^^
Ctif
TNini
MTV
DWli
Khaug
Viet
I—;
Jiea^S^-(AB.BC.CA)
Suy
ra SH =
VSA^
-

AH^
= ^— ^
Ha
AHlBC=i>AH±(BCC'B'),
Vi
AA7/(13CC'B')
^d(AA',(BCC'B'))
=
d(A,(BCC'B')) = AH = a
Ha
CKl AC, vi AB± AC
va
ABIAA'
=>AB1(ACCA')
=>AB1CK =:>CKl(ABC)
=>CK
=
d(C,(ABC))
= b.
Ta CO
ABl(ACCA')=i>CA2'
la
goc giira hai mat phang
(ABC)
va
(ABC)
=>
CAC
= cp.
Do

do:
VsABC
=
^SH.S^ABC
=
Goi
p la nira chu vi tarn
giac
ABC, ta co: p = a
Nen
- p(p - AB)(p - BC)(p - CA)
16a2s2-64a^os2^cos2Pcos2l
a p y
cos—+
cos- + cos-
2 2 2;
smtp
smcp
cos 9 coscp
1111
sin^cp
b2-a^sin^(p
AB^
AH^ AC^ b^
a^b^
=
a
Vay
V.
SABC

a p
cos—+
COS —
2 2
a^k
vol
-cos^I
2
cos^I-
2
a p
cos
COS —
2 2
j^VAABC
^R'he tic
1
1
AB.AC
= -
ab
AB
=
ab^
ab
a^
sin^
(p
^Vb^-a^sin^cp'si"^
2sin(pVb2-a^

sin^ cp
le tich lang try
ABC.A'
B' C la:
.1 '^^l:
ab^
ab^
^
2 sin (px/b^-a^sin^cp sin lifyjb^-a^ sin^cp I.'
2 a
2
P
2
7
-4cos
—
COS
-cos r
Vidu
2.4.10.Cho
lang try dung
ABCA'B'C,
co day
ABC
la tam
giac
vuong
tai A. Khoang each tir
AA'deh
(BCCB')bang

a, khoang each tir C den
(ABC)
bang b, goc giiia hai mat phSng
(ABC)
va
(ABC)
bang
cp
.
1)
Tinh
the tich kho'i lang try
ABCA'B'C
theo
a,b va (p.
2) Khi a = b khong doi, hay xac dinh cp de the tich kho'i lang try
ABCA'B'C
nho nhat.
156
2)Khia
=
b=>V
= —.
s| 2sin(pcos cp
Do
^sincpcos^ (pj
=i.2sin^(p.cos^(p.cos^(p<^
2
^273
,,^33^73

|,
=>sin(pcos
(p<—^=>V>—-—.
2sin
(p
+
2cos
(p
3
,
1 1
Dang
thiic xay ra khi 2
sin''
9 = cos''
cp <=>
tan
(p
= o 9 =
arctan
-j=.
Vay
khi (p =
arctan-^
thi V dat gia tri nho nhat. ,;;
V2
157
Phucmg
phdp gidi Toan Hinh hgc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
Vidu 2.4.1-0.Cho lang try

dung
ABCD.A'B'C'D' c6 day ABCD la
hinh
thoi
canh 2a . Mat phang (B'AC) tao voi day mot goc
2>QP
, khoang
each
tu B de'n
mat
phang
(D'
AC)
bang |.
Tinh
the
tich
khoi
hi
di^n
ACB'
D'.
JUsfi
gidi.
Gpi
O la giao cua hai duong
cheo
AC va BD, ta c6 AC1(B'OB)=>B^ = 30"
GQI
H la

hinh
chieu cua B len B'O, suy ra:
BH
= d(B,(B'AC)) =
d(B,(D'AC))-|
^' ^'
Dodo:
BO = n = a
sin
30°
.
OC = N/BC^ - BO^ = a^/3,
BB' =
BOtan300=^
•SABCD
=
2^^.60
= 2BO.CO = 2a2V3 B
2a^V3- 2a^
M$t
khac Vg.^Bc - VQ^^^D -
^CB'C'D'
- ^AA'B'C - ^^ABCD.A'B'CD'
1
23^
Nen suy ra V^CBD' = gVABCD.ABC'D' = —•
Vi
du 2.4.11. Cho
hinh
hpp ABCD.A'B'C'D' c6 cac mat ben va mat (A'BD) hgp

voi
day goc
60",
biet goc B^ = 60°,AB =
2a,BD
= a\/7.
Tinh
Va^^-B'Ciy
•
Xgigidi.
Gpi
H la
hinh
chieu cua A'
tren
(ABD), J, K la
hinh
chieu cua H
tren
AB,
AD.
Ap
dyng
djnh
li cosin cho
AABD:
BD^ =
AB^
+
AD^

- 2AB.AD.cosBAD
=>
AD^
- 2a.AD - Sa^ = 0 o AD = 3a
S^^BD
=
^AB.AD.sinBAD
-
^
Tu gia thiet suy ra
hir\
chop
A'
.ABD
c6 cac mat ben
hqip
day goc 60"
Nen
H la
each
deu cac canh cua
AABD
158
1:
Neu H nam
trong
AABD
thi H
la
tam duong

tron
npi tiep
AABD
.
Goc
giiia
mat ben (ABB'A')
va
day bang
A^H
=
6O''.
Gpi
r la ban
kinh
duong
tron
npi
tjg'p
AABD
thi:
SAABD 3\/3a
Cty
TNHH
MTV I)
VVII
Khang Vi^t
r
=
5 + V7

=>A'H
= r.tan60" =
-
/ /
\/5V/7 _
5 + V7
D
Tudo,
1
VABCD.A'B'C'D' -6VA'.ABD
=6-O'^'^-SAABD
^ 1^-
^
5 + V7
TH
2: Neu H nam ngoai
AABD
thi H la tam duong
tron
bang tiep goc BAD
ciia
AABD.
Gpi
Tg
la ban
kinh
duong
tron
bang tiep
AABD

tuong
ling
thi:
p-BD 5_77
Tudo,
V^BCD.A'B'C'D'
=6V^,ABD
-6.iA'H.S,A3i, =?^^
^du
2.4.12.
Cho lang tru ABC.A'B'C c6 the
tich
bang the
tich
khoi
lap
Phuang canh a. Tren cac canh AA',BB' lay M,N sao cho — = ^ =
AA'
BB' 3
Gpi
E,F Ian
lupt
la giao diem cua CM vai C
A'
va CN vai C'B'.
1) Mat phang (CMN) chia
khoi
lang try thanh hai phan.
Tinh
ti so the

tich
hai
phan do.
j)Tmh
the
tich
khoi
chop
C'CEF
.
JCgrigidL
^)
Do \ 1 \ c T7 2,, 2a'
ABC = 3 VABCA'BC - Suy ra VC.ABBA' = 3 VABC.A'B'C =^
M^t khac S
0:;
ABNM
•SA'B'NM
V - ^ V ^
^C'A'B'NM
"2 ^C'ABB'A' =
153
Phumig
pUiipgidi
Toan
With
hoc
titeo
chmien
de-

Nguyen
Phii
Kluinit,
Nguyen
Tti't Thu
Suy ra
VCCABNM
=
2a-
Vay
^C'A'BNM
^C'CABNM
2)
Taco:
1
'\CF.F _ 2
CE.CF.sinECF
1
ACAB
'CA.CB.sinACB
EA MA 2 CE „
BC CC 3 CB 2 S
^ACAB
9 3a
Suy ra CCCEF = 2^CXAB =
Pi
du
2.4./3.
Cho hinh chop deu
S.ABCD

c6
M,N,E
Ian luat la trung diem
cac canh AB, AD, SC. Tinh ty so the tich hai phan cua hinh chop dugc cat
boi
mat phang
(MNE)
.
.Cgi
gidi
Duong
th3ng MN cat BC va
CD tai K va L; EL cat SD tai P; EK
cat SB tai Q. Mat phang (MNE) cat
hinh
chop
theo
mat cat la ngu
giac
NMPEQ.
Dat AB = a,SO = h.
Ta CO KB = DL -
2
Ha EH//SO=>EH
la duang trung binh ^
cua
ASOC
nen EH =
—
.

2
'ACKL
= -CK.CL = 4-
1
3a 3a 9a
2 2 2 2
Ta
CO
Q la trung diem ciia EK nen
VECKL-3EH.SCKL-3-2- 8
1
h 9a^ Sa^h
16
V,
KCEL
KB.KQ.KM
_ 1 1 1 ^J_^v
KC.KE.KL • • "
KCEL
96
160
Cty
TNHH
MTV DWH
Khang
Viet
Tuong tu
VLNDP
=
a^

96
Suy ra = VBCDNMQEP = ^ECKL -fV,
GQ'I
V2 la phan the tich SEQMANP ta c6:
KBMQ
+ ^LDNpl-
Sa^h a^h a^h
16
48
Suy ra V2 = VSABCD - Vj =
a^h a^h a^h
X>i
dijL
2.4.14.
Cho hinh chop
S.ABCD
c6 day la hinh vuong canh a,
ASC = 90°,SA lap voi day goc a (0° < a < 90°) va mat phang (SAC) vuong
^6c voi mat phang (ABC). Tinh khoang
each
tu A den
(SBC).
J[gigidi.
Taco
VA.sBc=^d(A,(SBC)).SBcs
nen
d(A,(SBC))
=
SBCS
Vi

(SAC) 1 (ABC) nengpi H la hinh chieu
ciia S tren canh AC thi SH1 (ABC), hinh
chieu cua SA tren mat phang (ABCD) la AH
nen (SA,(ABCfD)) = SAH = a.
Taco
ASC = 90° nen SA =
AC.
cos a = 72.a. cos a.
Do do SH =
SA.sina
=
\/2.a.cosasina
f
l
-JT.
Nen
Vs^B(;
=-SH.S^B(
=—.a^.cosasina.
C
3 '^"^6
^
GQI
K la trung diem ciia SC thi OK la duong trung binh ciia tam
giac
SAC
nen OK
/
/SA ^ OK 1 SC. Ma BD1 (SAC) ^ BD1 SC nen BK1 SC.
Taco

SC =
AC.
sin
a = 72.a.
sin
a nen BK =
VBC^^^CK^
=;
2 -
sin^
a
1 2
sin^a
.
Vay khoang
each
can tim la:
d(A,(SBC))
=
3. — .a . cos a sm a.
yl2.a.
cos a
-a^
sina.\/2-sin2a
V2-sin^a
^idvL
2.4.15.
Cho hinh hop dung ABCD.A'B'C'D' c6 day la hinh thoi canh
^tam
giac

ABD la tam
giac
deu. Goi M,N Ian lugt la trung diem cua cac
161
Phuang
phdp
gidi
Todn
Hinh
liQC
theo
chuyen
de-
Nguyen
Pku
Khdnh,
Nguyen
Tat Thu
canh
BCC'D'.
Tinh khoang
each
tif D den mat phang (AMN)
biet
rang
MNIB'D.
JUffi
gidi.
Dat AA' = x,AB = y, AD = z.
Ta

CO
tarn
giac
ABD la tarn
giac
deu nen :
D'
N
C
x.y = x.z =
0,y.z=
y
.cos60°=-
A'.
Ta CO DB' = DD' + DC + DA = x + y - z.
Vi
M, N la trung diem ciia
BC,C'D'
nen MN - MC + CC + C'N hay
1U , ^ 1
MN-|AD
+ CC + -C'D' = -z + x y. ^
/ 1 / in
/I
/ ii\
/ 1 /
11
^
fB'
1

/ !'\
1 / 11^
1 / i \
1/
11
/ 1 1
/ ^
/ / \
' ' ^- \
\
\
\
5
Theo
bai ra MN 1 B'D nen MN.DB' = 0
Do do (x + y - z)
,2
1,.
1.
-z
+ X —
y
2 2^
= 0
2
1 a^ 1 2 1 2 1 a n
ox^+
.a^—.a^+ —=
0 <»x = —a.
2 2 2 2 2 2 2

Taco:
d(D,(AMN)) = ^^^^^N .
D~e
thay S^MD =
^SABCD
=
SABD
= ^a'-
-5
AMN
^
•fl
Goi H la trung diem cua DC thi NH 1 (ABCD),NH = —a nen
re
'^D.AMN
-^N.AMD -
^NH.S^MD
" 24
a3.
Ke HK 1 AM ta c6 NK i
AM.
Theo
djnh li ham so
cosin
AM^
= BA^ + BC^ -
2BA.BC.cosl20°
- ^a^ => AM - ^
a.
3 3yf^ 2

Ta CO S=
SABCD
"
(SADH
+ ^CHM + ^ABM ) = g
SABCD
= ^
Nen HK
AM
28 14
AM
1
2
Suy ra 3^^,^ =
^NK.AM
= ^a^, do do d(D,(AMN)) = ^a.
22
Vay khoang
each
tu diem D den mat phang (AMN) la ——a.
162
Cty
TNHH
MTV DWH
Khang
Viet
p
Bai tap
BOi
2.4.1,

Cho hinh
chop
S.ABCD
c6 day ABCD la hinh thoi va AB = BD = a,
SA = a>/3 , SA 1 (ABCD). Ggi M la diem tren
canh
SB sao
choBM
= - SB, gia
3
sCr N la diem di dong tren
canh
AD. Tim vi tri ciia diem N de BN 1 DM va
Ichi
do tinh the tich cua khoi tii dien BDMN.
Jiu&ngMngidi
, p ^ , ,
Tu gia thiet, ta suy ra tarn
giac
ABD deu
canh
a. Dat AN = xAD ^' '
Ve MI//SA=:>MIl(ABCD):^MIlBN.Dod6 BNlDMoBNlDI
Ta c6: DI =-AD + AI =
i
AB - AD, BN =-AB + xAD
3
Suyra
DLBN = AB^ - xAD^ +
3

. 3
AB.AD
1
2 2 1
3 2
2
5 2 1 ;
a
=—a^x
+ -a
'1
Dodo
DIlBN<»DI.BN
= Oc^x =
i:r>AN
= ia
5 5
Taco:
MI = |sA = ^,
S,3ND=|S^3D~
VM.BDN-^MLS^BND
1
2a73
a^V3 _2a^
•3" 3
- _ 5 15
'^<^i
2.4.2.
Cho hai khoi
chop

S.ABCD
va 5'.ABCD c6 chung day ABCD la
f»Ot
hinh vuong
canh
a (S va S' nam ve cimg mpt phia cua
(ABCD)
). Goi H, K
lugt
la trung diem cua AD va BC,
biet
SH = S'K = h va
SH,S'K
ciing
163
Phucmg
phdp
gidi
To&n Hinh hgc
theo
chuySn AJ-
Nguyen
Phu Khdnh,
Nguyen
Tat Thu
vuong
goc vai
(ABCD)
.
Ti'nh

the tich phan chung ciia hai khoi chop
S.ABCD
va S'.ABCD
theo
a va h
.
Jiudrng
ddn
gidi
Tir
gia
thiet
de
bai,
ta suy ra cac
tii
S
giac
SDCSSABS'
la cac hinh binh hanh.
Gpi
E,
F
Ian luc^t la tam ciia cac hinh binh
hanh
SDCS',
SABS'.
Ta c6
phan chung
cua hai khoi chop S.ABCD

va
S'.ABCD
la khoi da dien: ABCDEF.
Ta c6:
Vyj^gj-DEF
-
^S'ABCD
~
^S'BCEF
•
Ma:
S'ABCD
=
is'K.S
ABCD
ah.a^=
a^h
>i B
_S'F
_1 1
VS'BCF
-^r^-Vs'ABC Vs'ABCD
VS'CEF
- g
Vg.ABCD
24
a^h
Suy ra VS.BCEF =
^s.gcF
+

^sxEF
=
•
^ay VABCDEF
=
Sa^h
Bdi
2,4.3,
Cho hinh chop
S.ABC
c6
day la tam
giac
vuong tai A, AB = a, AC = 2a.
Mat
phSng (SBC) vuong goc voi day, hai mat phang (SAB) va (SAC) cimg tao
voi
mat phang day goc 60". Tinh the tich kho'i chop
S.ABC
theo
a.
^
Jiudrng
ddn
gidi
Goi
H
la hinh chieu cua
S len BC; E,F
Ian lugt

la hinh chieu cua
H
len AB, AC suy ra SH1 (ABC)
va HE = HF nen AH la phan
giac
ciia goc BAC
Ta c6:
AB
•HF
=
HF
HC
AB.AC
BC
, BH , AB
•
=
1
+ =
!
+
•
CH
AC
2a
AB+AC
3
Suy ra SH
-
HF.

tan
60° =
,
2a^^/3
W
=
I
AB.AC
= a2. Vay
Vs.A3C
-•
Bdi
2.4.4.
Cho hinh chop tam
giac
deu
S.ABC.
Tinh the tich khoi chop
S.ABC
biet:
1) Canh day bang
a
va mat ben tao voi day mpt goc
60°
2) Canh ben bang
2a va
SA 1BM, voi
M
la trung diem SC
.

164
Cty
TNHH MTV DWH Khang Viet
Jiitong
ddn
gidi
Goi
O
la tam ciia day,
I
la trung diem BC
1) Ta
CO
BC
1 (SIO)
=>
slo =
((SBC),(AB"C))
= 60°,
lO
=
iAI
= SO = IOtan60° =
-,
3
6 2
'AABC
^r"
w
Icr^c

la a^V3 a^Vs
Vay
VsABC
-
3SO.SAABC
= 3
•
2 = '
2) Goi E,F,P Ian lugt la trung diem ciia AB, BS, SM, ta
c6:
(SA/BM)
=
(EF^) =^ EF
1
FP
. Dat
AB
= x
Ta c6: EF
-
a, BM
^J^^2_2(EC^+ES^)-SC^
2
2(BS2
+
BC2)-SC2
x^+2a^
,FP
=
'3X!

+
SA2-AE2^
-SC^
BM
4a2+x2
gp2
^
2(SE^+EM^)-SM^
_9a^ / , , ,
4
~ 16
Tam
giac
EFP vuong tai
F
nen EP^ =
EF^
+
FP^
o -
8a^ x = 2aV2
f'Bdi
2,4.5.
Cho lang tru tam
giac
ABCAjB^Ci
c6
ta't ca cac canh bang a, goc tao
boi
canh ben va mat phang day bang 30° .

Hinh
chieu
H
ciia diem A tren mat
phang (AiBjCi) thugc doan thSng BjCj. Tinh
the
tich khoi lang
tru
ABC.AjBjCj va khoang
each
giiia hai duong thang AAj va
BjCj
theo
a .
Jiuang
ddn
gidi
Ta
CO
AiH
la hinh chieu ciia AAi len mat phang (AiBiCi)
nen suy ra
AAjH
= 30°
Xet tam
giac
vuong
AHAj
ta
c6:

165
Phuontg
phdp
gidi
Todn
Hinh
hoc
theo
chuyen
de-
Nguyen
Phii
Khdnh,
Nguyen
Td't Thu
AH
= AAj.sin30" =|, AjH =
AApCosSO"
=
Ma
tarn
giac
AjBjCj deu
nen
H la
trung
diem ciia B[C|.
The
tich
kho'i

lang tru la:
V = AH.S^ABC
a a^Vs
Ve duong cao HK
ciia
tarn
giac
AHAj.
Ta
CO
BjCj
l(AHAj)
nen BjCj IHK
Suy ra d(AA,, BX,) = HK = ^"'^^^ = ^.
^ ^ ^ AAj 4
Bdi
2.4,6.
Cho
tii
dien gan deu ABCD (c6 cac cap canh doi bang nhau) va mat
phang
(a) luon
song song
voi AB va CD . Tim vi
tri
cua (a) de (a) chia
tii
dien
thanh
hai phan c6 the

tich
bang nhau .
Jiu&ng
dan gidi A
Vi
cac mat cua tu di?n c6 dien
tich
bang
nhau
nen cac duong cao bang nhau.
Mat
phang (a)
song song
voi AB va CD
nen
FG//HE//DCva EG//FH//AB.
Ta
can tim vj
tri
cua mp(a) de
^F.HDCE
+
^F.GCE
^F.AEGB
+
^F.AHE
F.ADC
~
2-Vp,AHE
-

^F.ABC
~
^-Vp^^EC
OV
A.FDC
1-2.
V,
F.AHE
V,
F.AHE
J
=
v,
A.FBC
1-2.
*'F.ABC
j
1-2.
^FDC
^FBC
1-2.
fGEC
BF
1-2.(^^^)2
^EA^
^AHE
'AHE
1-2
AH
AD

AH
AD2-2.AH2
o Mat phang a di qua
trung
diem cua canh AD.
166
CtyTNHH
MTV DWH
Khang
Viet
gdi
2.4.7.
Cho
hinh
chop
S.ABCD
c6 day ABCD la
hinh
chii
nhat AB = a, AD = 2a,
c^nh SA vuong goc voi day, canh SB tao voi mat phang day mgt goc 60". Tren
c^nh SA lay diem M sao cho AM = ^ . Mat phSng (BCM) cat canh SD tai N.
Tinh
the
tich
khoi
chop
S.BCMN. ,^
JIuang
dan gidi

Ta
c6: MN / /AD;BC 1 SA va BCl AB => BC 1 (SAB) ' ' ' '
BC 1 BM => BCMN la
hinh
thang vuong tai B va M . ,
Ta
c6: SA =
AB tan
60*^ = a Vs,
MN
SM 2 4a
=
= - => MN = —
AD
SA
II
BM
=
VAB^TAM^
= ~
Dien
tich
hinh
thang
BCMN:
2 3V3
Ha
SH 1 BM =^ SH 1 (BCMN) :^ SH
la
duong cao cua

khoi
chop
S.BCMN .
Do AMHS ~ AMAB nen suy ra:
MS.MA
aS B
MH.MB
= MS.MA MH = -
C
,i
MB
BH
= BM + MH = aV3=>SH =
VsB2-BH2
=\lAa^-3a^ =;
Vay
Vg.BCMN
=-S.SH
= ^.a =
loVSa^
3"" 3"3V3 27
Bai
2,4.8.
Cho
tii
dien ABCD c6 AC = AD = aV2 , BC = BD - a, khoang
each
tu
B den mat phSng (ACD) bang .
Tinh

goc
giiia
hai mat phang (ACD) va
V3
(BCD).
Biet the
tich
cua
khoi
tu dien ABCD bang
a^Vl5
27
Jiu&ng
dan gidi
Goi E la
trung
diem cua CD, ke BH1 AE .
Ta
CO
AACD can tai A nen CD 1 AE. Tuong tu CD 1 BE
Suy ra CD 1 (ABE) CD 1 BH
Ma
BH 1 AE => BH 1 (ACD) BH - . Goi a = ((ACD),(BCD).
167
Phuongphiipgiai Todn Hinh hoc theo chuyen de- Nguyen Phu Khdnit, Nguyen Tat Thu
1
-3
The
tich
cua

khoi
tii
dien
ABCD la V = -BH.S^ACD =
3 27
„ a^Vs .„^„ a'
'
^ASCD
= —— ^
AE.DE
=
3 9
Mat
khac:
AE^
+
DE^
= 2a^ =^
AE^,
DE^
la
hai nghiem cua
phuong
trinh
2
„ 2 5a^ ^ 5a^
X
-2a x + =
0<=>x
=

—,x=-
.AE2.DE2
= ^^
Vi
DE<ar^DE = —,AE =
3 ' 3
avTs
Xet
ABED
vuong
tai E nen BE =
N/BD^-DE^
= —
Xet
ABHE
vuong tai H nen sin a =
BH
1
•
a = 45'
0
BE ^
Vay
goc giua hai
mp(ACD)
va (BCD) la a = 45°.
Bdi
2.4.9.
Cho
hinh

chop S.ABCD c6 day ABCD la
hinh
thoi
canh a, BD = a.
Tren
canh AB lay M sao cho BM =
2AM
. Goi I la giao
diem
ctia AC va DM SI
vuong
goc voi mat phang day va mat ben (SAB) tao voi day mot goc 60".
Tinh
the
tich
cua
khol
chop S.IMBC.
Jluongddngidi
Goi
H la
hinh
chieu cua I len AB, suy ra AB 1 (SIH)
=>
SHI la goc giCra mat
ben
(SAB) va mat day. Do do SHI = 60°.
s
M
11

2a
Do
tam giac ABD deu nen suy ra ABD - 60 va BD — nen ap
dung
djnh
3
li
CO
sin cho tam giac BDM ta c6:
168
Cty
TNHH
MTV DWH Khang Viet
Is^D^=BM^
+ BD^-2BM.BD.cos60''=
—=>MD
= ^^ ; i*
9 3
MD
BM . ^-r- BM.sin60'' (s 2^;^^ 4
Mat
khac: = r=- =>
sm
BDM
= — = -
=>
cos"^ BDM = -
sin60'^
sin
BDM

MD V7 7
->tanBDM
=
(jiem
AO
-^:^-l=^=>OI
=
OD.tanBDM
=
^^I
la
trung
cos^BDM
2 4 "
-
AATTT
AAr^D
AI
AI.OB
aS
Ta
CO
AAHI
-
AAOB
=
=>
IH
=
•

Suy ra SI = IHtan60° =
OB AB AB
3a
8
SAMI
_ AM AI _ 1 11 _1_ _a\/|
S^~ AB'AO"3-2"6
^^^^'"6
'^^^O"
48 0 ; v
^'
c _c c
a2V3_lla2V3
UO
ao
S^iICB
-
^AABC
~
'^AAMI
- " I p
w
lore
1 3a "a^Vs lla^Vs ^,
Bai
2.4.10.
Cho
hinh
chop S.ABCD c6 day ABCD la
hinh

vuong
canh a .
Hinh
chieu
cua S
trung
voi
trpng
tam tam giac
ABD.
Mat ben (SAB) tao voi day mot
goc 60° .
Tinh
theo a the
tich
cua
khoi
chop S.ABCD.
Jiu&ng ddn
gidi
Gpi
H la
trong
tam cua tam giac
ABD
=>
SH 1 (ABCD)
s
^
^S.ABCD

= 3 SH.SABCD
"J"-^^
Ke HK1AB=> AB1(SHK)=>SK1AB.
^
Suy ra ((SAB),
(ABCD))
=
(SKJIK)
=
SKH
= 60°
Gpi
O la tam ciia day, ta
CO
AAKH
-
AAOB
,^ . „ ^
/'^
\
^
AHKH
AH.OB
_ 2 OB^ _ a iV~^^c-"\'/
I'^AB'OB"'
AB 3AB
3^/^ """"
^
^^'-V^
i'

J3
f Trong tam giac
vuong
SHK ta c6: SH =
HK.
tan
60° = ^,
,3
Vs.ABCD
a"^/3
169
Phuang
phdp gidi
Toiin
Hinh
hoc
theo chuyen
de-
Nguyen Phu
Khdnh,
Nguyen
Tat
Thu
Bai
2.4.11.
Cho
hinh
chop
S.ABCD
c6 day

ABCD
la
hinh
vuong
tarn
,)
AB
= a. Goi
M,
N
Ian
lugt
la
trung
diem
cua cac
canh OB,
SD.
Mat phark>
(NAC)
tao
v6i mat phang
day
mot
goc
60°; hai mat phang (SAM)
va
(SCl^^j
ciing
vuong

goc
vol mat phang day.
Tinh
the
tich
cua
kho'i
chop
S.ACN.
Jiixang
ddn gidi
^
Vi
hai mat
phang (SAM)
va
(SCM) cung
vuong
goc voi
mat phang
day nen
giao tuyen
SM
ciia hai mat phang
do
vuong
goc
voi day.
Goi
H

la
trung
diem cua MD
suy ra NH//SM=> NH
1
(ABCD)
0O
DO
1
AC ACl(NHO) => NOH
=
60°
Ta c6:
HD
= -DM = -OD:
2
4
H0
= i0D = ."^
4
Trong
tam
giac
vuong NHO, ta c6: NH
=
OH tan 60°
=
.SH
=
•^NACD

*'SACD
^SACN
^SACD
-SH.S
AACD
1
aV6
a^
6
4
•
2
N6_
48
Bai
2.4.12.
Cho
hinh
chop
S.ABC
c6
mat phang (SAC) vuong
goc voi
mat
phang (ABC), SA = AB = a, AC = 2a va ASC = ABC = 90°.
Tinh
the
tich
khoi
chop

S.ABC
va cosin ciia
goc
giua hai mat phang (SAB), (SBC).
Jiu&ng
ddn gidi
Ke SH vuong
goc
AC (H
e
AC) => SH
1
(ABC)
SC = BC = aV3, SH =
^
,S
AABC
V<
1
S.ABC
-
gS^ABC-SH
- —
Goi
M la
trung
diem
SB v^
(pla
goc

giua
hai mat phang (SAB) va (SBC).
^
Ta c6: SA = AB = a, SC = BC =
aS
cos AMC
AM
1
SB va CM
1
SB, suy ra
coscp
=
ax/3
ASAC
=
ABAC => SH
-
BH
=
•
SB
=
Jx/6
AM
la
trung
tuyen ASAB nen:
170
Cty

T'iVHH
MTV DWH
Khang
Viet
7
2AS^+2AB^-SB^
lOa^
aVlO
AM
=— =
=>AM
=
4
16 4
Tuongtu:
CM
=
=>cosAMC
=
-
i>
2.AM.CM
VTos
35
N/IOS
V^y:
cos(p
= ^^
gjjj
2.4.13.

Cho
lang
try
ABC.A'B'C
c6 day ABC la tam
giac
can
^B
=
AC
=
a,BAC
=
120°va
AB'
vuong
goc
voi
day
(A'B'C).
Goi
M,
N
Ian
\^xQt
la
trung
diem
cac
canh

CC va
A'B', mat phang (AA'C)
tao
vai mat
phang (ABC) mot
goc
30°
.
Tinh
the
tich
khoi
lang tru ABC.A'B'C
va c6
sin
cua
goc
giira
hai duong thang AM
va
C'N
.
Jiu&ng
ddn gidi
Ta c6: BC^
=
AB^
+
AC^
-

2AB.ACcos
A =
3a^ => BC
=
aVs
Goi
K
la
hinh
chieu cua B' len A'C, suy ra A'C
1
(AB'K)
Do
do
AKB'
=
((A'B'C'),(AA'C'))
=
30°
Trong
tam
giac
A'KB'
CO
KA^'
=
60°,A'B'
= a,
nen B'K
=

A'B'sin60°
=
Suy ra AB'=
B'K.tan30°
- ^
The
tich
kho'i
lang
tru:
V
= AB'.S
AABC
- g •
Goi E la
trung
diem cua AB',
suy ra ME//C'N, nen (C'N,AM) = (EM,AM)
Vi
AB'
1
C'N => AE
1
EM =^
(C'N,
AM)
= AME
|Ta c6:
AE
a

AB'
=
i;EM^
=
C'N^
=
2(CB''.C'A'^)-A'B-
^
2
4 4 4
'2
7a2
EM
=
l AM^ = AE^ +
EM^
=
293^ aV29
r-r^ ME - /7
__=,AM
=
—.Vay cos AME
= — =
2^ ^^^
171
Phuong
phap
gidi
Todn Hinh hgc
theo

chuyen
de -
Nguyen
Phu Khdnh,
Nguyen
Tat Thu
Bai
2.4.14.
Cho
hinh
lang tru deu ABC.A'B'C, M la
trung
diem cua canh CC
Mat
phSng (A'B'M) tao voi mat phang (ABC) mpt goc 60" va tam
giac
A'MB'
di^n
tich bang
13
. Tinh the tich khoi chop
AMA'B'.
Jivcang
dan
gidi
Gpi
N la
trung
diem cua A'B', ta c6
CNIA'B'

Matkhac: A'M = B'N ^ MN 1 A'B',
suy ra A'B'l(MNC')
Do do MNC la goc giiia mat phSng
(A'MN)
voi (A'B'C).
Nen
MNC' = 60°. Dat AB = x
Suyra: CN = -
^ MN = xV3
•MC'
= C'Ntan60" =
.0 3x
5AMA'B'
= 2^N-A'B' = —
" =>x = a,—
13 V13
1
3x^
Taco:
CC - 2C'M = 3x, suy ra S^^B' =
2^^'-'^'^'"~2~
d(M,(AB'A'))
= d(C',(AA'B')) = C'N =
Vay V
1
xVs 3x2
AMA'B'
3 2 2 4 338
Bai
2.4.15.

Cho
hinh
chop
S.ABCD
c6 day ABCD la
hinh
vuong tam O.
Hinh
chieu cua S len mat day
trimg
voi diem H la
trung
diem ciia AO. Mat phang
(SAD) tao voi day mpt goc 60° va SC = a . Tinh
Vg^Bj^D
va d(AB,SC).
J^it&ng ddn
gidi
DatAB =
x,x>0
^
Ve
HKIAD,
suy ra AD1(SHK)^SKH la goc giua mat ben (SAD) va
mat day nen SKH = 60°
Taco:
HC = -AC = ^^; HK = -DC = —.
4 4 4 4
Trong
tam

giac
vuong SHC, ta c6: SH^ = SC^ -
HC^
=a^-^^
8
172
c
B
C
Trong
tam
giac
vuong SHK, ta c6: SH = HK tan 60° =
3XN/3
Dan toi
27x^
16
= a2-
9x2 ^^^2
8
45
X =
V
-^QHc
_13XV3
2_>/3x'
Vs.ABCD
- -^n.bABCD - 3 ~
4a
3>/5 '

16Vl5a
675
Bdi
2.4.16.
Cho
hinh
chop
S.ABC
c6 day ABC la tam
giac
vuong can tai B,
AB = BC = 2a; hai mat phang (SAB) va (SAC) ciing vuong goc voi mat phang
(ABC). Gpi M la
trung
diem cua AB; mat phang SM va song song voi BC, cat
AC
tai N. Biet goc giiia hai mat phSng (SBC) va (ABC)
bSng
60°. Tinh the tich
khoi
chop S.BCNM va khoang
each
giiia hai duang thang AB va
SN
theo a.
Jiicang
dan
gidi
Do hai mat phang
(SAB)

va (SAC) cat nhau
theo giao tuye'n SA va cung vuong goc voi
(ABC)
nen SA 1 (ABC), hay SA la duong cao
ciia khoi chop S.BCNM.
Ta CO : SBCNM = S^BC " S^MN
i
=2a2-lMA.MN-2a2-ia2=.^
i
2 2 2
Do
BCIAB
.(SAB)lBC.
BCISA
'Nen
SBA chinh la goc giua hai mat phang (SBC) va (ABC), the thi theo
thiettaco
S"BA
= 60°.
173
Pltumig
phlif}
gidi
Toiin
Hiiih
hoc theo chuycn
de-
Nguyen Phu
KItdnh,
Nguyen Tat Thu

'
Trong
tam
giac
vuong
SAB ta c6 SA
= ABtaneo" =
2a73
.
Vay Vs
3CNM
=
^SA.SBCNM
=\-2^S.~-
=
Sa^
(dvtt)
Goi
P la
trung
diem ciia
BC thi
AB//NP,ABcr
(SPN)
nen
AB//(SPN)
do
do
d(AB,SN)
=

d(AB;(SPN))
=
d(A;(SPN))
PN
1 AE
Tu
A ha
AE1NP,E6PN
thi ^ , „^
PN
1
(SAE);ha
AHXSE
thi
AH
1
(SPN)
d(A;(SPN))
=
AH.
•
Taco
AE
=
NP
=
a;SA
= 2aV3^^—= -—+ =
13
AH^

AS^ AE^ 12a2
•AH
= ajH
Vl3
Vayd(A;(SPN))
=
aj|.
Bdi
2.4.17,
Cho
lang
try
ABCD-AjBiCiDj
c6 day
ABCD
la
hinh
chu
nhat
AB
= a,
AD
=
asl3
.
Hinh
chieu vuong
goc cua
diem Ai tren mat phang (ABCD)
trung

voi giao diem AC
va
BD.
Goc
giua hai mat phang
(ADDJAJ)
va
(ABCD)
bang
60'^.
Tinh
the
tich
khoi
lang
try da cho va
khoang
each
tu
diem
Bi
den
matphSng
(A[BD)
theo
a .
Jiuong
ddn
gidi
Goi

O =
ACn BD,I
la
trung
diem canh AD.
Ta CO AD
1
(AOI)
=>
Alio
=
((ADDi
AI
),(ABCD))
- 60'
Vi
01
=
I, non
ta^
suy
ra
A^I
= 20I
- a
=>AiO
=
OI.tan60"-'^.
2
Dodo

V^BCD.A,HAD1
=AIO.SABCD
=a.aS.^
=
^
Goi
B2 la
hinh
chieu ciia
B^
xuong mat phang
(ABCD)
Do BiC//A,D:r> BjC/
/f
A,BD)d(Bi,(AiBD))
=
d(C,(AiBD))
=
CH
Trong
do CH la
duang
cao cua
tam
giac
vuong
BCD
174
Cty
TNHH

MTV
DWH
Khang Viet
•pa
c6:
CH
=
CD.CB
CD^
+ CB'
:^.Vayd(Bi,(A,BD))
= ^
gCii
2A.
18. Cho
hinh
chop
S.ABC
c6 day
ABC
la
tam
giac
vuong tai
B, BA =
3a,
pC
=
4a;
mat

phang
(SBC)
vuong
goc
voi
mat
phMng (ABC). Biet
SB = 2aV3
5BC
=
30'^.
Tinh
the
tich
khoi
chop
S.ABC
va
khoang
each
tu
diem
B den
in?t phang
(SAC)
theo
a.
J^udrng
ddn
gidi

^
GQI
H la
hinh
chieu ciia
S
xuong
BC .
Vi
(SBC) ±
(ABC)
nen
SH1 (ABC).
Ta CO
SH
=
aS.
Do
do
Vs.ABCD
=|SH.S^ABC
=2a^^/3
Ta
CO
tam
giac
SAC
vuong tai
S.
Vi

SA =
aV2T,
SC =
2a, AC
=
5a
va
S^^Q = a V21 nen ta c6
dug-c:
d(B,(SAC))
= ^^ = ;|.
'AS
AC
Bdi
2.4.19.
Cho
hinh
chop
S.ABCD
c6
day
ABCD
la
hinh
vuong canh
a.
Gpi
M
va N
Ian lu(?t

la
trung
diem
cua cac
c^nh
AB va
AD;
H la
giao diem ciia
CN
va
DM. Biet
SH
vuong
goc
voi
mat
phang
(ABCD)
va
SH
= aVs .
Tinh
the tich
khoi
chop S.CDNM
va
khoang
each
giiia hai duang thang

DM va SC
theo
a. 5 ;
Jiuamg
ddn
gidi
Ta
c6: Vc
=
-SH.S
3
^3
SMNDC
=SABCD
"S^AMN
"S^MBC
_
2 _5a^
8
4 " 8
Nen
I
V
S.CDNM
5a^
sVsa
Lai
thay:
DM.CN
=

i(2DA
-
DC).i(2DC
-
DA)
=
DA^
-
DC' =
0.
A.
175
Phuong
phdpgidi Todit Hinh hoc theo chut/en de- Nguyen Phu Khdnh, Nguyen Tai Thu
V^y CN
J_
DM
tu do
SC
±
DM
boi vay:
d(SC;DM)
=
d(H;SC)
= = = , .
SC
Lai
CO-
CH =

^^"^"^M"
=
^(^ABCD
^^AMD
SAQMB
j ^ ^ /4
VSH^+CH^
0
DM DM
Thay
len tren ta c6 khoang each can tinh la:
2a J—
.
Bdi
2.4.20.
Cho
hinh
lang tru tarn giac deu
ABC.A'B'C
c6 AB = a, goc
gic,g
hai
mat phang
(A'BC)
va
(ABC)
bang 60". Goi G la trong tam tarn giac A'Bc
Tinh
the tich khoi lang try da cho va tinh ban
kinh

mat cau ngoai tiep tu dien
GABC
theo
a.
Goi
H la trung diem ctia
BC,
theo
gia
thuyet ta c6 :
.VHA
= 60*^
Taco: AH -
—,A'H
= 2AH = aV3
va AA'
=
3a
/
/
G
/
,-'-'71
.
M
Vay
the tich khoi lang try: c
a^Vi
3a Sa^Vs ,.
=

GQI
I la tam cua tam giac
ABC,
suy ra
GI
/
/ AA' GI
±
(ABC)
Gpi
J la tam mat cau ngoai tiep hr dien
GABC
suy ra J la
giao
diem ciia
GI
voi
duong trung
true
doan GA; M la trung diem
GA,
ta c6:
r^-KKi^h.
/^T/^T
n
GM.GA
GA^ 7a
GM.GA
=
GJ.GI

x> R =
GI
= = = —.
GI
2GI 12
Bai
2.4.21.
Cho
hinh
chop
S.ABCD
c6 day
ABCD
la
hinh
vuong canh a, canh
ben
SA = a;
hinh
chieu vuong goc cua dinh S tren mSt phang
(ABCD)
la diem
AC
H
thuoc
doan
AC, AH
- — .
Gpi
CM la du-ong cao cua tam giac

SAC.
Chung
minh
M la trung diem ciia SA va tinh the tich khoi tu dien
SMBC
theo
a.
Jiu6ng
dan
gidi
•
Chung minh M la trung diem SA
Ta
c6:
176
Cty
TNHH
MTV DWH Khang Viet
,
av2 I
AH=
=^SH
HC
=
4
3aV2
=VSA2-AH2
=
•
SC

=
VsH^+CH^
= aV2 .
Suy ra SC = AC =>
AACS
can tai C
rven
M la trung diem SA.
.
Tinh
y^MliC
•
Vi
M la trung diem
SC
nen
S^CM
=
2^SAC
suy
ra
V^uec
=
\^SABC
=
-^''H-SAABC
=
a^Vli
48
Bai

2.4.22.
Cho
hinh
chop
S.ABCD
c6 day
ABCD
la
hinh
thang vuong tai A
va
D; AB =
AD
= 2a,CD = a; goc giiia hai mat phang (SBC) va
(ABCD)
bang
60". Goi I la trung diem cua canh
AD.
Biet hai mat phang (SBl) va (SCl) cung
vuong goc voi mat phSng
(ABCD),
tinh the tich khoi chop
S.ABCD
theo
a.
,/firdng
ddn gidi
Vi
hai mat phang (SBl) va (SCl) cimg vuong goc voi mat phang
(ABCD)

nen
giao
tuyen cua hai mat phSng do SI vuong goc voi mat phang
(ABCD)
hay la SI 1 (ABCD) nen
Vg
^^^0 =
^SLS^BCD
^
, _
AD(AB
+
CD)
- 2
Taco:SABCD=
- =
^^
•
G(?i
H la
hinh
chieu cua I xuo'ng BC
BC
1
(SIH)
SHI
=
((SBC),(ABCDJ)
= 60°.
Taco:

BC
=
JAD^ +
(AB-DC)^
=
aS)
=
a
'
^ABCI
~
^ABCD
(^AIDC
,=>IH =
2S
AIBC
BC
^=^SI
= IH.tan600=^
177
Phucrtig
phdp
gidi
Todn
Hiuh
hoc
theo
chm/en
de-
Nguyen

Phu
Khduh,
Nguyen
Td't Thu
Bai
2.4.23.
Cho hinh lang tru tam
giac
ABC.A'B'C c6 BB' = a, goc giO-
duong
thSng
BB' va mat pli5ng (ABC)
bang
600; tam
giac
ABC vuong tai C v;,
BAC = 60*'.
Hinti
chieu vuong goc ciia diemB' len mat phang (ABC) tmrig
voi
trgng tam cua tam
giac
ABC. Tinh the tich khoi tu dien A'.ABC
theo
a.
Jiuang
dan
gidi
Goi D la trung diem AC, G la trong tam AABC
=>B'G±(ABC)=>B'BG

= 60^'
BG = -^BD = —.
2 4
B'G =
BB'.sinB'BG
=
Trong AABC, ta c6:
BC = ^,AC = ^
CD-
AB
BC^ + CD^ = BD^
3AB^ AB'
13
4 16
3aVl3
26
-;S
AABC
1
9a
The tich khoi
tudi^n
A'.ABC: V^.^BC = VB.ABC =
g^'CS^ABC
=^
•
Bai
2.4.24.
Cho hinh lang tru dung ABC.A'B'C c6 day ABC la tam
giac

vuong tai B, AB = a,
AA'
2a, A'C = 3a . Gpi M la trung diem cua doan thang
A'C,
Ila
giao
diem cua AM va A'C . Tinh
theo
a the tich khoi tu dien lABC
va khoang
each
tu diem A den mat phang (IBC).
JIu&ng
dan
gidi
A'
Ha IH1 AC(H e AC) IH 1 (ABC);
IH
la duong cao cua tu dien lABC
IH//AA':
=>IH
=
-AA'
3
IH
CI
AA'
CA'
~ 3
2

3
AC
=
VA'C^-A'A^
= aVs,
BC = VAC^-AB^ = 2a
178
Cty TNllU MTV DWH
Khang
Vij-t
pi^n
tich tam
giac
ABC : S^^^BC = -
AB.BC
= a^.
The tich khoi tu dien lABC : V = -IH.S^ABC = —
•
'• "'
AKlA'B(KeA'B). Vi
BCI(ABB'A')
nen AK 1 BC =^ AK 1 (IBC).
IChoang
each
tu A den mat phang (IBC) la AK
AK
=
2S
AAA'B
AA'.AB

2aV5
^'^
VA'A^+AB^
5
Bai
2.4,25.
Cho lang tru ABC.A'B'C c6 do dai
canh
ben
bang
2a, day ABC la
tam
giac
vuong tai A, AB = a, AC = aVs va hinh chieu vuong goc cua dinh A'
tren mat phang (ABC) la trung diem ciia
canh
BC. Tinh
theo
a the tich khoi
chop
A'.ABC va
tinh
cosin
cua goc giua hai duong thang AA', B'C. .
Jiuang
ddn
gidi
Gpi
H la trung diem BC A'H 1 (ABC)
vaAH

= iBC = -7a2 + 3a' =a
2 2
Do do A'H^
-A'A^-AH^
=3a
•A'H
=
aN/3
Vay
1
^A'.ABC
=-A'H.S^ABC
•(dvtt).
Trong tam
giac
vuong A'B'H c6 HB' = VA'B'^ +
A'H^
= 2a
|en tam
giac
B'BH can tai B'. Dat (p la goc gii>a hai duong thMng
AA'
va
2.2a ~ 4
•
•
|thi: (p
=
B'BH.
Vay

cos(p
=
-
1.4.26.
Cho hinh
chop
S.ABCD
c6 day ABCD la hinh vuong
canh
2a,
fa, SB = ary3 va mat phang (SAB) vuong goc voi mat ph^ng day. Goi
f Ian lugt la trung diem cua cac
canh
AB, BC. Tinh
theo
a the tich cua
i6p S.BMDN va
tinh
cosin
cua goc gii>a hai duong thJing SM, DN.
Jiic&ng
ddn
gidi
"^oi
H la hinh chieu ciia S tren AB, suy ra SH1 (ABCD).
179
Phucntg phdpgiai Toan Hinh hoc
theo
chuyen dc-
Nguyen

IVtii
Kluiiih,
Nguyen
Tii't
Tim
Do
do SH la duong cao cua
hinh
chop
S.BMDN
.
Ta
c6:
SA^ + SB^
=
a^ +
3a^ =
AB^
•
ASAB
vuong tai S
SM
= — = a
Do
do tarn giac deu, suy ra SH =
Di^n
tich lu giac
BMDN
la:
1

'BMDN
'A BCD
The
tich khoi chop
S.BMDN
:
V
=-SH.Sij^,D^j (dvtt).
Ke ME / /DN (E
€
AD)
AE =
Dat q)
la goc
giCra
hai duong thang
SM
va
DN.
Ta c6:
(SM,ME)
= 9.
Theo djnh ly ba duong vuong goc ta c6:
SA
1 AE
;WSA2
+ AE2=^,
ME=.7AM2+AE^ ^''^
•SE^
Do

ASME
can tai E nen SME = 9 va coscp =
a
2
Bdi
2,4.27.
Cho lang try dung
ABC.A'B'C
c6 day ABC la tarn giac vu6n&
AB
=
BC =
a, canh ben
AA'
-
aVz
. Goi M la trung diem cua canh BC.
Tinh
theo
a the tich cua khoi lang try
ABCA'B'C
va khoang each giua hai duong
thang AM, B'C. '
Jiitang dan gidi
Tu
gia thiet suy ra tarn giac
ABC
vuong can tai B.
The
tich khoi lang try la:

E
^ABC.A'B'C
-^^'-^ABC '
^a3(dvtt).
GQI
E la trung diem ciia BB'.
Khi
do mat phang
(AME)
/
/B'C
^
nen d(AM,B'C) =
d(B'C,(AME))
=
d(C,(AME)).
Nhan
thay
d(C,(AME))
=
d(B,(AME))
= h
. .i
Cty
TNHH MTV DWH Khang Vift
potudien
BAME
c6
BA,BM,BE
doi mot

vuong
goc nen:
1 ] 1
-
+
^
+
-
BA^
BM^ BE^
h
=
>V7
\jay
khoang each giua hai duong thang
AM
va B'C la —y
0
2,4.28.
Cho
hinh
chop
S.ABCD
co day la
hinh
vuong canh a, mat ben
I'T
t^"^ S'^c deu va nam trong mat phang vuong goc voi day. Gpi
M,N,P
Ian li"?'

''^ trung diem cua cac canh
SB,BC,CD.
Chung minh AM vuong goc
voi
BP va tinh the tich khoi tu dien
CMNP.
Jiuang
dan gidi
Gpi
H la trung diem cua
AD.
Ta
CO
tam giac SAD deu nen SH 1
AD
.
Do (SAD)
1
(ABCD)
=>
SH
1
(ABCD)
=>SH1BP ' (1)
Ta
CO
ABCD
la
hinh
vuong nen:

ACDH
=
ABCP
=> BP 1CH (2).
Tif
(1) va (2) suy ra: BP 1
(SHC)
Mat khac: /;'
MN
/
/SC;
AN /
/HC
=>
(AMN)
/ /
(SHC) =>
BP 1 AM
Goi
K =
BH
n
AN
. Ta co
MK
la duong trung binh ciia tam giac SBH
Suy ra MK / /SH ==> MK 1
(CMN);MK
= -SH ^ VSa
2 4

Dien
tich tam giac
CMN:
S^MN
=
-CM.CN
- —
2 8
'The
tich
khoi
tu
di?n
CMNP: V^MNP
=
-MK.SCMN
3
Si
96
(dvtt).
^^i
2.4,29.
Cho
hinh
chop tu giac deu
S.ABCD
co day la
hinh
vuong c^inh a.
E

la diem do'i xiing cua D qua trung diem ciia SA, M la trung diem ciia
^E/
N la trung diem ciia
BC.
Chiing
minh MN vuong gck voi
BD
va tinh
(theo
khoang
each giua hai duong thang MN va
AC.
Jk Jlu&ng dan gidi
•^pi
P la trung diem ciia
SA.
Ta co MP la duong trung binh ciia tam giac
"EAD
=> MP / /AD
=>
MP /
/NC
va
MN
= ^ AD = NC .
Suy
ra
MNCP
la
hinh

binh hanh
MN
/ /CP
=> MN
/
/(SAC).
Phumtg
phapgiai
Toan
HhtU hqc
theo
chiiycn
ilc -
Nguyen
Phii Kltiiiilt,
Nginfen
Tat Tin,
Ta de chiing minh
duoc
BD 1 (SAC) => BD 1 MN
Vi
MN//(SAC) nen:
d(MN,AC) =
d(N,(SAC))
=
ld(B,(SAC))
= lBD = 4^
2 4 4
Vay d(MN,AC) =
N

C
Bdi
2.4.30.
Cho hinh
chop
S.ABCD
day la hinh thang, ABC = BAD =
90°
BA = BC = a, AD 2a .
Canh
ben SA vuong goc voi day va SA = a\/2 . Goi
la hinh
chieu
ciia A len SB. Chung minh tarn
giac
SCD vuong va
tinh
(theo
a) khoang
each
tu H den
mp(SCD).
Jiuangdangiai
^
Goi I la trung diem ciia AD.
AD
Ta CO CI - lA = ID = —
2
suy ra AACD vuong tai C => CD 1 AC.
Ma SA 1

(ABCD)
=^ SA 1 CD //
nen ta c6 CD 1SD hay
ASCD
vuong.
Goi di, d2 Ian lugt la khoang
each
tu
B,H den
mp(SCD)
Ta c6:
ASAB
~ ASHA:
SA^SB
SH^SA^^2
, SH^d
SH SA SB ~ SB^ ~ 3 "^^ SB ~ d
— = —- = - ma — = ^ =
-3>do
= -di
3 2.1
The tich khoi tu dien
S.BCD:
V.
Ta c6:
1
V2a3
SBCD
=
3SA AB.BC

= —
SC = VSA^ + AC^ =
2a,CD
- Vci^ + ID^ = 72a
S^^
=
^SCCD
= V2a^
Ta c6: W^BCD =
^^I-SSCD
^ di = = ^
3.
>/2a3
Vay khoang
each
tu H den
mp(SCD)
la d2 =
3
182
Cty TNHH MTV DWH
Khang
Viet
§ 5. MAT CAU - MAT TRU
TRON
XOAY
I
ivlcit
cau
j_

Mdl cau
ngoai
tiep
hinh
chop:
Hinh
chop
S.A,A2 A„
noi
tiep cau khi va chi khi day la
(Ja
giac
noi tiep. Khi do de xac
(Jinh
tarn mat cau ngoai tiep, ta
lam nhu sau:
• Dung duong thang A vuong
goc voi day tai tam cua day ^'
• Dung mat phang trung true
(p) ciia mot
canh
ben
•
Giao
diem I =
A
n (a) la tam
mat cau ngoai tiep hinh
chop.
2. Mat cau

ngoai
tiep
hinh
Idng
try.
Hinh
lang try c6 mat cau
ngoai tiep khi va chi khi do la
ISng
tru dung va day la da
giac
noi
tiep. Khi do tam mat cau
ngoai tiep la trung diem doan noi
tam ciia hai day.
3. Vi tri
tuang
doi cua mot
hinh
phdng
mi mat cau.
Cho mat cau
S(0,R)
va mot
mat
phSng
(P) bat ki trong
khong gian. Goi H la hinh
chieu
ciia O len (P).

• Neu OH > R thi (P) khong cat mat cau
• Neu OH = R thi (P) va (S) eo mpt diem chiing duy
nhat
la H.
Khi
do ta noi: (P) tiep xiic voi mat cau va(P) goi la mat phang tiep dien, H
goi la tiep diem.
• Neu OH < R thi (P) cat mat cau
theo
mpt duong tron (C) c6 tam H ban
kinh
r = 7R^
-OH^
. ' ' '"''^ .
Neu O n3m tren (P) thi (C) goi la duong tron Ion va c6 ban
kinh
R. ^ ^
183
Phuintg gh'ii Toiiti llhili hoc
theo
chiiyeu de -
Nguyen
Phii Khanh,
'Nguyen
Tai Thu
4. Vi
tri
tuang doi cua mot
duang
thAng

v&i
mat cdu
Cho
mat cau S(0,R) va mot duong d ba't ki trong khong gian. Goi H 1^
hinh
chieu cua O len d.
• Neu OH > R thi d va mat cau khong c6 diem chung.
• Neu OH = R thi d va mat cau (S) c6 mot diem chung duy nhat la H.
Khi
(JQ
ta
noi d tie'p xuc voi mat cau va d goi la tie'p tuye'n ciia mat cau, H goi la tie'p
diem.
• Neu HO < R thi d va mat cau c6 dung hai diem chung. Khi do ta noi d cat
mat cau tai hai
dieiVi
phan biet.
5. i)ien
tich
mat cdu vd the
tich
khdi cdu.
Dien
tich
hinh
cau ban
kfnh
R : S = 47rR^ .
The
tich khoi cau ban

kinh
R: V = -
TIR^
.
3
II.
Mat tru, mat
non
tron
xoay
Cong
thuc tinh dien tich va the tich.
•
Dien
tich xung quanh
hinh
non S^^ =
:tRl
•
Di^n
tich toan phan cua
hinh
non =
S^^
+ =
7iR(l
+ R).
• The tich khoi non V =
-TiR^h.
3

•
Dien
tich mat xung quanh cua
hinh
tru
S^^
= 27tRh.
•
Di^n
tich toan phan ciia
hinh
try: Sjp =
Sxq
+ 2S^ = 27:R(R + h).
• The tich khoi tru : V ^
TtR^h
.
Vi
dxjL
2.5.1.
Cho mat chop tam giac deu
S.ABC
c6 canh ben bang 2a, mat
ben
tao voi day mot goc 60" .
1)
Tinh
the tich khoi cau ngoai tie'p
hinh
chop

S.ABC
2)
Tinh
the tich va dien tich xung quanh cua
hinh
non c6 dinh S va day la
duang tron ngoai tie'p tam giac
ABC.
jCgigidi.
1) Dat AB =
X
. Goi O la tam ciia day, suy ra
SO
i (ABC)
Goi
M la
trung
diem doan AB, suy ra BC1
(SMA)
nen goc SMA la goc
giija
mat ben va mat day ciia
hinh
chop, suy ra SMA = 60" .
Goi
F la trung diem canh SA, trong mat phang (SMA) duong trung
trifC
doan SA cat SO tai I. Ta c6 I la tam mat cau ngoai tie'p
hinh
chop

S.ABC
va ban
kinh
R = SI.
184
Cty
TNIm
MTV DWH Khang Viet
_ . xVs _ . 2 XN/S ^. . 1 xVs
Taco AM = , OA = -AM = ——-,
OM
= -AM =
2 3 3 3 6
Trong
tam giac
SAO,
ta c6:
SO^
=
SA^
-
AO^
= 4a^ - —
Trong
tam giac
SOM,
ta c6:
SO
= OM.tan60" = |
Tu

do suy ra:
, 2 7x2
— = 4a = 4a => x = .
4 3 12 7
Ta
CO
tam giac SFI dong dang voi tam giac
SOA
nen
ta c6:
, SI SF . ^
SA^
2a^ a
SA
SO
2SO
2aV2T
721 '
4 „3
47ta
Vay
the tich khoi cau ngoai tie'p
hinh
chop la: V = —
7tR
= .— .
3
63V21
2) Ta
CO

ban
kinh
duong tron ngoai tie'p tam giac
ABC
la r =
4aV7
2
167ia^
Suy
ra dien tich duang tron ngoai tie'p tam giac
ABC
la: Sj = 7tr = ——
Vay
the tich khoi non ngoai tie'p
hinh
chop la:
V
=
isoS
=
^
-^^"^^ ^
327ia^V2T
• 3 • 3" 7 • 7 147
Vidu 2.5.
J.
Trong
hinh
phang (P) cho nua luc giac deu
ABCD

noi tie'p duong
tron duong
kinh
AD
= 2R. Qua A ke duong thSng Ax vuong goc vai (P), tren
Ax
lay diem S sao cho goc giiia hai mat phang
(SDC)
va (P) bang 60". Xac
djnh
tam va ban
kinh
hinh
cau di qua nam diem
S,A,B,C,D.
-
t
JCgigiai
Gpi
O la trung diem doan thang
AD,
ta c6 O la tam duang tron ngoai tie'p
giac
ABCD.
Ke Ox song song vai SA, ta c6 Ox la tryc duong tron ngoai tie'p
,tu giac
ABCD.
Trong tam giac
SAC
tu trung diem J ciia canh SA ke Jy song

Song
voi
AC,
ta c6 Jy la duang trung true canh SA. Goi I la giao diem ciia Ox
vajy.
; Ta
CO
I thuoc Ox nen lA =
IB
=
IC
=
ID
va I thuoc Jy nen
IS=IA
v^y I la tam
•hinh
cau di qua nam diem S, A, B, C,
D.
Va ban
kinh
hinh
cau nay la r =
lA.
185
Phucmg phtip giiii Toiin Hinh hoc
theo
chuycn dc -
Nguyen
Phi'i

Khanh,
Nguyen
Tai Thu
Theo
gia thie't ta c6:
SCA =
((SCD),(P))
= 60" .
=> AC = ADsin60" = N/SR ,
SA = ACtan60" =3R .
Tir
giac
AOIJ la
hinh
chir nhat
nen AI = JAJ^ + AO^ =
•
Vay ban
kinh
mat cau la r =
Vl3R
D
Vi
dy
2.5.3.
Cho
hinh
chop
S.ABCD
c6 day la

hinh
thang vuong tai A, D,
AB = AD - a, CD - 2a. Canh ben SD 1 (ABCD) va SD = a. Goi E la
trung
diem
ciia DC. Xac djnh tam va
tinh
ban
kinh
mat cau ngoai tiep
hinh
chop
S.BCE.
Loi
giai.
Vi
AB = DE = AD = a va DAB = Iv nen ABED la
hinh
vuong.
Tam
giac
BDC c6 EB = ED
= EC = a nen vuong tai B,
BE 1 CD nen
trung
diem M
ciia BC la tam duong
tron
ngoai tiep tam
giac

EBC.
Dyng
A la true duong
tron
ngoai tiep tam
giac
EBC
thi
A song song voi SD.
Dung
mat phang
trung
tryc canh SC, mat
phang do cat A tai I.
Diem
I la tam
m|t
cau ngoai tiep
hinh
chop
S.BCE.
KeSN//DMc3t MI tai N,
taco
SDMN la
hinh
chi> nhat voi SD-a va
^^,2 DB^+DC^ BC^ AB^+AD^+DC^
EC^+EB^
Sa^
DM

= = = .
2 4 2 4 2
Ta
CO
SI^ = SN^ +
NI^
= SN^ + (NM -
IM)^
= ^a^ + (a -
IM)^
186
Cty
TNIIII
MTV DWH Khang Vift
Ma
IC^ =
IM^
+ MC^ =
IM^
+ — va R = IC = IS
nen ^a^ +(a-IM)2
=IM2
+y «IM = |a
Vay ban
kinh
mat cau ngoai tiep
hinh
chop
S.BEC
la: R = JlM^ +— =^!^a

Vidu
2.5.4.
Cho
hinh
chop
S.ABCD
c6 day ABCD la
hinh
chu nhat AB = a,
AD
= -^. Mat phang (SAB) 1 (ABCD) va SA = SB = a. Xac djnh tam va
3
tinh
the tich khoi cau ngoai tiep
hinh
chop S.ABD.
JCffigidi.
Vi
tam
giac
SAB can tai S va
(SAB) 1 (ABCD) nen goi H la
trung
diem
cua AB thi SH 1 (ABCD).
Goi
O la tam
hinh
chu nhat
ABCD thi O la tam duong

tron
ngoai
tiep
tam
giac
ABD, true duang
tron
ngoai tiep tam
giac
ABD la duong
thang A song song voi SH.
Goi
G la trong tam tam
giac
deu
SAB. ^
Vi
HO 1 (SAB) nen true duong
tron
ngoai tiep tam
giac
SAB la duong
th^ng
qua G, song song voi HO cat A tai I thi I la tam mat cau ngoai tiep
hinh
chop S.ABD. Tu
giac
GHOI la
hinh
ehi>

nhat. i'^ :
TacoGH = -SH = —a ,
3 6
-'^"•''i
:i,n'
•!
Nen
lA =
^J\0^
+OA^ = JGH^ +
AC^
282
2 a + - a
a 3
— + — ^ a.
12 4
Vay ban
kinh
ciia mat cau ngoai tiep
hinh
chop
S.ABD
do la R = lA, nen
Ithe tieh khoi cau V = — R3 = — al
Vi du
2.5.5.
Cho tu dien ABCD
c6AB
= 6, CD = 8, cac canh con lai bang
\f74 . Hay tim ban

kinh
hinh
cau ngoai tiep tu dien ABCD .
187
Phmnig
plriipgiiii
Toiin Iliuh hgc
theo
chttyen
de-
Nguyen
Phii
Khdnh,
Nguyen
Tat Thu
J:fflgidi.
Goi O la tarn mat cau ngoni tiep fu dien
ABCD. Do O
each
deu AB nen O thuoc
mat phang trung true
canh
AB. Do chinh
la mp(MCD). Voi M la trung diem AB.
Tuong tu O
each
deu CD nen O thuoc mp
trung
true
canh

CD la mp (NAB), voi N la
trung
diem CD. Vay O nam tren
giao
tuyen MN cua mp(MCD) va MP (NAB). ^
DoMN
1 AB va MN 1 CD .
Ta CO OA = OD = R ;
OM =
VR^-AM^
=
VR^-9,
ON = VR^-ND^ = VR^ -16 . ^
Ta CO DM^ = AD^ -
AM^
= 65 =>
MN^
= DM^ -
DN^
- 49 => MN = 7 .
• Neu O nam trong hinh
chop
thi MN = OM + ON
C^7
=
VR^-9
+
VR^-16=>R2=25=>R
= 5.
• Neu O nam ngoai hinh

chop
thi MN = OM - ON <=> 7 = 7R^ -9 -
VR^
-16
v6 nghiem.
Vay ban
kinh
R = 5 .
Vi du
2.5.6.
Cho hai tia Ax, By
cheo
nhau va vuong goc, nhan AB la doan
vuong goc chung, AB 2a . Tren Ax, By Ian
lugt
lay cac diem C, D.
1) Chung minh CD tiep xuc voi mat cau duong
kinh
AB <=> CD = AC + BD.
2) Vdi dieu kien 6 cau 1, chung to
rang
diem tiep xuc cua CD voi mat cau
duong
kinh
AB thuoc mot duong tron co
dinh
va goc tao boi CD vdi mat
phang
chua
duong tron do khong ddi.

3) Neu CD thda man dieu kien AC^ + BD^ = b^ (b la so cho trudc). Hay tim
lien
h^ giOa a va b deCD tiep xuc vdi mat cau duong
kinh
AB.
JCffigidi.
l)Dat AC = c, BD = d.
Goi I la trung diem ciia ABva H la hinh chieu cua I len CD.
Ta CO lA = IB = a . i
Cdx:hl
• Neu CD tiep xuc vdi mat cau dudng
kinh
AB thi ta cd H la tiep diem. Ta
CO
CH = CA, DH = DB nen CD = CH + DH = AC + BD.
• Ngu(?c lai neu CD = AC + BD thi ta chung minh CD tiep xuc vdi mat can
dudng
kinh
AB.
188
Cty TNUH MTV DWH
Khang
Viet
Do in 1 CD nen ta can chung minh IH =
AB
= a
That vay
Neu IH < a . Ap dung djnh ly pitago cho hai A vuong lAC va IHC
CH
=

7ci2
_
ipj2
> ^ci2 -
lA^
=
AC
(1).
Ap
dung djnh ly pitago cho hai A vuong IBD va IHD
(2).
A
x' ,
DH -
VDI^-IH^
>
VDI^-IB^
= BD
Tu(l)va (2) suy ra
CD = CH + DH > AC + BD
trai
vdi gia thie't. M'
* Neu
IH>a
chung minh
tuang tu ta cd CD < AC + BD
trai
vdi gia thie't.
Vay IH-a hay CD tiep
xuc vdi mat cau dudng

kinh
AB.
Cdch
2. Lay diem M la diem tren tia doi cua tia Ax sao cho AM = BD.
Ta de chung minh duoc OM = OD va CM = CD => ACOM = ACOD.
Suy ra lA = OH = a (Hai dudng cao tuong ung).
2) • Chung minh H thuoc dudng tron co djnh. Ke tia Bx'
song song
va cung
chieu vdi Ax. Ggi C va H' Ian lugt la hinh chieu cua C, H len mp( Bx'y). Ta cd
BH'
la phan
giac
ciia goc x'By. Nen H' co
dinh
thuoc tia phan
giac
Bz cua gdc
" x'By. Vay H thuoc mp(ABz) cddjnh. Mat
khac
H lai thuoc mat cau codinh. Nen
H
thupc dudng tron co'djnh la
giao
cua mp (ABz) vdi mat cau dudng
kinh
AB.
• Chung minh gdc giCra CD va mp (ABz) khong
ddi.
' '

GQI
K la hinh chieu cua D len Bz ta cd gdc DHK la gdc giua CD va mp
(ABz). Ta cd: ABDK = AHDK ^ DHK = 45° (khong ddi) , , >
3) He thuc lien h? giija a va b.
Do CD tiep xiic vai mat cau nen
theo
cau 1 ta cd CD = AC + BD = c + d
Suy ra cd = 23^ (1)
Ta cd AC^ + BD^ = b^ ^ CD = ^4a^ + b^ ^ c + d = V4a^ + b^ (2).
Tir
(1) va (2) ta cd c, d la nghiem cua phuong
trinh:
x^ - V4a^ + b^ x + 2a^ =0
Vay h^ thtfc lien giira a va b de CD tiep xiic vdi mat cau dudng
kinh
AE
lab>2a.
Phuonig phlipgilii Todtt Hinh
hoc
theo
chuyeu
ile
-
Nguyen
P/iii
Khmih,
Nguyen
Tat Thu
Vi
du

2.5.7.
Cho
hinh
chop
tu
giac
deu
S.ABCD
c6
canh
day
bang
a va
ASB =
a.
1)
Xac
djnh
tam va ban
ki'nh
mat cau ngoai tiep
hinh
chop
a '
iA'
2)
Xac
djnh
tam va ban
kinh

mat cau npi tiep
hinh
chop
' "
3) Tim gia trj
cua a
detam mat cau npi tiep, ngoai tiep
trung
nhau.
Xffi
gidi.
1) Tam va ban
kinh
mat cau ngoai tiep.
Gpi
H la
tam ciia
hinh
vuong ABCD.
Ta c6
SH
la
tryc du-ong
tron
ngoai tiep
day.
Trong mat ph5ng (SHA) ke
dirong
trung
true canh

SA
cat SH tai O.
Ta c6
O
la
tam mat cau ngoai tiep
hinh
chop S.ABCD, ban
kinh
R
= SO
.
Gpi
N la
trung
diem SA,
ta c6:
ASNO
~ ASHA:
SN
SH
so
=
SN.SA
SO
SA SH 2SH
Ap
dung
djnh
ly sin cho tam

giac
can SAB
ta c6 :
SA
AB „. AB
sin(
180'
,0
.SA =
•a
)
sma
2sin^
SHWSA^-AH^="^
2sin'^
2
Vay
R
= SO
=
a
47cosa.sin
—
2
2) Tam va ban
kinh
mat cau npi tiep.
Ta
CO
tam

I
ciia
mat cau
npi tiep thupc duang thSng SH. Gpi
M la
trung
diem
cua
AB,
ta c6
AB
1
(SHM)
tai
M. Gpi
I la
chan duong phan
giac
trong
ciia
goc
SMH (I
e
SH)
Ta c6 I la
tam ciia mat cau npi tiep
hinh
chop. Ban
kinh
r

=
IH.
De
tinh
ban
kinh
r ta c6
the
tinh
theo hai
each
sau:
Cdch
l.Dua vao
tinh
chat ciia duang phan
giac
ta c6
IS
MS SH
MS
+
MH
IH
MH IH MH
Vay ban
kinh
mat cau npi tiep la:
r
= IH

=
SH.MH
a a
^^^^
a
IH
=
— ;
SM = -cot—, MH =

MS
+
MH
2 2 2
aVcosa
2(sm —+
COS—)
Cdch
2. Dua vao cong thuc
r =
3V
IS-
190
Cty
TNHH
MTV DWH
Khang Vi$t
1 ,
^ a^Vcosa
The

tich
khoi
chop:
V
= -SH-S^BCD
=
f
3.2
sin"
2
I^J^S
=
4S^AB +
SABCD
=
4 SM.AB
+
a^
=
1
2,
. a a.
^
a
(sin +
cos-)
sin
a
teankinh
r

=
3V
IS
avcosa
2(sin~ +COS-)
2
2
^ Tam ciia hai
hinh
cau
trung
nhau
<=>
R
+
r
= SH
.
aVcosa aVcosa
Hay:
4\/cosa.sin^
2(sin^+cos^)
2sin^
1
cos a
cos a
1
a
„
. a .a a .a

2sin—
sin
+cos-
sin
2
2 2 2
2sin-
2
a cos a
+
cos—
sin
—
=
a
2 2 . a
sin-
<=>
1
+ sina
-
2sin^
—
=
2cosa
o
sina =
cosa
<=>
a

= 45°.
2
Vay khi
a
= 45" thi tam mat cau ngoai tiep
va
tam mat cau npi tiep ciia
hinh
chop
trung
nhau.
Vi
du
2.5.9.
Ben trong
hinh
tru
c6
mot
hinh
vuong ABCD canh
a
npi tiep
ma
hai
dinh
lien
tiep A,
B
nam tren duang

tron
day thu nhat ciia
hinh
tru,
hai
dinh
con
lai nam tren duong
tron
day thu hai ciia
hinh
tru.
Mat phang
hinh
vuong
tao
voi
day
hinh
tru mot
goc 45".
Tinh
dien
tich
xung quanh
va the
tich
ciia
hinh
try do.

JCgigidi.
Ta
CO
DC
1
BC,CE
la
hinh
chieu ciia duang xien
=i>DClCE(djnh
li
ba
duong vuong
goc)
do
do
BCE = 45°
la goc
giiia
hinh
vuong ABCD
Va
day
hinh
try.
Trong
tam
giac
BEC ta
c6

BE
-
BC.sin45° =
—;
BD
-
V2a.
2
Trong
tam
giac
vuong BED
ta
c6: DE^ = DB^
-
BE^
=
191