内容正文:
书书书
(= - )23
2
(+ - )23
2
(+ )23槡
2
= 槡23
3
,
!"
PF
#$%&
槡23
3
.
20.(1)
!"
:
'
PA
#(
)
H,
*+
HE,BH,
,-
11.
./
E
/
PD
#()
,
!
"
HE
/ △APD#(01,
!"
HE∥AD,
2
HE= 1
2
AD.
3
AD∥BC,BC=
1
2
AD,
!"
HE∥BC,HE=BC,
!"456
BCEH
/78456
,
!"
CE∥BH.
./
BH 79 ABP,CE:;79 ABP<,
!"
CE∥ 79 ABP.
(2)
#
:
=>?@ △PAD/ABCDED6,45
6
ABCD
/CDF6
.
'
AD
#()
F,
*+
BF,PF,
./
AD=2BC=4,
!";79G56
PABCD
(
,P,F,B
E)H1
,
2
PF=BF=2,
!"IJK
,PF⊥AD,BF⊥AD,PF∩BF=F,
!"
DF⊥ 79 PBF,!" BC⊥ 79 PBF.
./
PB 79 PBF,!" BC⊥PB.
;CDED6
PBC
(
,PC= 槡22,BC=2,
!"
PB=2,
!" △PBF/A5ED6.
'
BF
#()
O,DC
#()
M,
*+
PO,OM,
L
PO⊥BF,
./
DF⊥ 79 PBF,!" DF⊥PO.
3
DF∩BF=F,!" OP⊥ 79 ABCD.
"
O
/M)
,OB,OM,OP
!;#C1NO/
x
P
,
y
P
,z
PQRSTCDUVW
,
L
B(1,0,0),P(0,0,槡3),A(-1,-2,0), (E -12,1,
槡3)2 ,!"→ (AE= 12,3,槡3)2 ,→AB=(2,2,0),→BP=
(-1,0,槡3).
X79
ABP
#YZ[/
n=(x,y,z),
L
n·→AB=0,
n·→BP=0{ ,\
x+y=0,
-x+槡3z=0
{
,
]^'
n=(3,-3,槡3),
!"
cos〈n,→AE〉= n·
→AE
|n|· →|AE|
=-槡210
35
,
!" C 1
AE
_ 7 9
ABP
! ` D # a b c /
槡210
35
.
21.(1)
!"
:
./456
ABCD
&78456
,
2
∠BCD=
3π
4
,
!" ∠ABC=π4
,
=dbef
,
g
AC2=AB2+BC2-2AB·BC·cosπ
4
=AB2+2AB2-2AB2 =AB2,
!"
AC=AB,∠ACB=π4
,∠BAC=π2
,
\
AC⊥AB.
./
PB⊥ 79 PAC,AC 79 PAC,
!"
PB⊥AC,
3
PB79PAB,AB79PAB,PB∩AB=B,
!"
AC⊥ 79 PAB.
(2)
#
:
=
AB= 槡22,BC=槡2AB,g BC=4,
./
BC∥AD,!" ∠PCB&h9C1 PC_ AD
!`#D
,
!" ∠PCB=π6
.
3
PB⊥ 79 PAC,PC 79 PAC,
!"
PB⊥PC,; Rt△BPC(,
PB=BC·sin∠PCB=4×
1
2
=2,
i@
PB⊥PA,!"; Rt△PAB(,PA2 =AB2 -
PB2 =8-4=4,
!"
PA=2,
'
AB
#()
E,
*+
PE,
L
PE⊥AB,2 PE=
1
2
AB=槡2,
=
(1)
@
AC⊥ 79 PAB,2 PE 79 PAB,
!"
AC⊥PE,
3
AB79ABCD,AC79ABCD,AC∩AB=A,
!"
PE⊥ 79 ABCD.
,-
12,
"
A
/UVM)
,
AB,AC
!;#C1NO/
x
P
,
y
P
,
j)
A
2_
PE
78#C
1/
z
P
,
QRSTCDUV
W
,
L
B(槡22,0,0),C(0,槡22,
0),D(- 槡22,槡22,0),P(槡2,0,槡2),
!"
→PC=(-槡2,槡22,-槡2),
→BC=(-槡22,槡22,
0),→DC=(槡22,0,0).
X79
PBC
#YZ[/
m =(x,y,z),
L
m·→PC=-槡2x+ 槡22y-槡2z=0,
m·→BC=- 槡22x+ 槡22y=0
{
,
'
x=1,
L
y=z=1,
!"
m =(1,1,1).
X79
PDC
#YZ[/
n=(a,b,c),
L
n·→PC=-槡2a+ 槡22b-槡2c=0,
n·→DC= 槡22a=0
{
,
L
a=0,
'
b=1,
L
c=2,
!"
n=(0,1,2).
!"
cos〈m,n〉= m·n
|m|·|n|
=槡15
5
,
=-^@k9D
B-PC-