夜雨聆风学习资料网 资料列表 开通 VIP 用登录 / 注册 ⌄
PDF

大题03空间向量与立体几何(5大题型)(原卷版)

查看资料信息、公开预览和下载状态。

共 23 页PDF · 1.0MB
下载

文档在线预览

公开展示前 10 页
100%
1
《大题03空间向量与立体几何(5大题型)(原卷版)》第1页预览
2
《大题03空间向量与立体几何(5大题型)(原卷版)》第2页预览
3
《大题03空间向量与立体几何(5大题型)(原卷版)》第3页预览
4
《大题03空间向量与立体几何(5大题型)(原卷版)》第4页预览
5
《大题03空间向量与立体几何(5大题型)(原卷版)》第5页预览
6
《大题03空间向量与立体几何(5大题型)(原卷版)》第6页预览
7
《大题03空间向量与立体几何(5大题型)(原卷版)》第7页预览
8
《大题03空间向量与立体几何(5大题型)(原卷版)》第8页预览
9
《大题03空间向量与立体几何(5大题型)(原卷版)》第9页预览
10
《大题03空间向量与立体几何(5大题型)(原卷版)》第10页预览

本文档共 23 页 剩余 13 页未展示,请下载后查阅

下载

文档文本内容

      
      
 03  
 
 !"#$%&'()*%&+$,&-./)01,2345)67&8
9:);<5=>?9)??9)@AB?9)CDE5FGHIJKL@AM/:CNO 2025
#$%&+)P23@??9QR*B?9$SG45C
题型一  空间几何中的线面角
1TU24-25 %VW·XY·Z[\]^_`)=Vab
1
1
1
ABC
A B C
-
+)
^
BC
c?
1
1
1
,
ACC A AC
AB
^
T
  
(1)de
c?
1A BC ^ c? ABC f
      
      
(2)g
1
2
2
AC
AC
BC
=
=
)dh>
1
AB c?
1
1
BCC B ij9$klmT
U1^n>od>?9Upqhro^
stuv>c?)wx>?$y/ B Ivzf
w>=?{$6/ A)|/ A c?a }n>)tunz Of~vznzIv> AB =?a €$‚f
‚ BO v> AB ƒ$„9I>?9f…‚ BO v> AB †x6‡V9S+ˆ‰d0UŠ‹Œ
ŽluQRh9V9S^T
U2^God>?9Upqo^
o)d4v> PA =?{$6/ P x?$‘’)ŒV
9S$klGˆ‰d0TGI
𝑠𝑖𝑛𝜃=
ℎ
𝑙)“+𝜃*v>c?ij$9)ℎ*n>[$”)𝑙*v>[
$”T
2•od>?9
–
1n
ur
*h> l $F)
2n
uur
*c?a $o)h>c?$„9Iq .—
1
2
1
2
1
2
sin
cos
,
n n
n n
n n
q
×
=
<
> =
ur uur
ur uur
ur uur T
1U2025·X˜™š·6›^_`)=œa S
ABCD
-
+)
/ /
BC
AD )
1
AB
BC
=
= )/ E
= AD €)ž
,
1,
2
SE
AD AE
DE
^
=
=
T
      
      
(1)/ F =>[ SE €)ž
//
BF
c? SCD )eŸ
 F I>[ SE $+/f
(2)g AB ^c?
,
SAD SD c? SAB ij$9$ŽlmI 10
10
)d SD $” T
题型二:空间几何中面面角
1U2025·X˜¡¢·6›^_`)=£? DABCE +)
2,
2
AB
AC
BC
AD
DB
DC
EB
EC
=
=
=
=
=
=
=
=
) BC $+/IO .
(1)de
 ,
, ,
A D O E œ/¤?f
(2)gh> DE c? ACD ij9$klmI
7
7
)dc? ABC c? BCE „9$¥.
      
      
  o d ? ? 9 
 g
1
2
,
n n
ur uur
H ¦ I c ?
,
a b $ o   ) q I c ?
,
a b $ „ 9 ) —
1
2
1
2
1
2
cos
cos
,
n n
n n
n n
q
×
=
<
> =
ur uur
ur uur
ur uur T§ŠC¨??9$©ª* 0 x 90
U2025·«¬­®¯°·6›^_`)EA± DC ²nh³c? ABC )ž
2
EA
DC
=
)F *
>[ BE €6/.
(1)g
/ /
DF
c? ABC )eŸ F * BE $+/f
(2)g AB
BC
CD
=
=
) AB
BC
^
)c? DEF c? ACF „9$ŽlmI
2
3
)d BF
FE .
      
      
2U24-25 %VW·´µ·¶&·^_`i¸)=¹ºO O
¢
+)œ»S ABCD I|O O
¢
$¹º¼?)
AEF
△
I
½?¹ O $¾rkV9S.
  
(1)eŸ
c?CEF ^ c? ABCD f
(2)g
2
2
AD
AB
BC
=
=
)dc? ABF ±c?CEF „9$Žlm.
3U2025·´µ¿X·6›^_`)=Vaº
1
1
1
ABC
A B C
-
+)
1
AA ^ ½?
1
1
,
1,
2,
ABC A B
AB
AC
D
=
=
=
I BC $
+/)
1
AC
C D
^
T
(1)eŸ
 AC
AB
^
f
(2)g
1
2
AA =
)dc?
1
1
BCC B c?
1
ADC „9$ŽlmT
      
      
题型三  空间几何中的二面角
U2025·XY·6›^_`)À-œa P
ABCD
-
+)½? ABCD IÁS)
60
ABC
Ð
=
° )
2
AB
PA
=
=
) PA ^c? ABCD )
,
,
E M F H¦*
,
,
CD PD PC $+/T
(1)eŸ
c? PAE ^c? PCDf
(2)dB?9 F
AE
M
-
-
$klmT
odB?9
U1^uÂoUa€6/Ãn>o^
=B?9$a€w6‡ÄÅ/)=ƇÇc?¾H¦|È/Énh³
a$Ê>T
U2^Vn>oU?€6/Ãn>o^
ËB?9$6‡?€6/Ì{6‡?Én>)ÍÎnzaÉn
>Ïxa€$/U§vz^)vz±?€6/$~>vz±nz$~>i„$9)§IB?9$c?9T
U3^n?oU6/n?o^
|6/Éanh$c?)¼B?9ÏÆÐÊ>)ÑÆÐÊ>ij
$9Ò*B?9$c?9T
U4^ʂ?o
À-c? b ¾6‡£»S$?I S)Ó=c?a ¾$ʂ`S$?I𝑆射影)
      
      
c?a ±c? b ij$B?9$¥Iq )—𝐶𝑂𝑆𝜃=
𝑆射影
𝑆.чFoÔ³ÕaB?9$d0ÖרT
2•odB?9
g
1
2
,
n n
ur uur
H¦Ic?
,
a b $o)q Ic?
,
a b $„9)—
1
2
1
2
1
2
cos
cos
,
n n
n n
n n
q
×
=
<
> =
ur uur
ur uur
ur uur T.B?9$©ª* 0 x 180
1TU2025·ÙÚÛ·B›^_`)=œa P
ABCD
-
+) PA ^½?
,
ABCD BC ∥
,
,
1,
2
AD AB
BC AB
BC
AD
AP
^
=
=
=
=
)/ E Üz
(0
1)
DE
DP
l
l
=
<
<
uuur
uuur
.
(1)gCE ∥c? PAB )dl $mf
(2)g
1
3
l =
)dc? ACE c? PCDij$B?9$klm.
题型四:空间几何中的折叠问题
      
      
1U2025·ÝÞ·V›^_`)ߏàh9V9S ABC áv» BC €$+> AD âã)Ïx
œ? PABD T
(1)eŸ
 AD ^ c? PBD f
(2)g
2,
2 3
AD
PB
=
=
)dc? PAD c? PAB „9$ŽlmT
.U2025·äåæÂ·›çNè^À-é¹


2
2
2
2
:
1
0
x
y
C
a
b
a
b
+
=
>
>
$ê•ëì/H¦I
1F •
2
F )’íîI
2
2 )ï|/
1F žðv9I
π
0
2
q
q
æ
ö
<
<
ç
÷
è
ø$h>l é¹y³ A • B Æ/U“+/ A = x ñ€
F^)ž
1
2
AF F
△
?$òmI 2Tßc? xOy á x ñ€ãó)ôB?9
1
2
A
F F
B
-
-
IhB?9)_`
1)2 i¸T
(1)dé¹C $õöF÷f
      
      
(2)ø
π
4
q =
ù)dãóú)c?
1
2
AF F c?
1
ABF ijûB?9$Žlmf
(3)dãóú
1
ABF
V
?$òmT
1.	


 !"#$%&'
 ()*+,-.
2. ,-/01234
516,-789:/;<=># =>?
526@ABCDE!"# FG=>
3 IJKLMN1OP*Q	
,-RST!"U
VWXJYZ,-[\]^NI
JOP_`Q	
,-$%a8.
1 U24-25 %VW·´ÞÞü·¶&·^_` 1 =h9ýS ABCD +) AD
BC
∥
)
π
2
BAD
Ð
=
)
1
AB
BC
=
= )
2
AD =
)E * AD $+/)O * AC  BE $y/)ß ABE
V
á BE ãþx` 2
+
1A BE
V
$ÿĀ)Ïxœa
1A
BCDE
-
T
      
      
  
(1)eŸ
CD ^ c?
1AOC f
(2)øc?
1A BE ^ c? BCDE )dc?
1A BC c?
1ACD „9$Žlm.
2U24-25 %VW·āÞß·Z[\]^_`ă)=àh9
ABC
V
+)
90
A =
o)á½»$% AD  ACD
V
$
+ÿ> EF )H¦ß
ABD
△
±
CEF
Ą
ãþx
PAD
△
±
QEF
△
$ÿĀ)_`ą)ãó|÷Ćć DP ∥EQ .
(1)eŸ
,
, ,
A F P Q œ/¤?f
(2)dh> AE c? PDF ij9$kl$òm.
      
      
题型五:空间几何中的动点问题
1U24-25 %VW·µĈĉ·Z[\]^_` 1 i¸)=àýS ABCD )
//
BC AD )
CE
AD
^
)nzI E )
3
3
AD
BC
=
=
)
1
EC = )ß DEC
V
á EC ãþx
1
D EC
Ą
$ÿĀ)_` 2 i¸)/G
Ia
1
AD €6‡M/T
(1)de
1
CE
AD
^
f
(2)gc?
1
D EC ^ c? ABCE f 
Ui^dh>
1
CD c?
1
ABD ij9$klmf
Uii^=a
1
AD €*Ċċ=/G )ô
//
BG
c?
1
D EC )gċ=)d4
1
AG
AD $m)gČċ=)čϟÎT
      
      
2U24-25 %VW·ďĐ·Z[\]^_`)=œa P
ABCD
-
+)½? ABCD Ih9ý
S)“+ AD ∥
,
2
,
,
,
BC AD
BC AB
AD M N
=
^
H¦I
,
CD PA $+/)/Q Üz
2
PQ
QB
=
uuur
uuur
.
  
(1)eŸ
CQ ∥MN f
(2)g PA ^½?
,
,
ABCD PA
AB
CDA
PBA
Ð
Ð
=
=
)—a PD €*Ċċ=/ E )ôÏc? BCE c? PDC $
„9I30o đgċ=)d DE
DP $mfgČċ=)čϟÎ.
b2c7de,-fXgbehi]^NjkXlmn5xyz6rUsIJftLXu<
vw>#jX^Nxvwyz{|})#~.Q€‚>#ƒ#<„…j†.
1.5L‡6VX<Lw>#r„Uskˆ1Xlm?‰Š‹Š	
Œ#MŽ——𝑎//𝑏⟹ 
∃𝜆∈Ru’ 𝑎= 𝜆𝑏
2.	
5“‡6VX<Ww>#r„Uskˆ1Xlm.
3.‰Š	
Œ#”•MŽ,𝑎, 𝑏=4	
Vt—LwŒ#𝑐˜gb𝑥p™‘u’𝑐= 𝑥𝑎+𝑦𝑏
      
      
1U2025·ÙÚÛ·6›^_`i¸)kV9S ABC $»”I 2) D ) E ) F H¦*Ē»
$+/)5ß
ADE
V
)
BEF
Ą
) CDF
V
H¦á DE ) EF ) DF ãþ)ôÏ
ADE
V
)
BEF
Ą
) CDF
V
i=c
?ē½? DEF nh.
  
(1)de
c? ABC ∥c? DEF f
(2)dB?9C
DA
E
-
-
$klm.
16TU2025·ĔYĕĖ·›çNè^_` 1)À-
ADE
V
I»V9S)œ»S ABCD Ic‰œ»S)
1,
2,
5
BC
BD
BA
=
=
=
.…
ADE
V
á AD €ãþ)ô/ E xė/ P ÿĀ)ôÏc? PAD ^c? PBD )_`
2 i¸.
(1)eŸ
 PA
BD
^
f
(2)dc? APB c? PBC „9$Žlmf
(3)ø/Q =>[ PA UĘęĚ/^€ěMù)–h> BQ c? PAD ij$9Iq )d tanq $Ĝm©ª.
      
      
 
6•01
1TU2025·ÛYÞĝ·6›^_`)=Va P
ABC
-
+) PA ^c?
,
1,
120 ,
ABC AB
BC
ABC
PA
AC
=
=
Ð
=
°
=
) D I PC $+/T
  (1)de
 BD
AC
^
f
(2)d BD c? PAB ij9$klmT
2TU24-25 %VW·ÛĞ·¶&·^_`)=Va P
ADE
-
+)C Ia PD €6/)
2 3
3
1,
,
3
3
AC
AE
CE
=
=
=
)ž AC
PD
^
) PD
DE
^
T
(1)eŸ
 AC ^c? PDE f
(2)dœ? ACDE ${rğ$f
(3)gc?CAE c? ADE „9$ŽlmI 3 13
13
)d DE $”T
      
      
3TU2025·ÙÚÛĠġĢ·6›^_`)=œa P
ABCD
-
+) PA ^c?
π
,
,
/ /
2
ABCD
DAB
AB
CD
Ð
=
)
2,
1,
PA
AB
AD
CD
M
=
=
=
=
*>[ PB €$M/.
(1)ø M *>[ PB +/ù)de
CM ∥c? PAD f
(2)– N * PD $+/)øB?9 P
AC
M
-
-
$ŽlmI
30
6
ù)d/ N xc? ACM
$‘’.
4TU2025·´µĆu·›çNè^_`)Vab
1
1
1
ABC
A B C
-
$ģ?
1
1
ACC A ģ?
1
1
BCC B ēIkFS)D I
AC $+/)
1
1
1
B D
AC
^
.
(1)eŸ
1
CC ^ c? ABC f
(2)eŸ
c?
1
BC D ^c?
1
1
ACC A f
(3)dB?9
1
1
B
C D
B
-
-
$klm.
      
      
5TU24-25 %V€·ÛĞÕĤ·ĥĦ^_`)=œa P
ABCD
-
+)œ»S ABCD IkFS)
6
AB =
)
57
PC
PD
=
=
)B?9 P
CD
A
-
-
$¥I π
6 )/ M I>[ PD €6/.
(1)eŸ
c? PAB ^ c? ABCD .
(2)g
2
MD
PM
=
)dœa M
ABCD
-
$.
(3)/ M I>[ PD €6M/)dh> MD c? AMC ij9$kl$òm.
6TU24-25 %VW·ħ˜·¶&·^_`)œa P
ABCD
-
$½?*ĨS)
2
2
,
AD
PC
AB
PD E
=
=
=
I
BC $+/) PB
AE
^
.
  
(1)eŸ
 PD ^ c? ABCD f
(2)=a PB €*Ċċ=ĩ³Ě/
,
P B $/ F )ôÏc? AEF c? BEF $„9I 45o đgċ=)d4/ F $
ÿĀfgČċ=)čϟÎ.
      
      
7TU24-25 %VW·X˜¡¢·¶&·^–V9S ABC Üz 

1 cos
cos
b
C
c
B
-
=
)“+ , ,
a b c H¦I , ,
A B C
$Ô»T
(1)eŸ
 AC
BC
=
f
(2)–V9S ABC +) D I» AB Ī" A $VH/)ž
3
AB = )
2π
3
ACB
Ð
=
)ßV9S ABC €âãU_
`^)gc? ACD ^ c? ACB )dB?9 B
AD
C
-
-
$ŽlmT
8TU24-25 %V€·X˜īÝ·ĥĦ^_`)=@
, ,
,
,
,
A B C D E F IĬ/$£?+)c? ABCD ^ c?
,
AEFB AD
AB
^
) AB ∥CD ∥
,
4,
2,
EF AB
AD
CD
EF
AE
BF
M
=
=
=
=
=
=
I AD $+/
  
(1)eŸ
 AF / / c? EMC f
(2)=a BF €*Ċċ=6/ P )ôÏh>CP c? EMC ij9$¥I π
6 .gċ=)d BP
BF $mfgČċ=)
čϟÎ.
      
      
9TU2025·X˜·6›^_`)œ? ABCD +)
ABC
V
I»V9S)ž
2
AB =
)
ADC
Ą
Iàh9V9
S)ž
90
ADC
Ð
=
° T
(1)ø
7
BD =
ù)
Uⅰ^dB?9 D
AC
B
-
-
$klmf
Uⅱ^ø P I>[ BD +/ù)dh> AD c? APC ij9klmf
(2)ø
2
BD = ù)g
(0
1)
DP
DB
l
l
=
<
<
uuur
uuur
)ž PH ^c?
,
ABC H Inz)CD +/I
,
M AB +/I N fh> MN
c? APC $y/IG )øVa P
ACH
-
òù)dGM $mT
6•01
1TU2024·ÛĞ·%&ĭ^_`)œa P
ABCD
-
+)PA ^½? ABCD)
2
PA
AC
=
=
)
1,
3
BC
AB
=
=
T
(1)g AD
PB
^
)eŸ
//
AD
c? PBC f
(2)g AD
DC
^
)žB?9 A
CP
D
-
-
$klmI
42
7
)d AD T
      
      
2TU2024·«ĮõⅡį·%&ĭ^_`)c?œ»S ABCD +)
8
AB =
)
3
CD =
)
5 3
AD =
)
90
ADC
°
Ð
=
)
30
BAD
°
Ð
=
)/ E)F Üz
2
5
AE
AD
=
uuur
uuur
)
1
2
AF
AB
=
uuur
uuur
)ß
AEF
△
á EF âãİ
PEF
!
)ôÏ
4 3
PC =
T
(1)eŸ
 EF
PD
^
f
(2)dc? PCD c? PBF ij$B?9$klmT
3TU2024·ıIJăį·%&ĭ^_`)=@ A)B)C)D)E)F IĬ/$ij?+)œ»S ABCD œ»
S ADEF ēIàýS)
/ /
,
/ /
EF
AD BC
AD )
4,
2
AD
AB
BC
EF
=
=
=
=
)
10,
2 3
ED
FB
=
=
) M I AD
$+/T
(1)eŸ
/ /
BM
c?CDE f
(2)dB?9 F
BM
E
-
-
$klmT
      
      
4TU2023·«ĮõⅠį·%&ĭ^_`)=kœab
1
1
1
1
ABCD
A B C D
-
+)
1
2,
4
AB
AA
=
=
T/
2
2
2
2
,
,
,
A B C D
H¦=a
1
1
1
,
,
AA BB CC ,
1
DD €)
2
2
2
2
1,
2,
3
AA
BB
DD
CC
=
=
=
=
T
  
(1)eŸ
2
2
2
2
B C
A D
Ĵ
f
(2)/ P =a
1
BB €)øB?9
2
2
2
P
A C
D
-
-
I150°ù)d
2
B P T
5TU2023·«ĮõⅡį·%&ĭ^_`)Va A
BCD
-
+) DA
DB
DC
=
=
) BD
CD
^
)
60
ADB
ADC
Ð
= Ð
=
o )E I BC $+/T
(1)eŸ
 BC
DA
^
f
(2)/ F Üz EF
DA
=
uuur
uuur )dB?9 D
AB
F
-
-
$klmT
      
      
6TU2023·ıIJąį·%&ĭ^_`)=Va P
ABC
-
+) AB
BC
^
)
2
AB =
)
2 2
BC =
)
6
PB
PC
=
=
)BP)AP)BC $+/H¦I D)E)O)
5
AD
DO
=
)/ F = AC €) BF
AO
^
.
  
(1)eŸ
/ /
EF
c? ADO f
(2)eŸ
c? ADO ^ c? BEFf
(3)dB?9 D
AO
C
-
-
$klm.
7TU2023·ıIJăį·%&ĭ^_`)=Vab
1
1
1
ABC
A B C
-
+)
1AC ^ ½? ABC)
1
90 ,
2
ACB
AA
Ð
=
°
=
)
1A xc?
1
1
BCC B $‘’I 1T
  
(1)eŸ
1AC
AC
=
f
(2)À-
1
AA 
1
BB $‘’I 2)d
1
AB c?
1
1
BCC B ij9$klmT
      
      
8TU2022·ıIJąį·%&ĭ^_`)œ? ABCD +)
,
,
AD
CD AD
CD
ADB
BDC
^
=
Ð
= Ð
)E I AC $
+/T
(1)eŸ
c? BED ^c? ACD f
(2)–
2,
60
AB
BD
ACB
=
=
Ð
=
°)/ F = BD €)ø AFC
V
$?ò¥ù)dCF c? ABD ij$9$kl
mT
9TU2022·ıIJăį·%&ĭ^=œa P
ABCD
-
+) PD ^ ½?
,
,
1,
2,
3
ABCD CD
AB AD
DC
CB
AB
DP
=
=
=
=
=
Ĵ
T
(1)eŸ
 BD
PA
^
f
(2)d PD c? PAB ij$9$klmT
      
      
10TU2022·«%&ıIJⅠį·%&ĭ^_`)hVab
1
1
1
ABC
A B C
-
$I 4)
1A BC
V
$?I 2 2 T
(1)d A xc?
1A BC $‘’f
(2)– D I
1AC $+/)
1
AA
AB
=
)c?
1A BC ^ c?
1
1
ABB A )dB?9 A
BD
C
-
-
$klmT
11TU2022·«%&ıIJⅡį·%&ĭ^_`) PO *Va P
ABC
-
$%) PA
PB
=
) AB
AC
^
)E * PB
$+/T
  
(1)eŸ
/ /
OE
c? PAC f
(2)g
30
ABO
CBO
Ð
= Ð
=
° )
3
PO =
)
5
PA =
)dB?9C
AE
B
-
-
$klmT