重难点22立体几何必考经典解答题全归类十大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点22 立体几何必考经典解答题全归类【十大题型】

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(1)’m
𝑂𝐷∥Ú/𝑃𝐴𝐵
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Ĭk 1-1?2024·ŌōŎŏ·jA—Ž¶·𝑃−𝐴𝐵𝐶"W𝐵𝐴⊥𝐵𝐶W𝑃𝐵⊥Ú/𝐴𝐵𝐶W-𝐸—Ú/𝐴𝐵𝐶UWŋ
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𝐴𝐸⊥𝐵𝐸
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3 ›W’Ž¶·𝐸−𝑃𝐶𝐵#$Ÿ
      
      
Ĭk 1-2?2024·Őő·jňʼnŊAµ3W !𝐴𝐵𝐶𝐷𝐸𝐹"W­/𝐴𝐵𝐶𝐷wŒ(w 2 #œ±WÚ/𝐶𝐷𝐸𝐹⊥
Ú/𝐴𝐵𝐶𝐷WÚ/𝐵𝐶𝐹⊥Ú/𝐴𝐵𝐶𝐷W∠𝐷𝐴𝐵=
π
3.
(1)m‰
𝐶𝐹⊥Ú/𝐴𝐵𝐶𝐷
(2)ĝ𝐷𝐸=
13
2 WÚ/𝐴𝐷𝐸QÚ/𝐵𝐶𝐹#ô,w
π
6W’©¶·𝐸−𝐴𝐵𝐶𝐷#$.
Ĭk 1-3?2024·ŔŕŖŗŘŌ·jňʼnŊAµ3W©¶·𝑃−𝐴𝐵𝐶𝐷#­/𝐴𝐵𝐶𝐷Rř±W𝑃𝐷⊥Ú/
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2𝐴𝐷=
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(1)m‰
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3.
  
(1)’m
𝐵𝐹//Ú/𝐴𝐷𝐸
(2)ĝ¥&𝐸𝐶QÚ/𝐹𝐵𝐷×´,#Ċöîw4 10
15 W’&'𝐶𝐹#(.
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Ž,±W𝐷𝐶= 2𝐴𝐵= 2,𝐶𝐵= 𝐶𝐸W-𝐹w¶𝐵𝐸Ą#ğ-Ÿ
(1)m‰
𝐷𝐶⊥Ú/𝐵𝐶𝐸
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Ĭk 2-2?2024·Şş·jňʼnŊAµ3W𝐴𝐸⊥Ú/𝐴𝐵𝐶𝐷W𝐸,𝐹—Ú/𝐴𝐵𝐶𝐷#ŜÆW𝐴𝐸//𝐷𝐹W𝐴𝐷//𝐵𝐶W
𝐴𝐷⊥𝐴𝐵W𝐴𝐷= 𝐴𝐵=
1
2𝐵𝐶= 1.
      
      
(1)ĝ𝐵,𝐸,𝐹,𝐶©-—Ŝ`Ú/UW’&'𝐸𝐹#(
(2)ĝ𝐷𝐹= 2𝐴𝐸WÚ/𝐵𝐸𝐹QÚ/𝐵𝐶𝐹#ô,w30∘W’&'𝐴𝐸#(.
Ĭk 2-3?2024·Şő·jňʼnŊAµ3 1W—ŠŒ±𝐴𝐵𝐶𝐷𝑃"Wă¦ù,&𝐴𝐷W𝐴𝐷//𝐵𝐶W𝐴𝐷⊥𝐷𝐶W
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2AWŋ𝑃𝐵= 2 2,𝐸w𝐴𝐷#"-W𝑀w𝐵𝐶#"-W-𝑁—&'𝑃𝐸Ą.
  
(1)’m
Ú/𝑃𝐴𝐷⊥Ú/𝐴𝐵𝐶𝐷
(2)ĝÚ/𝐴𝑀𝑁¯Ú/𝑃𝐴𝐵#ô,#õöîw3 87
29 W’&'𝐸𝑁#(.
 3  *+,%
Ń 3?2024·ţŤ·jAµ3W—Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"Wד¶(ťqW𝐶𝐵1 ∩𝐵𝐶1 = 𝑂W∠𝐴𝐵𝐵1
= 60°W𝐶𝐵⊥𝐵𝐵1.
      
      
  
(1)m‰𝐴𝑂⊥Ú/𝐵𝐵1𝐶1𝐶.
(2)ĝ/,𝐶1−𝐴1𝐵1−𝐵#Ċöî.
Ĭk 3-1?2024·Ŧ;ŕŧ·ŽjAµ3W—©¶Ũ𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"W­/©Œ± ABCD wœ±W
∠𝐴𝐵𝐶= 60°,𝐴𝐵= 2𝐴𝐴1 = 2𝐴1𝐵1,𝐴𝐴1 ⊥Ú/ ABCD.
  
(1)m‰
𝐵𝐷⊥𝐶𝐶1
(2)ĝ M R¶ BC Ą#-WŋĨÿ
𝐵𝑀
𝐵𝐶=
2
3W’/,𝑀−𝐴𝐷1−𝐷#õöî.
Ĭk 3-2?2024·ŔŕŖĖŝ·ŽjAµ3W—©¶·𝑃−𝐴𝐵𝐶𝐷"W𝐴𝐷 // 
𝐵𝐶,∠𝐵𝐴𝐷= 90°,𝐴𝐷= 2𝐵𝐶= 4,𝐴𝐵= 2W𝑃𝐴= 2 2,∠𝑃𝐴𝑂= 45°Wŋ𝑂R𝐴𝐷#"-.
      
      
  
(1)’m
Ú/𝑃𝑂𝐶⊥Ú/𝐴𝐵𝐶
(2)ĝ/,𝑃−𝐴𝐷−𝐵#Ėƒw120∘W’¥&𝑃𝐵QÚ/𝑃𝐴𝐷×´,#õöî.
Ĭk 3-3?2024·ŐőũŇ·jňʼnŊAµ3×òW—Ūū±𝐴𝐵𝐶𝐷"W𝐴𝐵//𝐶𝐷W𝐴𝐷= 𝐴𝐵= 𝐵𝐶= 2W
𝐶𝐷= 4WE w CD "-WAE Q BD qâ^- OWX △𝐴𝐷𝐸š AE 4ŢWŬ- D þŭ- P #‡ˆ?𝑃∉Ú/
𝐴𝐵𝐶𝐸AŸ
(1)’m
Ú/𝑃𝑂𝐵⊥Ú/ PBC
(2)ĝ𝑃𝐵=
6WŮĠġ&' PB ĄRłļ—`- Q?Ĺçß-AWŬē¥& PC QÚ/𝐴𝐸𝑄×´,#Ċöî
w 15
5 Wĝļ—W’ Q —&' PB Ą#‡ˆĝĹļ—Wů‰ĒŸ
 4  *+-.&./#01%
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𝐴𝐴1 ⊥Ú/ ABCW- MWNWD z»w ABWACWBC #"-W𝐴1𝐵⊥𝐴𝐶1Ÿ
      
      
(1)m‰
𝐶𝐶1∥Ú/𝐴1𝑀𝑁
(2)’¥&𝐴1𝐷QÚ/𝐴1𝑀𝑁×´,#Ċöî
(3)’- D þÚ/𝐴1𝑀𝑁#01Ÿ
Ĭk 4-1?2024·Ųō·ŽjAµ3WŒ(w 4 #ávĊŽ,±𝐴𝐵𝐶W𝐵𝐶𝐷חÚ/ųqú¥W𝐸W𝐹z»
w𝐵𝐶W𝐶𝐷#"-W-𝐺—¶𝐴𝐷ĄW𝐴𝐺= 2𝐺𝐷W¥&𝐴𝐵QÚ/𝐸𝐹𝐺qâ^-𝐻.
(1)m‰
𝐵𝐷//𝐺𝐻
(2)’¥&𝐵𝐷QÚ/𝐸𝐹𝐺#01.
Ĭk 4-2?2024·ĄŤ·ŽjAµ3W—¥Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"W𝐴𝐴1 = 𝐴𝐵= 2W𝐴𝐶= 1W∠𝐴𝐶𝐵= 90°W
D R¶ AB Ą#`-Ÿ
      
      
(1)ĝ𝐴𝐷= 𝐷𝐵W’Ö/¥&𝐵1𝐷Q𝐴1𝐶1×´#,#Ėƒ
(2)ĝ𝐶𝐷⊥𝐵1𝐷W’- B þÚ/𝐵1𝐶𝐷#01Ÿ
Ĭk 4-3?2024·Ťő·jňʼnŊAµ3W—¥©¶¸𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"W­/©Œ±𝐴𝐵𝐶𝐷wū±W
𝐴𝐷//𝐵𝐶,𝐴𝐵= 𝐴𝐷= 2W𝐵𝐷= 2 2,𝐵𝐶= 4.
(1)m‰
 𝐴1𝐵1 ⊥𝐴𝐷1;
(2)ĝ¥& 𝐴𝐵QÚ/ 𝐵1𝐶𝐷1×´,#Ċöîw 
6
6 W- 𝑀w&' 𝐵𝐷Ą`-W’-𝑀þÚ/ 𝐵1𝐶𝐷1#01.
 5   !"#23%
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3.|- D w𝐴1𝐶1
Ą#`-Wā DWA 2Ú/𝐵𝐶𝐶1𝐵1#ú/𝛼W
      
      
(1)Ŷ°Ú/𝛼QĊŽ¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1¢/#â&?ŷŸ23Ź7WŜm‰A
(2)ĝ𝐴1þÚ/𝛼#01w 3
2 W’ AC QÚ/𝛼×´,#Ċöî.
Ĭk 5-1?2024·ŲōŲŵ·jňʼnŊA—©¶·𝑃−𝐴𝐵𝐶𝐷"W­/𝐴𝐵𝐶𝐷w¥,ū±W𝐶𝐷//𝐴𝐵W
∠𝐴𝐵𝐶= 90°W𝐴𝐵= 2𝐶𝐷WŽ¶·𝐵−𝑃𝐶𝐷#$w2 2
3 WÚ/𝑃𝐴𝐷QÚ/𝑃𝐵𝐶#â&w𝑙.
  
(1)’©¶·𝑃−𝐴𝐵𝐶𝐷#$Wé—KźĄŶ°â&𝑙?÷ŷŸ23Ź7A
(2)ĝ𝐴𝐵= 2𝐵𝐶= 4W𝑃𝐴= 𝑃𝐷WŋÚ/𝑃𝐴𝐷⊥Ú/𝐴𝐵𝐶𝐷W—𝑙ĄRłļ—-𝑁WŬÚ/𝑃𝐷𝐶QÚ/𝐷𝐶𝑁×´
,#õöîw 6
3 Żĝļ—W’𝑃𝑁#()ĝĹļ—Wżů‰Ē.
Ĭk 5-2?2023·ŲŅ·jňʼnŊAĿ ©¶·𝑃−𝐴𝐵𝐶𝐷"W­/𝐴𝐵𝐶𝐷w¥,ū±W𝑃𝐴⊥Ú/𝐴𝐵𝐶𝐷W
𝐴𝐷∥𝐵𝐶W𝐴𝐵⊥𝐴𝐷W𝑃𝐴= 𝐴𝐷= 4W𝐵𝐴= 𝐵𝐶= 2WM w𝑃𝐴"-Wā CWDWM #Ú/Ž©¶·𝑃−𝐴𝐵𝐶𝐷×
      
      
ē#Ž/w𝛼Ÿ
(1)ĝ𝛼Q¶𝑃𝐵â^- FWŶ°Ž/𝛼WŷŸ23Ź7?Ĺů‰ĒAW’- F #‡ˆ
(2)’Ú/𝐶𝐷𝑀QÚ/𝑃𝐵𝐶״ύ/,#õöîŸ
Ĭk 5-3?2024·ŲŅŐž·jňʼnŊAĿ ©¶·𝑃−𝐴𝐵𝐶𝐷"W­/𝐴𝐵𝐶𝐷w¥,ū±W𝑃𝐴⊥Ú/
𝐴𝐵𝐶𝐷W𝐴𝐷∥𝐵𝐶W𝐴𝐵⊥𝐴𝐷W𝑃𝐴= 𝐴𝐷= 4W𝐵𝐴= 𝐵𝐶= 2W𝑀w𝑃𝐴"-Wā𝐶W𝐷W𝑀#Ú/Ž©¶·𝑃−𝐴𝐵𝐶𝐷
×ē#Ž/w𝛼Ÿ
(1)ĝ𝛼Q¶𝑃𝐵â^-𝐹WŶ°Ž/𝛼WŷŸ23Ź7?Ĺů‰ĒAWém‰
𝑃𝐵
𝐹𝐵= 3Ÿ
(2)’ſ/𝐴𝐵𝐶𝐷𝑀𝐹#$Ÿ
      
      
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þŽ¶·𝑀−𝐵𝐶𝐷WÃ" △𝑀𝐵𝐷R45Ƃ# △𝐴𝐵𝐷Wā M 2𝐵𝐷#ú&Wúÿw HW𝑀𝐶=
10.
(1)’m
𝑀𝐻⊥𝐶𝐷
(2)ā H 2𝑀𝐵#ú&Wúÿw NW’- N þÚ/𝑀𝐶𝐷#01.
Ĭk 6-1?2023·ƃƄ·`jAµ3ƅ×ò#Ċ¤±𝐴𝐴′𝐴′1𝐴1"W𝐴𝐴1 = 12,𝐴𝐵= 𝐴1𝐵1 = 3,𝐵𝐶= 𝐵1𝐶1 = 4W
ù,&𝐴𝐴′1z»â𝐵𝐵1,𝐶𝐶1^-𝑃,𝑄WXĊ¤±𝐴𝐴′𝐴′1𝐴1š𝐵𝐵1,𝐶𝐶145Ŭē𝐴𝐴1Q𝐴′𝐴′1SsWƆ´µ3Ƈ×
ò#Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1.-𝑀—¶𝐴𝐶ĄWŋ𝐴𝑀=
15
7 .
(1)m‰
𝐵𝑀 ∥Ú/𝐴𝑃𝑄
(2)’Ž¶·𝑀−𝐴𝑃𝑄#$.
      
      
Ĭk 6-2?2024·Lƈ·jňʼnŊAµ3W—ū±𝐴𝐵𝐶𝐷"W𝐴𝐵∥𝐶𝐷,∠𝐵𝐴𝐷= 90°,𝐶𝐷= 2𝐴𝐷= 2,𝐴𝐵= 3,𝐸
w&'𝐴𝐵ĄƉc-𝐴#Žz-WX △𝐴𝐷𝐸šį𝐷𝐸45Wēþ©¶·𝐴−𝐵𝐶𝐷𝐸WŬÚ/𝐴𝐷𝐸⊥Ú/𝐵𝐶𝐷𝐸,𝑃
w&'𝐶𝐸Ą#-Ÿ
(1)’m
𝐴𝐷⊥𝐴𝑃
(2)Rłļ—-𝑃WŬē¥&𝐴𝑃QÚ/𝐴𝐵𝐸×´,#Ċöîw 6
6 ?ĝļ—W’°&'𝐸𝑃#(ĝĹļ—Wżů
‰ĒŸ
Ĭk 6-3?2024·ƊƋsƌ·ŽjAµ3`
Ū¥, △𝐴𝐵𝐶"𝐴𝐶⊥𝐴𝐵ŋ𝐴𝐶= 2Wz»šŽ,±ŽŒO
Ā2Ūū±𝐴𝐵𝐵2𝐴2,𝐵𝐶𝐶2𝐵3,𝐶𝐴𝐴3𝐶3Ŭē𝐴𝐴2 = 𝐵𝐵2 = 𝐶𝐶2 = 1,∠𝐶𝐴𝐴3 = ∠𝐵𝐴𝐴2 =
π
3WšŽŒ𝐴𝐵,𝐵𝐶,𝐶𝐴4
5WŬē𝐴2𝐴3,𝐵2𝐵3,𝐶2𝐶3WSs^𝐴1,𝐵1,𝐶1Wµ3
(1)’m
𝐴𝐴1 ⊥𝐵1𝐶1Ÿ
(2)’¥&𝐶𝐶1QÚ/𝐴𝐴1𝐵1𝐵×´,𝜃#ĊöîŸ
      
      
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UWŋĨÿÚ/𝑃𝐴𝐸⊥Ú/𝑃𝐵𝐸,𝐵𝐴ú¥^𝐵𝐶Ÿ
(1)Þ∠𝐴𝐵𝐸∈π
8 , π
3 ›W’-𝐸#67()
(2)ލ/,𝐸−𝑃𝐴−𝐵#õöîw 3
3 ›W’Ž¶·𝐸−𝑃𝐶𝐵#$Ÿ
Ĭk 7-1?2024·Lƈ·jňʼnŊAµ3W—¥Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"W𝐴𝐵= 𝐴𝐶= 𝐴𝐴1 = 3,𝐵𝐶= 3 2,𝐸R
Æ/𝐴𝐴1𝐶1𝐶U#ğ-?ƎƏŒƐAWD w𝐵𝐶1#"-W𝐵1𝐸⊥𝐴1𝐷Ÿ
(1)’m
- E #67w&'𝐴𝐶1
(2)’Ú/𝐴𝐷𝐸QÚ/𝐴𝐵𝐶ô,#ĖƒŸ
Ĭk 7-2?2024·Lƈ·jňʼnŊAµ3W©Œ±𝐴𝐵𝐷𝐶wģŨ𝑂1𝑂2#ƑŽ/W𝐴𝐶= 2𝐵𝐷WģŨ#ƒ&Q
      
      
­/×´#,w 45°Wƒ&(w 2W𝐸R𝐵𝐷#"-Ÿ
(1)Ŀ ģ𝑂2Uļ—-𝐺WŬē𝐷𝐸⊥Ú/𝐵𝐸𝐺W2°-𝐺#67?Ɠ°JāĦA
(2)-𝐾Rģ𝑂2Ą#`-?ĹŜ^𝐴W𝐶AW2𝐶𝐾= 𝐴𝐶W’Ú/𝐴𝐵𝐾QÚ/𝐶𝐷𝐾×´,#ĊöîŸ
Ĭk 7-3?2024·ƔőĤƕ·jňʼnŊAµ3W©/𝐴𝐵𝐶𝐷#Ɩà¶(«^ 2W𝑀，𝐺，𝑁z»R¶𝐴𝐵，
𝐵𝐶，𝐶𝐷#"-W𝑂，𝐸，𝐹z»w/𝐵𝐶𝐷W/𝐴𝐵𝐶W/𝐴𝐶𝐷#SƗ.
(1)’m
/𝑂𝐸𝐹///𝐴𝐵𝐷
(2)’Ú/𝑂𝐸𝐹QÚ/𝐴𝐵𝑁#ô,#õöî
(3)ŷƘ-𝐸，𝐹‡ˆĹĬW— △𝐵𝐶𝐷U?ƎƏŒƐAƙğ-𝑂WŬ¥&𝑀𝑁QÚ/𝑂𝐸𝐹ڙW’-𝑂67()
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"∠𝐸𝐵𝐶=
π
2Wŋ𝐴𝐵= 𝐵𝐶= 2𝐶𝐷= 4Ÿ
      
      
(1)|&'𝐵𝐸"-w𝐹Wm‰
𝐶𝐹∥Ú/𝐴𝐷𝐸
(2)—&'𝐴𝐵ĄRłļ—-𝑀WŬē-𝐵þÚ/𝐶𝐸𝑀#01^ 2
2 Wµƞļ—W’𝑀𝐵#(Ÿ
Ĭk 8-1?2024·ŴŵƟŅ·`jAµ3×òw¥©¶¸𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1,𝐴𝐵= 𝐴𝐷= 2 2,𝐶𝐵= 𝐶𝐷= 4,𝐴𝐴1
= 4W∠𝐵𝐶𝐷= 60°W𝑀,𝑀1z»R&'𝐵𝐶,𝐵1𝐶1#"-.
(1)m‰:𝐵𝐶⊥Ú/𝑀𝑀1𝐷
(2)’¥& BC QÚ/𝐵𝐷𝐴1×´,#ĊöîWéĠġ&' BC ĄRłļ—-𝑃WŬē𝑃𝐵1//Ú/𝐵𝐷𝐴1Wĝļ—W
’° BP #îWĝĹļ—Wżů‰Ē.
Ĭk 8-2?2024·ŔŕŖƠơƢ·`jAµ3 1W—Ú™©Œ±𝐴𝐵𝐶𝐷"W𝐷= 60°,𝐷𝐶= 2𝐴𝐷= 2WX △𝐴𝐷𝐶
š𝐴𝐶4ŢWŬ- D þŭ- P ‡ˆWŋ𝑃𝐶⊥𝐵𝐶Wă¦𝑃𝐵ēŽ¶·𝑃−𝐴𝐵𝐶Wµ3 2Ÿ
      
      
(1)m‰
Ú/𝑃𝐴𝐵⊥Ú/𝐴𝐵𝐶
(2)—&'𝑃𝐶ĄRłļ—- MWŬÚ/𝐴𝑀𝐵QÚ/𝑀𝐵𝐶#ô,#õöîw
5
8Wĝļ—W’°
|𝑃𝑀|
|𝑃𝐶|#îWĝĹļ
—Wżů‰ĒŸ
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𝑃𝐴= 𝐴𝐷= 3𝐷𝑄= 3W-𝐸.𝐹z»w&'𝑃𝐵.𝐶𝑄#"-Ÿ
(1)’m
𝐸𝐹//Ú/𝑃𝐴𝐷𝑄
(2)’Ú/𝑃𝐶𝑄QÚ/𝐶𝐷𝑄ô,#õöî
(3)&'𝑃𝐶ĄRłļ—-𝑀WŬē¥&𝐴𝑀QÚ/𝑃𝐶𝑄×´,#ĊöîR 42
7 Wĝļ—’°
𝑃𝑀
𝑀𝐶#îWĝĹļ—W
ů‰ĒŸ
      
      
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𝐴1𝐶1#"-.
(1)m‰
Ú/𝐴1𝐶𝐷⊥Ú/𝐴1𝐴𝐵𝐵1
(2)’¥&𝐸𝐹Q𝐴1𝐵1×´,#Ċöî.
Ĭk 9-1?2024·ńŅƣƀ·jňʼnŊAµ3W—©¶·𝑃−𝐴𝐵𝐶𝐷"W­/𝐴𝐵𝐶𝐷Rڙ©Œ±W
∠𝐴𝐵𝐶= 120∘W𝐴𝐵= 1W𝐵𝐶= 4W𝑃𝐵= 2 3W𝑃𝐷⊥𝐶𝐷W- E R𝐵𝐶#"-Wŋ𝑃𝐸⊥𝐸𝐷Ÿ
  
(1)’m
𝑃𝐸⊥𝐴𝐷
(2)’- E þÚ/𝑃𝐴𝐷#01Ÿ
Ĭk 9-2?2024·Lƈ·jňʼnŊAµ3W—©¶·𝑃−𝐴𝐵𝐶𝐷"W𝐴𝐵∥𝐶𝐷Wŋ𝐴𝐵⊥𝐴𝑃W𝐶𝐷⊥𝐷𝑃.
      
      
(1)m‰
Ú/𝑃𝐶𝐷⊥Ú/𝑃𝐴𝐷
(2)ĝ𝑃𝐴= 𝑃𝐷= 𝐴𝐵W𝑃𝐴⊥𝑃𝐷W’𝑃𝐵QÚ/𝐴𝐵𝐶𝐷×´,#Ėƒ.
Ĭk 9-3?2024·ƤŖ·jňʼnŊA.µ3W­/𝐴1𝐵1𝐶1𝐷1ƥD—­/𝛼Ą#ƦƧVƨƩwĊ¤±𝐴𝐵𝐶𝐷W
ƶ𝐴𝐴1W𝐵𝐵1W𝐶𝐶1W𝐷𝐷1qųڙ.
(1)m‰
­/©Œ±𝐴1𝐵1𝐶1𝐷1Rڙ©Œ±
(2)ĝĿ ©àƶú¥^/𝐴𝐵𝐶𝐷Wŋ𝐴𝐴1 = 𝐷𝐷1 = 4W𝐵𝐵1 = 𝐶𝐶1 = 𝐴𝐵= 2.ƪƫÈVƨ"ƧW’ÈVƨ
ÏĖƦƧ$𝑉Ƭè›Æ/𝐵𝐵1𝐶1𝐶Q­/𝛼×´,𝜃#õöî?Ƨ/ڙ^­/𝛼A.
 10  BC.DEF# !%
Ń 10?23-24 `F·©ƀ´«·ƭƮANƯ^ưÚ/"#õöDW“Žư*+"#Ž/,õöD
µ3 1WĒċ&𝑃𝐴W𝑃𝐵W𝑃𝐶Ɔ´#Ž/,𝑃−𝐴𝐵𝐶W∠𝐴𝑃𝐶= 𝛼W∠𝐵𝑃𝐶= 𝛽W∠𝐴𝑃𝐵= 𝛾W/,𝐴−𝑃𝐶−𝐵#
Ėƒw𝜃WĞcos𝛾= cos𝛼cos𝛽+ sin𝛼sin𝛽cos𝜃Ÿ
      
      
?1AÞ𝛼.𝛽∈0, π
2 ›Wm‰tĄŽ/,õöD
?2Aµ3 2WڙƱ/𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"WÚ/𝐴𝐴1𝐶1𝐶⊥Ú/𝐴𝐵𝐶𝐷W∠𝐴1𝐴𝐶= 60°W∠𝐵𝐴𝐶= 45°W
①’∠𝐴1𝐴𝐵#õöî
②—¥&𝐶𝐶1ĄRłļ—-𝑃WŬ𝐵𝑃//Ú/𝐷𝐴1𝐶1Żĝļ—W’°-𝑃#‡ˆĝĹļ—Wů‰ĒŸ
Ĭk 10-1?24-25 ŽĄ·ƤŖ·ƲŮAĿ ΩR¶(w 2#Ċ©/𝐴𝐵𝐶𝐷W|Ω#©vƳ-þÚ/
𝛼#01×Ɔ´#ƴsw𝑀Wĝ𝑀"Ƶƶ#v
w𝑘WĞû𝛼wΩ#𝑘Ʒ0Ú/W𝑀wΩ#𝑘Ʒ0ƴ.
(1)ĝ𝛼wΩ# 1 Ʒ0Ú/ŋ 1 Ʒ0ƴw{𝑎}W’𝑎#ד¬ĈîtƬq¿#𝛼#v

(2)Ŀ 𝛽wΩ# 4 Ʒ0Ú/Wŋ-𝐴Q-𝐵,𝐶,𝐷z»‡^𝛽#áÆ.ĝΩ# 4 Ʒ0ƴw{𝑏,2𝑏,3𝑏,4𝑏}WÃ"-𝐴
þ𝛽#01w𝑏W’Ú/𝐵𝐶𝐷Q𝛽ô,#õöî.
Ĭk 10-2?23-24 `F·Ŧ;Ž‰·ƭƮAƸƹ
ƺƻ
“|𝑃wſ/𝑀#`vƳ-WDEſ/𝑀
—-𝑃Ƽ#1ƽĤƾw 1−
1
2π(∠𝑄1𝑃𝑄2 + ∠𝑄2𝑃𝑄3 + ∠𝑄3𝑃𝑄4 + ⋯+ ∠𝑄𝑘−1𝑃𝑄𝑘+ ∠𝑄𝑘𝑃𝑄1)WÃ" 𝑄𝑖
𝑖= 1，2，⋯，𝑘，𝑘≥3 wſ/𝑀#דQ-𝑃qƿ#Ƴ-WŋÚ/𝑄1𝑃𝑄2WÚ/ 𝑄2𝑃𝑄3W…WÚ/
𝑄𝑘−1𝑃𝑄𝑘¯Ú/𝑄𝑘𝑃𝑄1wſ/𝑀#דt𝑃wǀ-#/. ”Ŀ —¥©¶¸𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"W­/
𝐴𝐵𝐶𝐷wœ±.𝐴𝐴1 = 𝐴𝐵. ?,#ĥpťǁǂ)ǃA
(1)ĝ𝐴𝐶= 𝐵𝐷W’©¶¸𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1—Ƴ-𝐴Ƽ#1ƽĤƾ
      
      
(2)ĝ©¶¸𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1—Ƴ-𝐴Ƽ#1ƽĤƾw
1
3W’𝐵𝐶1QÚ/𝐴𝐶𝐶1#ô,#Ċöî
(3)Žë©/𝐴1−𝐴𝐵𝐷WĝÈ©/—-𝐴1Ƽ#1ƽĤƾw
7
12，𝐴𝐶1QÚ/𝐴1𝐵𝐷â^-𝐺Wm‰
𝐴𝐺
𝐴𝐶1 =
1
3.
Ĭk 10-3?23-24 `F·Şő(DŽ·ƭƮA*+#DžĤ:R !ĺĻ#S…UVWĤƾdžŶ*+#Dž
Ĥ:WLJD
ſ/Ƴ-#Ĥƾ^2πQſ/—È-#/,_¯#LjWÃ"ſ/#/#U,ljĂſ/
#/,W,)ǂ)ǃ.ѵ
Ċ©/ƖvƳ-ť“ 3 v/,WƖv/,ťw
π
3WNJÃÇvƳ-#Ĥƾť
w2π−3 ×
π
3 = π.µ3W—¥Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"W-𝐴#Ĥƾw
2π
3 W𝑁W𝑀z»w𝐴𝐵W𝐶𝐶1#"-Wŋ
𝐴𝐵= 𝐴𝐶.
(1)m‰
𝐶𝑁⊥Ú/𝐴𝐵𝐵1𝐴1
(2)ĝ𝐴𝐴1 =
2𝐴𝐵W’/,𝐵1−𝐴𝑀−𝐶1#õöî
(3)¢/HāăNjĬ±¬tĬwnj/#ſ/ûwÓÔſ/.^ÓÔſ/“ǍǎǏǐD
|ÓÔſ/
#Ƴ-
w𝐷W¶
w𝐿W/
w𝑀WĞ“
𝐷−𝐿+ 𝑀= 2.óèDŮm‰
ÓÔſ/#ǑĤƾ?ſ
/“Ƴ-#Ĥƾ_¯AR–
.
`.Ô
      
      
1Ÿ?2024·UƚƛƎǒ·ŽjAµ3WĿ Ċ¤± ABCD wģ¸#ƑŽ/W𝐴𝐵= 𝐵𝐶= 2WEWF wĄ­/ģ
ǓĄ#ávğ-Wŋ EF āĄ­/#ģƗ GWĝ𝐴𝐵⊥𝐸𝐹WЎ¶·𝐴−𝐵𝐸𝐹#$w?    A
  
AŸ
2
3
BŸ
4
3
CŸ2 2
3
DŸ2 3
3
2Ÿ?2024·Lƈ·jňʼnŊAĿ Ċ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1#¶(w 4W-𝑀∈Ú/𝐴1𝐵𝐶𝐷1Wŋ
𝑀𝐵
𝑀𝐴=
1
3WĞ-
M #67#()w?    A
AŸ 34π
BŸ 17π
CŸ 34π
2
DŸ 17π
2
3Ÿ?2024·Ōōǔő·ŽjAµ3×òWĊ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1#¶(w 1W-𝐸,𝐹,𝐺z»w𝐵𝐶,𝐶𝐶1,𝐵𝐵1#"
-WĞFǕů@Ċü#R?    A
AŸ¥&𝐷1𝐷Q¥&𝐴𝐹ú¥
BŸ¥&𝐴1𝐺QÚ/𝐴𝐸𝐹ڙ
CŸŽ¶·𝐹−𝐴𝐵𝐸#$w
1
8
DŸ¥& BC QÚ/𝐴𝐸𝐹×´#,w45°
4Ÿ?2024·Lƈ·jňʼnŊAĿ  △𝐴𝐵𝐶"W𝐴𝐶= 1,𝐴𝐵= 2,𝐵𝐶=
3W—&'𝐴𝐵Ąë`-𝑀Wă¦𝐶𝑀Wµ
3①×òŸX △𝐴𝐶𝑀š¥&𝐶𝑀4ŢWŬē-𝐴þŭ𝐴′#‡ˆWè› △𝐵𝐶𝑀Uºļ—`-𝑁WŬē𝐴′𝑁⊥Ú
/𝐵𝐶𝑀,𝑁𝐶=
7
3 Wµ3②×òWĞ𝐴′𝑀#î¬Ĉw?    A
      
      
AŸ
2
5
BŸ
3
5
CŸ
4
5
DŸ1
5Ÿ?2024·Şş·jňʼnŊAµ3×ò#ſ/RĒ­/w𝐴𝐵𝐶𝐷#(¤ǖŽ/𝐴𝐸𝐶1𝐹׎ēþ#WÃ"
𝐴𝐵= 4W𝐵𝐶= 2W𝐶𝐶1 = 3W𝐵𝐸= 1WĞ-𝐶þÚ/𝐴𝐸𝐶1𝐹#01w?    A
AŸ 2
2
BŸ3 2
2
CŸ4 33
11
DŸ 33
11
6Ÿ?2024·ŲŅőǗ·`jA—Œ(w 4 #œ±𝐴𝐵𝐶𝐷"W∠𝐴𝐵𝐶= 120°ŸXœ±šù,&𝐴𝐶45´Ėƒw30°
#/,𝐵′−𝐴𝐶−𝐷Ÿĝ-𝐸w𝐵′𝐶#"-W𝐹wŽ¶·𝐵′−𝐴𝐶𝐷¢/Ą#ğ-WŋǑĨÿ𝐴𝐶⊥𝐸𝐹WĞ-𝐹67
#()w?    A
AŸ4+ 6−2
2
BŸ4+ 6+ 2
2
CŸ4 +
6−2
DŸ4 +
6 +
2
7Ÿ?2024·ŔŕŖƠơƢ·ŽjAĿ ©¶·𝑃−𝐴𝐵𝐶𝐷#­/wĊ¤±W𝑃𝐷⊥­/𝐴𝐵𝐶𝐷,𝑃𝐷= 𝐴𝐷W-𝐸R&
'𝑃𝐵Ą#ğ-WĞ¥&𝐷𝐸QÚ/𝑃𝐵𝐶×´,#ÏĖîw?    A
AŸ
π
6
BŸ
π
4
CŸ
π
3
DŸ
π
2
8Ÿ?2024·ţŤ·jňʼnŊAµ3W—Ċ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"W𝐸W𝐹W𝑀W𝑁W𝐺W𝐻z»w¶𝐴𝐵W𝐵𝐶W
𝐴𝐷W𝐶𝐷W𝐴1𝐵1W𝐶1𝐷1#"-W𝑃w𝐷𝐻#"-Wă¦𝐸𝐻W𝐹𝐺Ÿù^*+ª÷á-𝐼W𝐽Wĝ&'𝐼𝐽ĄĹļ—
җ&'𝐸𝐻W𝐹𝐺Ą#-WĞû𝐼W𝐽á-“¬Ķ”WĞQ-𝐵1“¬Ķ”#-w?   A
      
      
AŸ𝐷
BŸ𝑃
CŸ𝑀
DŸ𝑁
.ſ
9Ÿ?2024·ŞşǘŇ·jňʼnŊAµ3WĿ Ċ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1#¶(w 2W𝐸W𝐹W𝐺z»w𝐴𝐷W𝐴𝐵W
𝐵1𝐶1#"-WtFů@Ċü#R?    A
AŸŽ¶·𝐶1−𝐸𝐹𝐺#$w
1
3
BŸ𝐴1𝐶⊥Ú/𝐸𝐹𝐺
CŸ𝐵𝐶1∥Ú/𝐸𝐹𝐺
DŸ/,𝐺−𝐸𝐹−𝐶#õöîw 3
6
10Ÿ?2024·ŲōǙŌ·jňʼnŊAµ3W—Ž¶·𝑃−𝐴𝐵𝐶"WÚ/𝑃𝐴𝐶⊥Ú/𝐴𝐵𝐶Wŋ △𝑃𝐴𝐶¯ △𝐴𝐵𝐶ť
RŒ(w2#ŒŽ,±W𝐷,𝐸,𝐹z»w𝐴𝐵,𝐴𝐶,𝐵𝐶#"-W𝐺w𝑃𝐵Ą#ğ-?Ĺçß-AWÚ/𝐸𝐹𝐺â¥&𝑃𝐴
^𝐻WĞFǕů@Ċü#R?    A
AŸÞ𝐺ĥğ›WǑ“𝐻𝐺//𝐴𝐵
BŸÞ𝐺ĥğ›W-𝐺þ¥&𝐴𝐶01#σîw 6
2
CŸļ—-𝐺WŬē𝐶𝐷⊥Ú/𝐸𝐹𝐺
      
      
DŸÞ𝑃𝐺> 𝐺𝐵›W¥&𝑃𝐶,𝐺𝐹,𝐻𝐸â^Ŝ`-
11Ÿ?2024·Ōō·jAµ3W—¥Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"W𝐴𝐵= 2,𝐴𝐵⊥𝐵𝐶W𝑃,𝑄z»w¶𝐵𝐶,𝐴1𝐶1Ą#ğ
-Wŋ𝐵𝑃= 𝜆𝐵𝐶W𝐶1𝑄= 𝜆𝐶1𝐴1W𝜆∈(0,1)WĞ?    A
AŸļ—𝜆Ŭē𝑃𝑄⊥𝐴1𝐵
BŸļ—𝜆Ŭē𝑃𝑄//Ú/𝐴𝐵𝐵1𝐴1
CŸĝ𝐵𝐵1,𝐵1𝐶1()wDîWĞ𝜆=
1
2›Ž¶·𝐵−𝐴1𝑃𝑄$ÏĖ
DŸÞ𝜆=
1
2›W¥&𝑃𝑄Q𝐴1𝐵×´,#õöî#σîw2 2
3
Ž.ǚ*
12Ÿ?2024·ƊƋƍŞ·ŽjA—¶(w 4 #Ċ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"W-𝐸R¶𝐵𝐵1#"-WĞ©/𝐴1𝐶1
𝐸𝐵#Ā¦nj#$w      Ÿ
13Ÿ?2024·©ƀ´«·jAµ3W—Ú™©Œ±𝐴𝐵𝐶𝐷"W𝐷𝐶=
2𝐴𝐷=
2𝐴𝐶= 4W𝐴𝐵= 4𝐴𝐹= 4𝐸𝐶W
ŋ EF â AC ^- GWƪš4Ź AC X △𝐴𝐷𝐶4ŢW¥ǛĨÿàĩ𝐷𝐶⊥𝐵𝐶Wè› EF #()w      Ÿ
14Ÿ?2024·Lƈ·jňʼnŊA—¶(w 2 #Ċ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1"Wğ-𝑀W𝑁z»—¶𝐵𝐶W𝐴𝐵ĄWŋ
Ĩÿ𝐴𝑁= 𝐵𝑀WÞ𝑉𝐷−𝑀𝑁𝐵1#$σ›W𝐵1𝑀QÚ/𝐴1𝑀𝑁×´,#ĊöîR      Ÿ
©.JK
      
      
15Ÿ?2024·ńŅƊǜ·jňʼnŊAµ3WĿ ſ/RĒĊ©¶· P-ABCD QĊ¤𝐴𝐵𝐶𝐷−𝐴1𝐵1𝐶1𝐷1¼s
ħ´#Wŋ𝑃𝐶=
3
2 𝐴𝐵.
  
(1)’m
𝑃𝐶//Ú/𝐴𝐷𝐶1𝐵1
(2)ĝ𝐴𝐵= 3W’©¶·𝑃−𝐴𝐷𝐶1𝐵1#$.
16Ÿ?2024·ńŅƣƀ·jňʼnŊAµ3W—©¶·𝑃−𝐴𝐵𝐶𝐷"W𝑃𝐷⊥Ú/ ABCDW𝑃𝐷= 𝐶𝐷= 2W
𝐴𝐷= 𝐴𝐵= 1W𝐴𝐵⊥𝐷𝐴W𝐴𝐵//𝐶𝐷W- M R¶ PC #"-Ÿ
(1)’m
𝐵𝑀//Ú/ PAD
(2)’Ú/ PAB QÚ/ BMD ״ύ/,#õöîŸ
17Ÿ?2024·Ɣő·jňʼnŊAµ3WĿ ©¶·𝑆−𝐴𝐵𝐶𝐷"W𝑆𝐴⊥Ú/𝐴𝐵𝐶𝐷W∠𝐶𝐷𝐴= ∠𝐷𝐶𝐵= 90°W
𝐵𝐶= 2𝐴𝐷= 2𝐶𝐷= 4Ÿ
      
      
  
(1)’m
Ú/𝑆𝐴𝐶⊥Ú/𝑆𝐴𝐵
(2)ĝÚ/𝑆𝐴𝐵QÚ/𝑆𝐶𝐷×´,#õöîw 6
6 W’&'𝑆𝐴#(Ÿ
18Ÿ?2024·©ƀǝƊ·jňʼnŊAµ3W—Ž¶¸𝐴𝐵𝐶−𝐴1𝐵1𝐶1"W¶𝐴𝐶,𝐶𝐶1#"-z»w𝐷,𝐸,𝐶1—Ú/𝐴𝐵𝐶
U#ċĆw DW △𝐴𝐵𝐶RŒ(w 2 #ŒŽ,±Wŋ𝐴𝐴1 = 2W- F —¶𝐵1𝐶1Ąĥğ?ƎƏß-AŸ
(1)ĝ-𝐹w¶𝐵1𝐶1#"-W’-𝐹þÚ/𝐵𝐷𝐸#01
(2)’ύ/,𝐹−𝐵𝐷−𝐸#õöî#ëîïðŸ
19Ÿ?2024·ńŅǞǟ·jňʼnŊAµ3 1W—Ú/©Œ±𝐵𝐶𝐷𝑃"W𝑃𝐷//𝐵𝐶W𝐵𝐴⊥𝐴𝐷Wúÿw
𝐴,𝑃𝐴= 𝐴𝐵= 𝐵𝐶= 2𝐴𝐷WX △𝑃𝐴𝐵š𝐴𝐵Ǡ4þ △𝑆𝐴𝐵#‡ˆWŬēÚ/𝑆𝐴𝐵⊥Ú/𝐴𝐵𝐶𝐷Wµ3 2 ×ò.
      
      
  
(1)|Ú/𝑆𝐶𝐷QÚ/𝑆𝐴𝐵#â&w𝑙Wm‰
𝐵𝐶⊥𝑙.
(2)—&'𝑆𝐶ĄRłļ—`-𝑄?-𝑄ĹQß-SsAWŬē/,𝑄−𝐵𝐷−𝐶#õöîw 6
6 Żĝļ—W’°
𝑆𝑄
𝑄𝐶
#îĝĹļ—Wżů‰Ē.