ᠠᠷᠪᠠᠨ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠪᠥᠯᠥᠭ ᠪᠦᠷᠢᠨ ᠲᠡᠩᠴᠡᠬᠦᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠠᠷᠪ ᠠᠨ ᠬᠤ ᠶᠠᠳᠤ ᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦ ᠨ ᠪᠠ ᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯᠥᠭ᠄ ᠨᠡᠷ ᠡ ᠄ ︵90 ᠮ ᠢᠨᠦ᠋ᠲ ᠂ ᠨᠠᠢ ᠲᠡ 120 ᠬᠤ ᠪᠢ ︶ ᠨᠢᠭᠡ ᠂ ᠰ ᠤᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ( ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 30 ᠬᠤ ᠪᠢ ) 1. ᠵᠢᠷᠤ ᠭ 1 ᠮᠡᠳ ᠦ᠂ ᠥᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ∆AB C≌ ∆D EF , AC = 22 , CF = 4 ᠪᠣᠯ CD ᠶ᠋ᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ ᠄ ( ) A. 22 B. 18 C. 16 D. 4 2. ᠵᠢᠷᠤᠭ 2 ᠮ ᠡᠳᠦ᠂ ᠰᠤᠪ ᠤᠷᠭ ᠠ ᠶ᠋ᠢ ᠨ ᠰ ᠠᠭᠤᠷᠢ ᠳ᠋ᠦ ᠪᠠ ᠢᠭ ᠠ ᠨᠡᠪᠳ ᠡᠷᠡᠵᠦ ᠲᠡᠢ ᠯᠬᠦ ᠬᠤᠶ ᠠᠷ ᠴᠡᠭ A , B ᠶ᠋ᠢᠨ ᠬ ᠣᠭᠤᠷᠤᠨ ᠳᠤᠬᠢ ᠵᠠᠢ ᠶ᠋ᠢ ᠬᠡᠮᠵᠢ ᠬᠦ ᠳ᠋ᠦ ᠂ AO ᠶ᠋ᠢ C ᠦᠦᠷᠳ ᠡᠯ ᠡ ᠷᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠭᠦᠯᠬ ᠦ ᠳ᠋ᠦ ᠪᠠ ᠨ AO = CO ᠪ ᠣᠯᠭᠠᠨ ᠠ᠂ BO ᠶ᠋ᠢD ᠦᠦᠷ ᠳᠡᠯ ᠡ ᠷᠷᠭ ᠦᠯᠵᠢ ᠯᠡᠭᠦᠯ ᠬᠦ ᠳ᠋ᠦ ᠪᠠᠨ DO = BO ᠪᠣᠯ ᠭᠠᠨ ᠠ᠂ ᠢᠩᠭᠢ ᠬᠦᠯᠡᠷ ∆ COD ≌ ∆A OB ᠪᠣ ᠯᠤᠨ ᠠ᠂ ᠡᠭᠦᠨ ᠤ ᠤᠯᠠᠮ ᠡᠴᠡ CD ᠶ᠋ᠢ ᠬ ᠡᠮᠵᠢᠬ ᠦ ᠪᠡᠷ ᠳᠠᠮᠵ ᠢᠨ A , B ᠶ᠋ ᠢᠨ ᠬ ᠣᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠶ᠋ ᠢ ᠣ ᠯᠴᠢᠬᠠ ᠨ ᠠ᠃ ᠡᠭᠦᠨ ᠳ᠋ᠦ ᠪᠦᠷ ᠢᠨ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ ᠶ᠋ᠢᠨ ᠷᠨ ᠳᠦᠰ ᠦᠯᠡᠯ ᠨᠢ᠄ ( ) A. SA S B. AS A C. AA S D. SSS 3. ᠵᠢᠷᠤᠭ 3 ᠮᠡᠳᠤ᠂ ᠴᠡᠭ C , D ᠬᠡᠷᠴ ᠢᠮAB ᠲᠡᠭᠡᠷ ᠠ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠂ AC = DB , AE || BF ᠲ ᠠᠷᠠᠭᠠᠬ ᠢ ᠨᠥᠬ ᠦᠴᠡᠯ ᠢ ᠨ ᠡᠮᠡᠬᠦ ᠳ᠋ᠤ ᠮᠥᠨᠴ ᠦ ∆AE D≌ ∆B FC ᠪᠣ ᠯᠭᠠᠵᠤ ᠲᠡᠢ ᠯᠬᠦ ᠷᠭᠡᠢ ᠨ ᠢ᠄ ( ) A. ED = CF B. AE = BF C. ∠E = ∠F D. ED ∥ CF 4. ᠵᠢᠷᠤᠭ 4 ᠮ ᠡᠳᠤ᠂ ᠵᠥ ᠪ ᠲ ᠠᠪᠤᠨ ᠷᠨᠴ ᠦᠭᠳᠦ AB CD E ᠶ᠋ ᠢᠨ ᠲᠠᠰ ᠢᠭᠤᠪᠴᠢ BD , CE ᠨᠢ ᠴᠡᠭ F ᠳ᠋ᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ ᠲ ᠡᠭᠡᠪᠡ ᠯ ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠳ᠋ ᠦ᠋ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠵᠥᠪ ᠨᠢ᠄ ( ) A. ∠B CE = 36° B. ∆B CF ᠪᠣᠯ ᠲ ᠡᠭᠰᠢ ᠷᠨᠴ ᠦᠭᠳᠦ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ C. ∆B CD ≌ ∆C DE D. BD ⊥ AB 5. ᠵᠢᠷᠤ ᠭ 5 ᠮᠡᠳᠤ᠂ ∆ AB C ᠳ᠋ᠦ ᠂ ∠ C= 90° , O ᠨᠢ ∆AB C ᠶ᠋ᠢᠨ ᠭᠤᠷᠪᠠ ᠨ ᠥᠨᠴ ᠦᠭ ᠦᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠶ᠋ᠢ ᠨ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠯ ᠤ ᠨ ᠴ ᠡᠭ᠂ OD ⊥ BC , OE ⊥ AC , OF ⊥ AB ᠂ ᠴᠡᠭ D , E , F ᠲᠤᠰ ᠪᠤᠷᠢ ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠪᠥᠭᠡ ᠳAB = 10c m , BC = 8c m , CA = 6c m ᠪᠣᠯ O ᠡ ᠴᠡ AB ᠦᠦᠷ ᠬᠦ ᠵᠠᠢ ᠨ ᠢ᠄ ( ) A. 2cm B. 3cm C. 4cm D. 5cm 6. ᠵᠢᠷᠤ ᠭ 6 ᠮᠡᠳᠦ᠂ AP ᠨᠢ ∠N AM ᠶ᠋ ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠂ PC = PB , AB AC , PD ⊥ AB ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨᠢ D , ∠ DPB = 50 ° ᠲ ᠡᠭᠡᠪᠡᠯ ∠A CP = ( ) A. 12 0° B. 13 0° C. 14 0° D. 15 0° 7. ᠵᠢᠷᠤᠭ 7 ᠮᠡ ᠳᠦ᠂ ∆ AB C ᠳ᠋ᠤ∠ C= 90° , AD ᠨᠢ ∠C AB ᠶ᠋ᠢ ᠲ ᠠᠯᠠᠯᠠᠨ ᠬ ᠤᠪᠢᠶ ᠠᠨ ᠠ ᠂DE ⊥ AB ᠪᠠ ᠷ ᠪᠠ ᠢᠭᠤ ᠯᠪᠠᠯ ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ E ,DE ᠨᠢ ∠A DB ᠶ᠋ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠲᠡᠭᠡᠪᠡᠯ ∠B ᠶ᠋ᠢᠨ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨ ᠢ᠄ ( ) (3) (1) (2) (4) (5) (6) (7) A. 40° B. 3 0° C. 25 ° D. 22. 5° 8.ᠵ ᠢᠷᠤᠭ 8 ᠮᠡᠳᠤ᠂ MP ⊥ NP ,MQ ᠨᠢ ∆N MP ᠶ᠋ᠢ ᠨ ᠥᠨᠴ ᠦᠭ ᠤ ᠨ ᠲ ᠠᠯᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ᠂ ᠴ ᠡᠭ T ᠨᠢ MN ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠴ ᠡᠭ᠂ MT = MP , TQ ᠶ᠋ᠢ ᠬ ᠣᠯᠪᠤ ᠨ ᠠ᠂ ᠲᠠᠷᠠᠭᠠᠬ ᠢ ᠳ᠋ᠦ᠋ ᠩᠨᠡᠯ ᠲᠡ ᠡᠴᠡ ᠪᠤᠷᠤᠭᠤ ᠨᠢ᠄ ( ) A. ∠PM N = ∠N QT B . ∠M QT = ∠M QP C. ∠Q TN = 90° D. ∠N QT = ∠M QT 9. ᠵᠢᠷᠤᠭ 9 ᠮᠡᠳᠦ᠂ ∆ AB C ᠳ᠋ ᠤ᠂AB = 5 , AC = 4 , AD ᠨᠢ ∠B AC ᠶ᠋ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠂DE ᠨᠢ ∆AB D ᠶ᠋ ᠢᠨ ᠲᠤᠮᠳ ᠠ ᠱ ᠤᠭᠤᠮ ᠂ S ∆A DE : S ∆A CD =( ) A. 4 ∶5 B. 5 ∶4 C. 16 ∶25 D. 5 ∶8 10. ᠵᠢᠷᠤ ᠭ 10 ᠮᠡᠳ ᠤ᠂ ① BC = B ′ C ② AC = A ′C ③ ∠ A ′CA = ∠B ′CB ④AB = A ′B ′ ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠲᠦ᠋ ᠷᠪᠡ ᠡᠴᠡ ᠳᠤ ᠷ ᠠ ᠶ᠋ ᠢᠨ ᠭᠤᠷᠪᠠ ᠶ᠋ᠢ ᠰᠣᠩ ᠭᠤᠵᠤ ᠨᠥᠬ ᠦᠴᠡᠯ ᠪᠣᠯᠭᠠᠨ ᠷᠯ ᠡᠳᠡᠭᠰ ᠡᠨ ᠨᠢᠭᠡ ᠪᠡᠷ ᠨᠢ ᠳ᠋ᠦ᠋ᠩ ᠨᠡᠯᠲᠡ ᠪᠣᠯᠭᠠᠪᠠ ᠯ ᠬ ᠠᠮᠤᠭ ᠤ ᠯᠠᠨ ᠪᠦᠷᠢ ᠯᠳᠦᠭᠦ ᠯᠬᠦ ᠵᠢᠩᠬ ᠢᠨᠢ ᠥᠭ ᠦᠯᠡᠯ ᠤ ᠨ ᠪ ᠣᠳᠠᠭᠠᠯ ᠢ ᠲ ᠣᠭ ᠠ ᠨᠢ᠄ ( ) A. 1 B. 2 C. 3 D. 4 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠳ᠋ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 24 ᠬᠤ ᠪᠢ ︶ 11. ᠵᠢ ᠷᠤᠭ 11 ᠮᠡᠳ ᠦ᠂∆ AB C ᠳ᠋ᠤAB = BE ,AD = DE , ∠ A = 80° ᠪᠣᠯ ∠C ED =_____ 12. ᠵᠢᠷᠤ ᠭ 12 ᠮᠡᠲ ᠦ᠂AC = BC ᠡᠭᠦ ᠨ ᠤ ᠲᠡᠭᠡᠷ ᠠ ᠨ ᠢᠭᠡ ᠬᠣᠣ ᠰ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯ ᠠ ᠪᠤᠶᠤ ᠨᠢᠭᠡ ᠬ ᠣᠣᠰ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ ᠥᠨᠴ ᠦᠭ ᠪᠣᠯᠬᠤ ᠨᠥ ᠬᠦᠴᠡᠯ ᠨᠡᠮᠡᠵ ᠦAD = BE ᠪᠣᠯ ᠭᠠᠨ ᠠ᠃ ᠲᠡᠭᠡᠪᠡᠯ ᠨᠡᠮᠡᠬ ᠦ ᠨᠥ ᠬᠦᠴᠡᠯ ᠨᠢ ᠄ _____ __ 13. ᠵᠢᠷᠤᠭ 13 ᠮᠡᠳ ᠦ᠂∆ AB C ᠶ᠋ ᠢᠨ ᠦᠦᠷ ᠢᠶᠡᠨ ᠤ ᠤ ᠷᠳᠤ ᠨᠢ21 , BO ,CO ᠲ ᠤᠰᠪᠤ ᠷᠢ ∠A BC ᠪᠠ ∠ AC B ᠶ᠋ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠂OD ⊥ BC ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢD ᠪᠥᠭᠡ ᠳ OD = 4 , ᠪᠣᠯ ∆AB C ᠶ᠋ ᠢᠨ ᠲ ᠠᠯᠠᠪᠠ ᠢ ᠄_______ 14. ᠵᠢᠷᠤᠭ 14 ᠮᠡᠳ ᠤ᠂ BC ⊥ AB ,DC ⊥ BC ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨᠢ ᠲᠤᠰ ᠲ ᠤᠰ B, C ᠂ AB = 6 ᠂ BC = 8 ᠂ CD = 2 ᠂ ᠴᠡᠭ P ᠨᠢ BC ᠲᠠᠯ ᠠ ᠲ ᠡᠭᠡᠷᠡᠬ ᠢ ᠦᠥᠳ ᠡᠯᠬᠦ ᠴᠡᠭ ᠂ BP = _______ ᠷᠶ ᠡᠰ Rt ∆AB P≌ Rt ∆PC D 15. ᠵᠢᠷᠤᠭ 15 ᠮᠡᠳ ᠦ᠂ ᠥᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ∆AB C≌ ∆AD E᠂ BC ᠶ᠋ᠢᠨ ᠷᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤ ᠮ ᠴᠡᠭ E ᠶ᠋ᠢ ᠲᠠᠭᠠᠷ ᠢᠨ ᠠ ᠂∠ AC B= ∠A = 105° , ∠C AD = 10° , ∠B = 50° , ᠪᠣᠯ ∠D EF =____ 16. ᠵᠢᠷᠤᠭ 16 ᠮᠡᠳ ᠤ᠂ ᠲᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠳ᠋ᠦ ᠂ ∠A = 90° ,∠B DC = 90° , AD = 2 , ∠A DB = ∠C (11) (10) (9) (8) (12) (13) (14) (15) (16) ᠲᠡᠭᠡᠪᠡᠯ ᠴᠡᠭ D ᠲᠠᠯ ᠠ BC ᠦᠦᠷ ᠬᠦ ᠵᠠᠢ ᠨᠢ ᠄_______ 17. ᠵᠢᠷᠤᠭ 17 ᠮᠡᠳ ᠤ᠂ ᠨᠢᠭᠡᠨ ᠠᠳ ᠠᠯᠢ ᠬᠠᠵᠠᠭᠤᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠬᠠᠪᠳ ᠠᠰᠤ ᠶ᠋ᠢ ᠲᠡᠭᠰ ᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠢ ᠷᠢᠰᠢ ᠯ ᠤ ᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ ᠳ᠋ᠦ ᠵᠢᠷᠤᠪᠠ ᠂ᠥᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ A( 0,1 ) , B (− 5,5 ) ᠲ ᠡᠭᠡᠪᠡᠯ ᠴᠡᠭ C ᠶ᠋ ᠢᠨ ᠪ ᠠᠢᠷᠢ ᠰᠢᠯ᠄ __ ______ ______ 18. ᠵᠢᠷᠤ ᠭ 1 8 ᠮ ᠡᠳᠦ᠂ ∆AB C ᠳ᠋ ᠤ AB = AC , ∠B AC = 90° ᠂ ᠴ ᠡᠭ E ᠪᠣᠯ AC ᠲᠠᠯ ᠠ ᠲ ᠡᠭᠡᠷᠡᠬ ᠢ ᠴ ᠡᠭ᠂ BE ᠶ᠋ᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ ᠂ ᠴ ᠡᠭ A ᠶ᠋ᠢ ᠲᠠᠭᠠᠷ ᠢᠭᠤᠯ ᠤᠨ AD ⊥ BE ᠪᠠᠷ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ ᠡᠭᠴᠡ ᠶ᠋ᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ D ᠂ DC ᠶ᠋ᠢ ᠬ ᠣᠯᠪᠤ ᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ AD = 4 ᠪᠣᠯ᠂ ∆AD C ᠶ᠋ ᠢᠨ ᠲ ᠠᠯᠠᠪᠠ ᠢ ᠄___ ____ ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ ᠨᠠᠢ ᠲᠡ 66 ᠬᠤ ᠪᠢ ︶ 19. (10 ᠬᠤᠪ ᠢ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠰ ᠢᠯᠤᠭᠤ ᠨ MN ᠲᠡᠭᠡᠷ ᠠ ᠨᠢ ᠭᠡ ᠴ ᠡᠭ P ᠶ᠋ᠢ ᠪᠠ ᠢᠭᠤ ᠯᠵᠤ ᠰᠠᠴ ᠤᠷᠠᠭ OA ᠪᠠ OB ᠦᠦᠷ ᠬᠦ ᠵ ᠠᠢ ᠶ᠋ᠢ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠪ ᠣᠯᠭ ᠠ ᠃ (ᠱᠠᠭᠠᠷ ᠳᠠᠯᠭ ᠠ ᠨ ᠢ᠄ ᡂᠢ ᠭᠤᠷᠳ ᠢᠭ ᠢ ᠶ᠋ᠡᠷ ᠵᠢᠷᠤᠭ ᠪᠠᠢ ᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠣ ᠷᠤᠮ ᠷᠯ ᠡᠳᠡᠭᠡᠨ ᠡ᠂ ᠪᠠᠢ ᠭᠤᠯᠬᠤ ᠠᠷᠭ ᠠ ᠪᠠ ᠪᠠᠳ ᠤᠯᠠᠬᠤ ᠶᠠᠪ ᠤᠴᠠ ᠶ᠋ ᠢ ᠪᠢᠴ ᠢᠬᠦ ᠴᠢᠬᠤᠯ ᠠ ᠷᠭᠡᠢ )᠃ 20. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ A ,B ᠪᠣᠯ ᠡᠮᠦᠨ ᠡ ᠬ ᠣᠢᠨ ᠠ ᠴ ᠢᠭᠯᠡᠯ ᠤᠨ ᠨᠢᠭᠡ ᠰᠢ ᠯᠤᠭᠤ ᠨ ᠳᠡᠭᠡ ᠷ ᠡ ᠪᠠᠢ ᠷᠢᠯᠠᠬᠤ ᠴᠠᠬ ᠢᠯᠭᠠᠨ ᠤ ᠳᠠᠰᠤ ᠨ ᠪᠠ ᠭᠠᠨ ᠠ ᠪ ᠣᠯᠤᠨ ᠠ᠂ A , B ᠬᠣᠶ ᠠᠷ ᠴ ᠠᠬᠢᠯᠭᠠᠨ ᠪᠠᠭᠠᠨ ᠠ ᠶ᠋ ᠢᠨ ᠬᠣᠭᠤ ᠷᠤᠨᠳ ᠤ ᠨᠢ ᠭᠡ ᠭ ᠣᠣᠯ ᠪᠠᠢ ᠨ ᠠ᠂ ᠪᠠᠳᠤ A ,B ᠬᠣ ᠶᠠᠷ ᠴᠠᠬᠢ ᠯᠭᠠᠨ ᠪᠠᠭᠠᠨ ᠠ ᠶ᠋ᠢ ᠨ ᠬ ᠣᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠶ᠋ ᠢ ᠪᠠᠷᠤᠭ ᠴᠠᠭᠠᠯ ᠠᠬᠤ ᠶ᠋ᠢᠨ ᠲᠥ ᠯᠦᠭᠡ ᠤ ᠷᠢᠳᠠᠪᠠ ᠷ A ᠡᠴ ᠡ ᠠ ᠰ ᠪᠠ ᠷᠠᠭᠤᠨ ᠰᠢ 20 ᠠᠯ ᠬᠤᠮ ᠶᠠᠪ ᠤᠭᠠᠳ ᠪᠦᠳ ᠦᠭᠦᠨ ᠮᠣᠳᠤ C ᠲ ᠤ ᠦᠦᠷ ᠪᠡ᠂ ᠵᠠᠯᠭᠠᠭᠠᠳ ᠷᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠨ 20 ᠠᠯ ᠬᠤᠮ ᠶᠠᠪ ᠤᠭᠠᠳ D ᠭᠠᠵᠠ ᠷ ᠲ ᠤ ᠦᠦᠷ ᠪᠡ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠵ ᠡᠭᠦᠨ ᠮᠥ ᠷ ᠢ ᠶ᠋ ᠡᠨ ᠳᠠᠭᠠ ᠵᠤ 90 ° ᠢ ᠶ᠋ᠡᠷ ᠮᠤᠷᠤᠢᠵ ᠤ ᠷᠷᠭ ᠦᠯᠵᠢ ᠯᠡᠨ ᠶᠠᠪᠤᠭᠠᠳ B ᠴᠠᠬ ᠢᠯᠭᠠᠨ ᠪᠠᠭᠠᠨ ᠠ ᠂ ᠪᠦᠳ ᠦᠭᠦᠨ ᠮᠣᠳ ᠤ C ᠪᠠ ᠥᠪᠡᠷ ᠤ ᠨ ᠪᠡᠶ ᠡ ᠶ᠋ᠢᠨ ᠪᠠᠢ ᠷᠢE ᠨᠢ ᠶᠠᠭ ᠰᠠᠶ᠋ᠢ ᠬᠠᠨ ᠨᠢᠭᠡ ᠰᠢᠯ ᠤᠭᠤᠨ ᠳ᠋ᠦ ᠣᠷᠤᠰᠢ ᠬᠤ ᠪᠠᠢ ᠷᠢ ᠳ᠋ ᠤ ᠦᠦᠷ ᠬᠦ ᠷᠶ ᠡᠰ᠂ D ᠭᠠᠵᠠ ᠷ ᠡᠴ ᠡ E ᠭᠠᠵᠠ ᠷ ᠦᠦᠷ ᠳᠡᠯ ᠠ 100 ᠠᠯᠬᠤᠮ ᠶᠠᠪ ᠤᠪᠠ ᠃ ᠪᠠ ᠳᠤ ᠡᠳᠡᠭᠡᠷ ᠲᠣᠭ ᠠ ᠪᠠ ᠷᠢᠮᠲ ᠠ ᠪᠠ ᠨ ᠠ ᠰᠢᠭᠯ ᠠᠨ A , B ᠬᠣᠶ ᠠᠷ ᠴ ᠠᠬᠢᠯᠭᠠᠨ ᠪᠠᠭᠠᠨ ᠠ ᠶ᠋ᠢ ᠨ ᠬ ᠣᠭᠤᠷᠤᠨ ᠳᠤᠬᠢ ᠵᠠᠢ ᠶ᠋ᠢ ᠬ ᠡᠮᠵᠢᠨ ᠣᠯᠪᠠ ᠃ (1) ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠥᠭ ᠦᠯᠡᠭᠰ ᠡᠨ ᠬ ᠡᠮᠵᠢᠯ ᠲᠡ ᠶ᠋ᠢᠨ ᠠ ᠷᠭ ᠠ ᠶ᠋ᠢ ᠵ ᠢᠷᠤᠭ ᠲᠡᠭᠡᠷ ᠠ ᠵᠢ ᠷᠤᠵᠤ ᠢᠯ ᠡᠳᠭᠡ ᠃ (2) ᠬᠡᠷᠪᠠ ᠪᠠᠳᠤ ᠶ᠋ᠢᠨ ᠨᠢ ᠭᠡ ᠠᠯᠬᠤᠮ ᠪᠠ ᠷᠤᠭ 60 cm ᠪᠣᠯ A ,B ᠬ ᠣᠶᠠᠷ ᠴᠠᠬᠢ ᠯᠭᠠᠨ ᠪᠠᠭᠠᠨ ᠠ ᠶ᠋ᠢ ᠨ ᠬᠣ ᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠶ᠋ ᠢ ᠣ ᠯ᠃ ᠭᠤᠤᠯ ᠤᠮᠠᠷ ᠠ T1 65 (17) (18) 21. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ OC ᠨᠢ ∠M ON ᠶ᠋ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠂ A , B ᠲᠤᠰ ᠪᠤᠷᠢ OM , ON ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠴ ᠡᠭ ᠪ ᠥᠭᠡᠳ AC = BC , BO AO ᠃ ᠪᠠᠳᠤ ᠯᠠᠭᠳ ᠠᠬᠤ ᠨᠢ᠄ ∠O AC + ∠OB C= 180° 22. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ∠A = ∠D = 90° , BE ᠨᠢ ∠A BC ᠶ᠋ᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ᠂ ᠴᠡᠭ E ᠨᠢ AD ᠶ᠋ᠢᠨ ᠲᠤ ᠮᠳᠠ ᠴᠡᠭ᠃ ᠪᠠᠳ ᠤᠯᠠᠭᠳ ᠠᠬᠤ ᠨᠢ᠄ BC = AB + CD 23. ( 12 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ ∆AB C ᠳ᠋ ᠤ᠂ D ᠨᠢ BC ᠶ᠋ᠢᠨ ᠲᠤ ᠮᠳᠠ ᠴᠡᠭ᠂ ᠴᠡᠭ D ᠶ᠋ ᠢ ᠲ ᠠᠭᠠᠷᠢ ᠭᠰᠠᠨ ᠰᠢ ᠯᠤᠭᠤ ᠨ GF ᠨᠢ AC ᠲᠠᠢ ᠴᠡᠭ F ᠳ᠋ᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ AC ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠤ ᠭᠤᠨ ᠪᠣᠯ ᠬᠤ BG ᠲᠠᠢ ᠴᠡᠭ G ᠲᠤ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ DE ⊥ GF ᠪᠥᠭᠡ ᠳAB ᠲᠠᠢ ᠴᠡᠭ E ᠳ᠋ ᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ EG , EF ᠶ᠋ᠢ ᠬᠣ ᠯᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠳ ᠤᠯᠠᠭ ᠳᠠᠬᠤ ᠨᠢ᠄ BG = CF (2) BE + CF ᠪᠠ EF ᠶ᠋ᠢ ᠨ ᠬᠣᠭᠤ ᠷᠤᠨᠳ ᠤᠬᠢ ᠶᠡᠬᠡ ᠪᠠ ᠭ ᠠ ᠶ᠋ᠢᠨ ᠬᠣᠯ ᠪᠤᠭᠳ ᠠᠯ ᠢ ᠲᠠᠭᠠᠮᠠᠭᠯ ᠠᠵᠤ ᠰᠢᠯᠲ ᠠᠭᠠᠨ ᠢ ᠨ ᠢ ᠲ ᠣᠳᠤᠷᠬ ᠠᠢᠯᠠ᠃ 24. (1 2 ᠬᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ ∆AB C ᠶ᠋ᠢᠨ ᠭᠠᠳ ᠠᠨ ᠠ ∆AB D ᠪᠠ ∆A CE ᠢ ᠪᠠᠢ ᠭᠤᠯᠬᠤ ᠳ᠋ᠦ ᠪᠠᠨ AB = AD , AC = AE ,∠D AB = ∠E AC ᠂ BE ᠪᠠ CD ᠶ᠋ ᠢ ᠬ ᠣᠯᠪᠤ ᠪᠠᠯ ᠴᠡᠭ P ᠳ᠋ᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠪᠠᠳᠤ ᠯᠠᠭᠳ ᠠᠬᠤ ᠨᠢ᠄ ᠴᠡᠭ A ᠨᠢ ∠D PE ᠶ᠋ᠢᠨ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠲ ᠡᠭᠡᠷ ᠠ ᠣᠷᠤᠰᠢ ᠨ ᠠ᠃