ᠠᠷᠪᠠᠨ ᠬᠤᠶᠠᠳᠤᠭᠠᠷ ᠪᠥᠯᠥᠭ ᠪᠦᠷᠢᠨ ᠲᠡᠩᠴᠡᠬᠦᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠠᠷᠪ ᠠᠨ ᠬᠣᠶᠠ ᠳᠤᠭ ᠠᠷ ᠪᠥ ᠯᠥᠭ ᠦᠨ ᠪᠠ ᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯᠥᠭ᠄ ᠨᠡᠷ ᠡ ᠄ ︵90 ᠮ ᠢᠨᠦ᠋ᠲ ᠂ ᠨᠠᠢ ᠲᠡ 120 ᠬᠤ ᠪᠢ ︶ ᠨᠢᠭᠡ ᠂ ᠰ ᠤᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ( ᠰᠡᠲᠤ ᠪ ᠪᠦᠷ ᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 30 ᠬᠤ ᠪᠢ) 1. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠳ ᠤ᠂ ∆A CB ≌ ∆A ′CB ′, ∠BC B ′ = 32 0 ᠂ ᠲᠡᠭᠡᠪᠡᠯ ∠A CA ′ = ( ) A. 30 0 B. 32 0 C. 35 0 D. 45 0 2. ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠬᠡᠯ ᠡᠯᠭᠡ ᠳᠦ ᠄ ① ᠪᠦᠷ ᠢᠨ ᠲ ᠡᠨᠭᠴᠡᠭᠦ ᠦ ᠲ ᠦ᠋ᠷᠰ ᠦ ᠶᠢ ᠨ ᠬᠡᠯ ᠪᠡᠷᠢ ᠲᠦ᠋ ᠷᠰᠦ᠂ ᠶᠡᠬ ᠡ ᠪᠠ ᠭ ᠠ ᠨᠢ ᠪ ᠦᠷᠢ ᠮᠦᠰᠦᠨ ᠠᠳ ᠠᠯᠢ᠃ ② ᠬ ᠤᠶᠠᠷ ᠲᠠᠯ ᠠ ᠪᠠ ᠨᠢᠬ ᠡ ᠦᠨᠴ ᠦᠭ ᠨ ᠢ ᠬ ᠠᠷᠠᠭᠠ ᠯᠵᠠᠨ ᠲᠡᠨᠭ ᠴᠡᠬᠦ ᠬᠤᠶ ᠠᠷ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠨᠢ ᠪᠦᠷᠢ ᠨ ᠲ ᠡᠨᠭᠴᠡᠭᠦ ᠦ᠃ ③ ᠨᠢ ᠭᠡ ᠬ ᠤᠷᠴᠠ ᠦᠨᠴ ᠦᠭ ᠪᠠ ᠨᠢᠭᠡ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠ ᠨᠢ ᠬᠠᠷᠠᠭᠠ ᠯᠵᠠᠨ ᠲᠡᠨᠭ ᠴᠡᠬᠦ ᠬᠤᠶ ᠠᠷ ᠲ ᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠨᠢ ᠪᠦᠷᠢ ᠨ ᠲ ᠡᠨᠭᠴᠡᠬ ᠦᠦ᠃ ④ᠪᠦᠷ ᠢᠨ ᠲ ᠡᠨᠭᠴᠡᠬ ᠦᠦ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠤ ᠬ ᠠᠷᠠᠭᠠ ᠯᠵᠠᠭᠰ ᠠᠨ ᠲ ᠠᠯ ᠠ ᠲᠡᠭᠡ ᠷᠡᠬᠢ ᠲᠤ ᠮᠳᠠ ᠱᠤᠭᠤ ᠮ ᠨ ᠢ ᠬᠠᠷᠢ ᠯᠴᠠᠨ ᠲᠡᠨᠭ ᠴᠡᠭᠦ ᠦ᠃ ᠵᠦ᠋ᠪ ᠪᠤᠯ ᠬᠤ ᠨ ᠢ ( ) A. ① ② ③ ④ B . ① ③ ④ C . ① ② ④ D. ② ③ ④ 3.ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ᠂ ᠵᠢᠷᠤ ᠭ 2 ᠮᠡᠳ ᠤ᠂ ∆ AB C ᠳᠦ᠂ ∠C AD = ∠E AD , ∠A DC = ∠A DE , CB = 5c m , BD = 3c m ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ED ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 2c m B. 3cm C. 5cm D. 8c m 4. ᠲᠡᠭᠰ ᠢ ᠲ ᠦ᠋ᠷᠪᠡᠯ ᠵᠢᠨ ᠬᠠᠨᠠᠯ ᠵᠢᠨ ᠵᠢᠷᠤ ᠭ ᠲ ᠤ᠂ ∠A OB ᠶ ᠢᠨ ᠪᠠᠢ ᠷᠢ ᠨ ᠢ ᠵᠢ ᠷᠤᠭ 3 ᠳᠤ ᠦᠵᠡᠬ ᠦᠯᠦᠭᠰᠡᠨ ᠮᠡᠳ ᠤ᠂ ∠A OB ᠶᠢᠨ ᠬᠤ ᠶᠠᠷ ᠲᠠᠯ ᠠ ᠬᠦᠷ ᠬᠦ ᠵᠠᠢ ᠨᠢ ᠲ ᠡᠨᠭᠴᠡᠭᠦ ᠦ ᠪᠤ ᠯᠬᠤ ᠴᠡᠭ ᠨᠢ ( ) A. ᠴᠡᠭ M B. ᠴᠡᠭ N C. ᠴᠡᠭ P D. ᠴᠡᠭ Q 5. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠤ᠂ ᠬᠤ ᠶᠠᠷ ᠠᠰᠬᠠᠨ ᠵ ᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ 𝑙1 , 𝑙2 ᠨᠢ ᠴ ᠡᠭ O ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠵᠠᠰᠠᠮ ᠠᠯ ᠵᠠᠮ ᠤᠨ ᠬᠤᠶ ᠠᠷ ᠲᠠᠯ ᠠ ᠪᠡᠷ ᠭᠤᠷᠪᠠ ᠨ ᠦᠢ ᠯᠡᠳᠪ ᠦᠷᠢ A , B , D ᠶᠢ ᠪᠠᠢ ᠭᠤᠯᠤ ᠨ ᠠ᠂ AB = AD = 5.2 km , CB = CD = 5 km ᠂ ᠲᠤᠰ ᠬᠤᠨ C ᠡᠴᠡ ᠵᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ 𝑙1 ᠬᠦᠷᠬ ᠦ ᠵᠠᠢ ᠨᠢ 4 km ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠲᠤ ᠰᠬᠤᠨ C ᠡᠴᠡ ᠵᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ 𝑙2 ᠬᠦᠷᠬ ᠦ ᠵᠠᠢ ᠨᠢ ( ) A. 3km B. 4km C . 5km D. 5. 2k m 6. ᠵᠢᠷᠤ ᠭ 5 ᠮᠡᠳᠤ᠂ ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ AB = AC , AD = AE ᠂ ᠪᠠᠰᠠ ᠨ ᠢᠭᠡ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠨᠡᠮᠡᠭᠡᠳ ∆ AB D ≌ ∆A CE ᠪᠤᠯ ᠭᠠᠵᠤ ᠴᠢᠳ ᠠᠬᠤ ᠦᠭᠡᠢ ᠨᠢ ( ) A. ∠A BD = ∠A CE B. BD = CE C. ∠B AD = ∠C AE D. ∠B AC = ∠D AE (1) (2) (3) (4) (5) 7. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ᠂ ᠤᠷ ᠳᠤ ᠨ ᠢ ᠠ ᠳᠠᠯᠢ ᠬᠤᠶ ᠠᠷ ᠱ ᠠᠳᠤ ᠪᠤᠢ (B C= EF )᠂ ᠵᠡᠬᠦᠨ ᠲᠠᠯ ᠠ ᠶᠢ ᠨ ᠱᠠᠳ ᠤᠨ ᠤ ᠦᠨᠳ ᠦᠷ AC ᠪᠠ ᠪ ᠠᠷᠠᠭᠤᠨ ᠲᠠᠯ ᠠ ᠶᠢ ᠨ ᠱᠠᠳ ᠤ ᠶ ᠢᠨ ᠲᠦ᠋ ᠪᠰᠢᠨ ᠬ ᠠᠪᠳᠠᠭᠠᠢ ᠳᠠᠬ ᠢ ᠤ ᠷᠳᠤ DF ᠨᠢ ᠲᠡᠨᠭ ᠴᠡᠭᠦ ᠦ᠂ ᠲᠡᠭᠡᠪᠡ ᠯ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠳᠦ᠋ᠨᠭ ᠨᠡᠯᠳᠡ᠄ ( 1) AB = DE (2) ∠A BC + ∠D FE = 90 0 ( 3) ∠A BC = ∠D EF ᠳᠦ ᠵᠦ᠋ᠪ ᠪᠤᠯ ᠬᠤ ᠨ ᠢ ( ) ᠪᠤᠳᠠᠭᠠᠯ ᠢ ᠪᠤ ᠢ᠃ A. 1 B. 2 C. 3 D. 0 8. ᠵᠢᠷᠤᠭ 7 ᠮ ᠡᠳᠤ᠂ ∆AB E≌ ∆A CD ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠳᠦ᠋ᠨᠭ ᠨᠡᠯᠳᠡ ᠳ ᠦ ᠪᠠ ᠢᠭᠤ ᠯᠤᠭᠳ ᠠᠬᠤ ᠦᠭᠡᠢ ᠪᠤᠯ ᠤᠯᠴᠠᠭ ᠠ ᠲ ᠠᠢ ᠨ ᠢ ( ) A. AD = AE B. BD = AE C. DF = EF D. BD = CE 9. ᠵᠢᠷᠤ ᠭ 8 ᠮᠡᠳᠤ᠂ ∆ AB C ᠳᠦ᠂ AB =6 , AC =4 , AD ᠨᠢ ∠B AC ᠶᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬ ᠤᠪᠢᠶ ᠠᠨ ᠠ ᠂ DE ⊥ AB , ᠡᠭᠴᠡ ᠶᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ E᠂ BF ⊥ AC ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ F , DE =2 ᠂ ᠲ ᠡᠭᠡᠪᠡ ᠯ BF ᠶᠢ ᠨ ᠤᠷ ᠳᠤ ᠨ ᠢ ( ) A. 3 B . 4 C. 5 D. 6 10. ᠵᠢᠷᠤᠭ 9 ᠮᠡᠳ ᠤ᠂ ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ AB = AC , PB = PC ᠂ ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠳᠦ᠋ᠨᠭ ᠨᠡᠯᠳᠡ᠄ ①EB =E C ②AD ⊥ BC ③ AE ᠨᠢ ∠B EC ᠶᠢ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ ④ ∠PB C= ∠PC B ᠵᠦ᠋ᠪ ᠪᠤᠯ ᠬᠤ ᠳ ᠦ᠋ᠨᠭᠨᠡᠯ ᠳᠡ ᠨ ᠢ ( ) A. ① ② ④ B. ① ③ ④ C. ① ② ③ D. ① ② ③ ④ ᠬᠤᠶ ᠠᠷ᠂ ᠬᠤᠭ ᠤᠰᠤᠨ ᠳᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠃ (ᠰᠡᠳ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤ ᠪᠢ᠂ ᠨᠡᠢ ᠳᠡ 24 ᠬᠤ ᠪᠢ ) 11. ᠵᠢᠷᠤᠭ 10 ᠮ ᠡᠳᠤ᠂ ∆ AB C≌ ∆D CB , AF ⊥ BC , DE ⊥ BC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ ᠲ ᠤᠰ ᠲᠤᠰ F , E ᠃ BC =18c m , ∆ AB C ᠶ ᠢᠨ ᠲᠠᠯᠠᠪᠠ ᠢ ᠨᠢ 108 𝑐𝑚 2᠂ ᠲᠡᠭᠡᠪᠡ ᠯ DE = _____ __c m 12. ᠵᠢᠷᠤ ᠭ 11 ᠮᠡᠳ ᠤ᠂ ∠1 = ∠2 ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ∆AB C≌ ∆B AD ᠪᠤᠯ ᠭᠠᠬᠤ ᠬᠡᠪᠡᠯ᠂ (1) ︽SAS ︾ ᠢᠶᠡᠷ ᠦᠨᠳ ᠦᠰᠦᠯ ᠡᠯ ᠪᠤᠯᠭᠠᠬᠤ ᠭᠡᠪᠡᠯ᠂ ᠨᠡᠮᠡᠬ ᠦ ᠴᠢᠬᠤᠯ ᠠᠳᠠᠢ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠨᠢ _________ ( 2) ︽AAS ︾ ᠢᠶᠡᠷ ᠦᠨᠳ ᠦᠰᠦᠯ ᠡᠯ ᠪᠤᠯᠭᠠᠬ ᠤ ᠭᠡᠪᠡᠯ᠂ᠨ ᠡᠮᠡᠬᠦ ᠴᠢᠬᠤᠯ ᠠᠳᠠᠢ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠨᠢ ________ ᠂ ( 3) ︽ASA ︾ ᠢᠶᠡᠷ ᠦᠨ ᠳᠦᠰ ᠦᠯᠡᠯ ᠪᠤᠯᠭᠠᠬᠤ ᠭᠡᠪᠡᠯ᠂ ᠨᠡᠮᠡᠬ ᠦ ᠴᠢᠬᠤᠯ ᠠᠳᠠᠢ ᠨ ᠦ᠋ᠬᠦ ᠴᠡᠯ ᠨᠢ ______ ___ ᠃ 13. ᠵᠢᠷᠤ ᠭ 12 ᠮᠡᠳ ᠤ᠂ OC ᠨᠢ ∠A OB ᠶᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬ ᠤᠪᠢᠶ ᠠᠨ ᠠ ᠂ ᠴ ᠡᠭ P ᠪᠤᠯ ᠰᠠᠴ ᠤᠷᠠᠭ OC ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠨ ᠢᠭᠡ ᠴᠡᠭ᠂ PD ⊥ OB ᠂ ᠡ ᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ D ᠂ P D= 5᠂ ᠴᠡᠭ Q ᠨᠢ ᠰᠠᠴ ᠤᠷᠠᠭ OA ᠳᠡᠭᠡ ᠷ ᠡ ᠬᠦᠳ ᠡᠯᠬᠡᠬ ᠡᠨ ᠬ ᠢᠨ ᠡ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠭᠡᠷᠴ ᠢᠮ PQ ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠪᠠᠭ ᠠ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨᠢ ____᠃ 14. ᠵᠢᠷᠤ ᠭ 13 ᠮᠡ ᠳᠤ᠂ ᠴᠡᠭ B , E , C , F ᠨᠢ ᠵᠵᠢ ᠯᠢ ᠨᠢᠭᠡᠨ ᠰᠢ ᠯᠤᠭᠤ ᠨ ᠳᠡᠭᠡ ᠷ ᠡ ᠤᠷᠤᠰᠢ ᠨ ᠠ᠂ ∠A = ∠D ,∠B = ∠D EF ,BE = CF ᠬᠡᠷᠪ AC = 5 ᠪᠤᠯ (6) (7) (8) (9) (10) (11) (12) DF =___ ___ 15. ᠵᠢᠷᠤᠭ 14 ᠮᠡᠳ ᠤ᠂ ᠬᠤᠶ ᠠᠷ ᠲ ᠤᠭ ᠤ ᠨ ᠪᠠ ᠭᠠᠨ ᠠ ᠶᠢ ᠨ ᠬ ᠤᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠨ ᠢ 12 m᠂ ᠨᠢᠭᠡᠨ ᠬᠦᠮᠦᠨ ᠴᠡᠭ B ᠡᠴᠡ ᠭᠠᠷᠴᠤ BA ᠲᠠᠭᠠᠤ ᠴᠡᠭ Aᠵ ᠦ᠋ᠭ ᠶᠠᠪᠤᠨ ᠠ᠂ ᠬ ᠡᠰᠡᠭ ᠬᠤᠭᠤ ᠴᠠᠭ ᠠ ᠶᠠᠪ ᠤᠭᠰᠠᠨ ᠤ ᠲ ᠠᠷᠠᠭ ᠠ ᠴᠡᠭ M ᠳᠤ ᠬᠦᠷᠦᠨ ᠡ᠂ ᠡᠨᠡ ᠦᠶ ᠡᠰ ᠬ ᠤᠶᠠᠷ ᠲᠤᠭ ᠤᠨ ᠡᠰᠢ ᠨ ᠦ ᠤᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ C ᠪᠠ D ᠶᠢ ᠠ ᠵᠢᠭᠯ ᠠᠬᠤ ᠦᠶ ᠡᠰ ᠂ ᠬᠤ ᠶᠠᠷ ᠬᠠᠷᠠᠭᠠᠨ ᠱᠤ ᠭᠤᠮ ᠤᠨ ᠬᠠᠪᠴᠢ ᠭᠰᠠᠨ ᠦᠨᠴ ᠦᠭ ᠨ ᠢ 90 0 ᠪᠠ ᠢᠭᠰᠠᠨ ᠪᠦᠭᠡᠳ CM = DM ᠪᠠᠢ ᠪᠠ᠂ ᠲᠤ ᠭ ᠤᠨ ᠡᠰ ᠢ AC ᠶᠢᠨ ᠦᠨ ᠳᠦᠷ ᠨᠢ 3 m᠂ ᠡᠨᠡ ᠬᠦᠮᠦᠨ ᠦ ᠬ ᠤᠷᠳᠤ ᠴᠠ ᠨ ᠢ 1 m/ s᠂ ᠲᠡᠭᠡᠪᠡᠯ B ᠡᠴᠡ M ᠬᠦᠷᠲ ᠡᠯ ᠡ ___ __ s ᠬᠡ ᠷᠡᠭᠯᠡᠵᠡ ᠢ᠃ 16. ᠵᠢ ᠷᠤᠭ 15 ᠮᠡᠳ ᠤ᠂ ∆AB C ᠳᠦ ᠂AB = 6 , ∠B AC ᠶᠢ ᠨ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴ ᠢ ᠨᠢ BC ᠲᠠᠢ ᠴᠡᠭ D ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠂DE ⊥ AC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ E , DE = 4 ᠲᠡᠭᠡᠪᠡᠯ ∆ AB D ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠ ᠢ ᠨᠢ _____᠃ 17. ᠵᠢᠷᠤᠭ 16 ᠮᠡᠳ ᠤ᠂ BC // EF , BC = BE , AB = FB , ∠1 = ∠2 ᠂ ᠬᠡᠷᠪ ∠1 = 55 0 ᠪ ᠤᠯ ∠C = ___ _ 18. ᠵᠢᠷᠤᠭ 17 ᠮᠡᠳᠤ᠂ ᠬᠤ ᠷᠴᠠ ᠦ ᠨᠴᠦᠭᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ AB C ᠪᠠ ᠬᠤᠷᠴᠠ ᠦᠨᠴ ᠦᠭᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ A ′B ′C ′ ᠳᠦ᠂ AD , A ′D ′ ᠨ ᠢ ᠲ ᠤᠰ ᠲᠤᠰ BC , B ′ C ′ ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠦᠨ ᠳᠦᠷ᠂ AB = A ′B ′ , AD = A ′D ′ , ∆ AB C≌ ∆A ′B ′C ′ᠪᠤᠯ ᠭᠠᠬᠤ ᠬᠡᠪᠡᠯ᠂ ᠨᠡᠮᠡᠬ ᠦ ᠴᠢ ᠬᠤᠯᠠᠳ ᠠᠢ ᠨ ᠦ᠋ᠬᠦ ᠴᠡᠯ ᠨᠢ _ _____ (ᠨ ᠢᠭᠡ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠨᠡᠮᠡᠪᠡᠯ ᠪᠣᠯᠴ ᠢᠬᠠᠨ ᠠ) ᠭᠤᠷᠪᠠ᠂ ᠪᠤᠳ ᠤᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠳᠤ ᠪ᠃ (ᠨᠡᠢ ᠳᠡ 6 ᠪᠠᠭ ᠠ ᠰ ᠡᠳᠤᠪ ᠲᠡᠢ᠂ 66 ᠬᠤ ᠪᠢ) 19. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ᠴᠡᠭ B , F , C , E ᠨᠢ ᠵᠵᠢ ᠯᠢ ᠨ ᠢᠭᠡᠨ ᠰᠢᠯ ᠤᠭᠤ ᠨ ᠳᠡᠭᠡ ᠷ ᠡ ᠤᠷᠤᠰᠢ ᠨ ᠠ᠂ ᠴᠡᠭ A ᠪᠠ D ᠨᠢ ᠰᠢᠯᠤ ᠭᠤᠨ BC ᠶᠢᠨ ᠬᠤ ᠶᠠᠷ ᠲᠠᠯ ᠠ ᠳᠤ ᠪᠤᠢ᠃ AB = DE , AC = DF , BF = EC ᠃ (1) ᠪᠠᠳ ᠤᠯᠠ᠄ ∆AB C ≌ ∆D EF (2) ᠵᠢᠷᠤ ᠭ ᠲ ᠠᠬᠢ ᠲᠡᠨᠭ ᠴᠡᠭᠦ ᠦ ᠦᠨᠴ ᠦᠭ ᠦᠳ ᠢ ᠪᠢᠴ ᠢᠵᠦ ᠭᠠᠷᠭ ᠠ᠃ (13) (14) (15) (16) (17) 20. (10 ᠬᠤ ᠪᠢ) ᠬᠦᠪᠡᠭᠡ ᠨᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠪᠤᠯᠬᠤ ᠭᠤᠤ ᠯ ᠤᠨ ᠦᠷᠭᠡᠨ ᠢ ᠲ ᠤᠤᠷᠠᠬ ᠢ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠬᠡᠮᠵᠢ ᠬᠦ ᠬ ᠡᠵᠦ ᠪᠤᠢ᠂ ① ᠭ ᠤᠤᠯ ᠤᠨ ᠬᠦᠪᠡᠭᠡ ᠳᠡᠭᠡ ᠷ ᠡ ᠨᠢᠭᠡ ᠴᠡᠭ B ᠶᠢ ᠠᠪᠴᠤ᠂ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ ᠡᠰ ᠡᠷᠭᠦ ᠲᠠᠯ ᠠ ᠳᠤ ᠨᠢ ᠨ ᠢᠭᠡ ᠰᠠᠭᠤ ᠷᠢ ᠮ ᠤᠳᠤ A ᠶᠢ ᠰ ᠤᠨᠭᠭᠤ ᠪᠠ ᠃ ② ᠭᠤ ᠤᠯ ᠤ ᠨ ᠬᠦᠪᠡ ᠭᠡ ᠲ ᠠᠭᠠᠤ 20 m ᠶᠠᠪᠤᠭᠠᠳ ᠨᠦ᠋ᠭ ᠦᠭᠡ ᠰᠠᠭᠤ ᠷᠢ ᠮ ᠤᠳᠤ C ᠳᠦ ᠬᠦᠷᠴ ᠤ᠂ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠨ ᠤᠷᠤᠭ ᠰᠢ 20 m ᠶᠠᠪᠤᠭᠠᠳ D ᠭᠠᠵᠠ ᠷ ᠬᠦᠷ ᠪ ᠃ ③ D ᠡᠴᠡ ᠭᠤᠤ ᠯ ᠤᠨ ᠬᠦᠪᠡᠭᠡᠨ ᠳᠦ ᠡᠭᠴᠡᠯ ᠡᠵᠦ ᠶᠠᠪᠤᠭᠠᠳ ᠂ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ Aᠮ ᠤᠳᠤ ᠨᠢ C ᠮᠤᠳᠤ ᠨ ᠳ ᠤ ᠬᠠᠯᠬᠠᠯ ᠠᠭᠳᠠᠬᠤ ᠪᠠᠢ ᠷᠢ E ᠳᠦ ᠬᠦᠷᠴ ᠤ ᠵᠤ ᠭᠰᠤ ᠪᠠ᠃ ④ D E ᠶ ᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ 5m ᠪᠠᠢ ᠬᠤ ᠶ ᠢ ᠬᠡᠮᠵ ᠢᠨ ᠤᠯᠪᠠ ᠃ ᠤ ᠯᠬᠤ ᠨᠢ ᠄ (1) ᠭᠤ ᠤᠯ ᠤ ᠨ ᠦᠷ ᠭᠡᠨ ᠨᠢ ᠬ ᠡᠳᠦᠢ ᠮᠸᠲ᠋ ᠷ ᠪᠤ ᠢ ︖ (2) ᠡᠨᠡᠬ ᠦ ᠠᠷᠭ ᠠ ᠶᠢ ᠨ ᠵᠦ᠋ ᠪ ᠪᠤ ᠯᠬᠤ ᠶᠢ ᠪᠠ ᠳᠤᠯᠠ᠃ 21. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ CE ⊥ AB , BF ⊥ AC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ ᠲᠤᠰ ᠲᠤ ᠰ E , F ᠪᠦᠭᠡᠳ BD = CD ᠪᠠᠳᠤ ᠯᠠ᠄ (1) ∆B DE ≌ ∆ CD F (2) ᠴᠡᠭ D ᠨᠢ ∠B AC ᠶᠢᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠃ 22. ( 12 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ AC ⊥ BC , DC ⊥ EC , AC = BC , DC = EC ᠃ ᠪᠠᠳᠤ ᠯᠠ᠄ (1) AE = BD (2) A E ᠪᠠ BD ᠶᠢᠨ ᠪᠠᠢ ᠷᠢ ᠶ ᠢᠨ ᠬᠤ ᠯᠪᠤᠭᠳ ᠠᠯ ᠢ ᠮ ᠠᠭᠠᠳ ᠯᠠᠵᠤ᠂ ᠳ ᠠᠷᠠᠭ ᠠ ᠨ ᠢ ᠪᠠ ᠳᠤᠯᠠ᠃ 23. (12 ᠬ ᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ ᠬᠤᠶ ᠠᠷ ᠡᠭᠴᠡ ᠲ ᠠᠯ ᠠ ᠨᠢ ᠬᠠᠷᠢ ᠯᠴᠠᠨ ᠲ ᠡᠨᠭᠴᠡᠭᠦ ᠦ ᠪ ᠤᠯᠬᠤ Rt ∆AB C ᠶᠢᠨ ᠵᠢᠰᠢ ᠬᠦᠦ ᠲᠠᠯ ᠠ BC ᠲᠡᠭᠡ ᠷᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠦᠳ ᠡᠯᠬᠦ ᠴᠡᠭ D ᠶ ᠢ ᠲ ᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨ ᠰ ᠠᠴᠤᠷᠠᠭ 𝐴𝐷 ᠶᠢ ᠪᠠᠢ ᠭᠤᠯᠤ ᠨ ᠠ᠃ ᠲᠠᠬ ᠢᠨ ᠴᠡᠭ B , C ᠶᠢ ᠲᠠᠭᠠᠷ ᠢᠭᠤᠯ ᠤᠨ AD ᠳ ᠦ ᠡ ᠭᠴᠡ ᠪᠤᠯᠬᠤ ᠰᠢᠯᠤ ᠭᠤᠨ BE ᠪᠠ C F ᠶ ᠢ ᠲ ᠤᠰ ᠲᠤᠰ ᠪᠠᠢ ᠭᠤᠯᠤ ᠨ ᠠ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ ᠲ ᠤᠰ ᠲᠤᠰ E , F ᠃ ᠲ ᠡᠭᠡᠪᠡᠯ BE , CF , EF ᠶᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ ᠬᠤᠭᠤ ᠷᠤᠨᠳ ᠤ ᠪᠠ ᠨ ᠶ ᠠᠮᠠᠷ ᠬᠤᠯ ᠪᠤᠭᠳ ᠠᠯ ᠪᠠ ᠢᠬᠤ ᠪᠤᠢ ︖ ᠰ ᠢᠯᠳᠠᠭᠠᠨ ᠢ ᠨ ᠢ ᠲᠤᠳ ᠤᠷᠬᠠ ᠢᠯᠠ᠃ 24. (1 2 ᠬ ᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠳᠤ᠂ ∆AB C ᠳᠦ᠂ ∠B = ∠C ᠃ ᠪᠠᠳ ᠤᠯᠠ᠄ AB = AC ᠃ ᠪᠠᠳᠤ ᠡᠨᠡ ᠰᠡᠳ ᠤᠪ ᠢ ᠪᠠᠳᠤ ᠯᠠᠬᠤ ᠦᠶ ᠡᠰ᠂ BC ᠶᠢᠨ ᠲᠤᠮ ᠳᠠ ᠴᠡᠭ D ᠶᠢ ᠤ ᠯᠵᠤ ᠂ A D ᠶᠢ ᠬᠤᠯ ᠪᠤᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ᠂ ∆AB D ≌ ∆A CD ᠶᠢ ᠤ ᠯᠵᠤ ᠴᠢᠳ ᠠᠬᠤ ᠦᠬ ᠡᠢ ᠶᠢ ᠮ ᠡᠳᠡᠭᠡᠳ ᠪᠠᠭᠰᠢ ᠠᠴᠠ ᠪ ᠠᠨ ᠠ ᠰᠠᠭᠤ ᠭᠰᠠᠨ ᠳᠤ ᠂ ᠪᠠ ᠭᠰᠢ ᠨᠢ ᠲ ᠡᠭᠦᠨ ᠳᠦ ᠄ ᠪᠠ ᠰᠠ ᠬᠤᠶ ᠠᠷ ᠡ ᠭᠴᠡ ᠭᠡᠷᠴ ᠢᠮ ᠪᠠᠢ ᠭᠤᠯᠴ ᠢᠬᠠᠪᠠ ᠯ ᠪᠠᠳᠤ ᠯᠠᠭᠠᠳ ᠭ ᠠᠷᠭᠠᠴᠢ ᠬᠠᠨ ᠠ ᠭᠡᠪ ᠃ ᠴ ᠢ ᠪᠠ ᠳᠤ ᠶ ᠢ ᠬᠠᠪ ᠰᠤᠷᠤᠨ ᠪᠠᠳ ᠤᠯᠠᠵᠤ ᠦᠭ ᠭᠦᠭᠡᠷᠡᠢ᠃