ᠠᠷᠪᠠᠨ ᠭᠤᠷᠪᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠲᠡᠩᠭᠡᠯᠢᠭ ᠲᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠠᠷᠪ ᠠᠨ ᠭᠤ ᠷᠪᠠᠳ ᠤᠭᠠᠷ 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 ᠃ ︵ ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 30 ᠬᠤ ᠪᠢ ︶ 1. ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠲ ᠠᠪᠤᠨ ᠦᠨᠴ ᠦᠭᠳᠦ AB CD E ᠪ ᠤᠯ ᠵᠦ᠋ᠪ ᠲᠠᠪ ᠤᠨ ᠦᠨᠴ ᠦᠭᠳᠦ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠡᠳ ᠡᠭᠡᠷ ᠵᠢᠷᠤᠭ ᠲᠦ᠋ ᠷᠰᠦ ᠲ ᠤᠳᠤᠷ ᠠ ᠡ ᠴᠡ ᠲᠡᠩ ᠭᠡᠯᠢᠭ ᠲᠤ ᠲ ᠡᠭᠰᠢ ᠭ ᠡᠮᠳᠡᠢ ᠵᠢᠷᠤ ᠭ ᠲ ᠦ᠋ᠷᠰ ᠦ ᠨᠢ ( ) A. 2 B. − 2 C. 12 D. − 12 2. ᠭᠡᠷᠪᠡ ᠴᠡᠭ A (𝑎 − 2,3 ) ᠪᠠ ( −1 , b + 5) ᠨᠢ y ᠲ ᠡᠩᠭᠡᠯ ᠢᠭ ᠦᠨ ᠬᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠭᠡᠮ ᠳᠡᠢ ᠪ ᠤᠯ ᠴ ᠡᠭ C (𝑎 , b) ᠨᠢ ( ) ᠳ ᠤ ᠣ ᠷᠣᠰᠢ ᠨ ᠠ᠃ A. ᠨᠢᠭᠡᠳ ᠦᠭᠡᠷ ᠮ ᠦ᠋ᠴᠡ B. ᠬᠤ ᠶᠠᠳᠤ ᠭᠠᠷ ᠮᠦ᠋ᠴ ᠡ C. ᠭᠤ ᠷᠪᠠᠳ ᠤᠭᠠᠷ ᠮᠦ᠋ᠴ ᠡ D. ᠲᠦ᠋ ᠷᠪᠡᠳ ᠦᠭᠡᠷ ᠮᠦ᠋ ᠴᠡ 3. ᠵᠢᠷᠤ ᠭ 1 ᠮᠡᠳ ᠤ᠂ ∆AB C ᠳ ᠦ᠂ AB = AC ᠂ ᠰᠢᠯᠤ ᠭᠤᠨ DE ᠪᠤᠯ AB ᠶᠢᠨ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ᠂ ᠬ ᠡᠷᠪᠡ BE + CE = 12 , BC = 8 ᠪ ᠤᠯ ∆ AB C ᠶ ᠢᠨ ᠬᠦ ᠷᠢᠶᠡᠨ ᠦ ᠤ ᠷᠳᠤ ᠨ ᠢ ( ) A. 20 B. 32 C. 24 D. 36 4. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠳᠤ᠂ ∆A OD ᠶᠢᠨ ᠰᠢ ᠯᠤᠭᠤ ᠨ𝑙 ᠶᠢᠨ ᠬᠤ ᠪᠢ ᠳ ᠦ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠡᠢ ᠪ ᠣᠯᠬᠤ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠨᠢ ∆B OC ᠂ ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠬᠡᠯ ᠡᠯᠭᠡ ᠳᠤ ᠪᠤᠷᠤᠭᠤ ᠨᠢ ( ) A. ∠D AO = ∠C BO , ∠A DO = ∠B CO B. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑙 ᠨ ᠢ ᠬᠡᠷᠴ ᠢᠮ AB , CD ᠶᠢ ᠡ ᠭᠴᠡ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ᠃ C. ∆A OD ᠪᠠ ∆B OC ᠨᠢ ᠴ ᠦ᠋ᠮ ᠠᠳ ᠠᠯᠢ ᠬᠠᠵᠠᠭᠤ ᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠃ D. AD = BC , OD = OC 5. ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠵᠢᠷᠤᠭ ᠪᠤᠯ ᠮᠢᠭᠮᠡ ᠰᠢᠳ ᠡᠷ ᠡ ᠲᠤᠭᠯ ᠠᠵᠤ ᠪᠠᠢ ᠭ ᠠ ᠵᠢᠷᠤᠭ ᠪᠤᠯᠤ ᠨ ᠠ᠂ ᠰᠢᠲ ᠠᠷ ᠠ ᠶᠢᠨ ᠪᠠᠢ ᠷᠢ ᠶ ᠢ ᠲ ᠠᠷᠠᠭᠠᠯ ᠠᠯᠳᠤ ᠲᠤᠭᠠᠨ ᠬᠣ ᠣᠰ ᠢ ᠶᠠᠷ ᠢᠯᠡᠷᠬ ᠡᠢᠯᠡᠨ ᠡ᠂ ᠵᠢᠱᠢ ᠶᠡᠯᠡᠪᠡᠯ A( 5,1 ) ᠃ ᠲᠠᠬᠭ ᠭᠠ ᠨᠢᠭᠡ ᠬᠠᠷ ᠠ ᠰ ᠢᠲᠠᠷ ᠠ ᠲ ᠠᠯᠪᠢ ᠨ ᠡ ᠳᠡᠭᠡᠷ 8 ᠰᠢᠲ ᠠᠷ ᠠ ᠶᠢ ᠲ ᠡᠩᠭᠡᠯ ᠢᠭ ᠲ ᠦ ᠲ ᠡᠭᠰᠢ ᠬᠡᠮ ᠳᠡᠢ ᠵᠢᠷᠤᠭ ᠳᠦ᠋ ᠷᠰᠦ ᠪᠤᠯᠭᠠᠪᠠ ᠯ ᠬ ᠠᠷ ᠠ ᠰᠢᠲ ᠠᠷ ᠠ ᠲ ᠠᠯᠪᠢ ᠬᠤ ᠪ ᠠᠢᠷᠢ ᠶᠢᠨ ᠪ ᠤᠷᠤᠭᠤ ᠪ ᠣᠯᠬᠤ ᠨᠢ ( ) A. (2 ,3) B. (3 ,2) C. (3 ,4) D. (3 ,1) 6.ᠵ ᠢᠷᠤᠭ 3 ᠮᠡᠳᠤ᠂ ∆AB C ᠳ ᠦ᠂ AB = AC , AD , CE ᠲᠤ ᠰᠪᠤᠷᠢ ∆AB C ᠶᠢᠨ ᠳᠤᠮᠳ ᠠ ᠱ ᠤᠭᠤᠮ ᠪᠠ ᠦᠨᠴ ᠦᠭ ᠦᠨ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴ ᠢ᠂ ∠A CE = 35° ᠦᠶ ᠡᠰ ∠B AD ᠶᠢᠨ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠤᠨ ᠲ ᠣᠭ ᠠ ᠨᠢ ( ) A. 55° B. 40° C. 35° D. 20° 7. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠤ᠂ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ AC DB ᠳᠦ᠂ ∠A = 110° ᠂ ᠬᠡᠷᠪᠡ ᠴᠡᠭ D ᠨᠢ AB , AC ᠶᠢ ᠨ ᠡ ᠭᠴᠡ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠲ ᠡᠭᠡᠷ ᠠ ᠤᠷᠤᠰᠢ ᠪᠠᠯ ∠B DC ᠨᠢ ( ) (1) (2) (3) (4) A. 90° B. 110° C. 120° D. 140° 8. ᠵᠢᠷᠤᠭ 5 ᠮᠡᠳ ᠤ᠂ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠ ᠭᠤᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ∆ AB C ᠳ ᠤ AB = AC , ∠A BC ᠪᠠ ∠A CB ᠶᠢᠨ ᠦᠨᠴ ᠦᠭ ᠦᠨ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬ ᠤᠪᠢᠶ ᠠᠭᠴᠢ ᠨᠢ ᠴ ᠡᠭ O ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ ᠴ ᠡᠭ O ᠶᠢ ᠲ ᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨ DE // BC ᠪᠠᠷ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ AB , AC ᠲᠠᠢ ᠲᠤ ᠰ ᠲᠤᠰ ᠴᠡᠭ D , E ᠳᠤ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ ∆ AD E ᠶ ᠢᠨ ᠬᠦᠷᠢ ᠶᠡᠨ ᠦ ᠤᠷᠳ ᠤ ᠨᠢ 18 ᠪᠣᠯ AB ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨ ᠢ ( ) A. 8 B. 9 C. 10 D. 12 9. ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ᠬᠡᠯ ᠪᠡᠷᠢ ᠳᠦ᠋ ᠷᠰᠦ ᠨᠢ ᠠ ᠳᠠᠯᠢ ᠦᠭᠡᠢ ᠴᠠᠭᠠᠰ ᠤᠨ ᠬᠠᠪᠲ ᠠᠰᠤ ᠶᠢ ᠨᠢᠭᠡ ᠰᠢ ᠯᠤᠭᠤ ᠨ ᠲᠠᠭᠠᠤ ᠬᠠᠢ ᠴᠢᠯᠠᠵᠤ ( ᠨ ᠢᠭᠡ ᠯᠠ ᠤᠳᠠ ᠭ ᠠ ) ᠬ ᠤᠶᠠᠷ ᠠᠳ ᠠᠯᠢ ᠬᠠᠵᠠ ᠭᠤᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠤᠯ ᠵᠣ ᠲ ᠡᠢᠯᠬ ᠦ ᠦᠭᠡᠢ ᠨᠢ ( ) A. B. C. D. 10. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ᠂ ∆AB C ᠳ ᠤ ∠A CB = 90° ᠂ D ᠪᠣᠯ AB ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠴᠡᠭ᠂ ᠴ ᠡᠭ D ᠶᠢ ᠲᠠᠭᠠᠷ ᠢᠭᠤᠯ ᠤᠨ DE ⊥ AB ᠪᠠ ᠷ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ BC ᠲᠠᠢ ᠴᠡᠭ F ᠲ ᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ AC ᠶᠢ ᠨ ᠦ ᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲᠠᠢ ᠴᠡᠭ E ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ CD ᠶ ᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠃ ∠D CA = ∠D AC ᠪᠣ ᠯ ᠲ ᠠᠷᠠᠭᠠᠬ ᠢ ᠳ ᠦ᠋ᠩᠨᠡᠯ ᠲᠡ ᠳᠦ ᠵᠦ᠋ᠪ ᠨᠢ ( ) ① ∠D CB = ∠B ②CD = 12 AB ③∆ AD C ᠪ ᠣᠯ ᠲᠡᠴ ᠴᠡᠬᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ④ ᠬᠡᠷᠪᠡ ∠E = 30° ᠪᠣᠯ DE = EF + CF A. ① ② ③ B. ① ② ④ C. ② ③ ④ D. ① ② ③ ④ ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠳ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ( ᠰ ᠡᠲᠤᠪ ᠪᠦᠷ ᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢ ᠲᠡ 24 ᠬᠤ ᠪᠢ ) 11. ᠵᠢᠷᠤᠭ 7 ᠮᠡᠳ ᠦ᠂ ∆ AB C ᠳᠦ ᠂ AB = AC , ∠A = 36° , AC ᠶᠢᠨ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠨᠢ AB ᠲᠠᠢ ᠴᠡᠭ E ᠳᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ ᠡ ᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨᠢ D ᠂ EC ᠶᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ ᠃ ᠲ ᠡᠭᠡᠪᠡᠯ ∠B EC = ( ) 12. ᠵᠢᠷᠤᠭ 8 ᠮᠡᠳ ᠤ᠂ ᠨᠢᠭᠡᠨ ᠬᠦᠯ ᠬᠡ ᠣᠩ ᠭᠤᠴ ᠡ ᠪᠠ ᠷᠠᠭᠤᠨ ᠠᠴᠠ ᠵᠡᠬᠦᠨᠰ ᠢ ᠠ ᠶᠠᠯᠠᠵᠤ ᠪᠠᠢ ᠨ ᠠ᠂ A ᠭᠠᠵᠠ ᠷ ᠠᠴ ᠠ ᠬ ᠡᠮᠵᠢᠨ ᠤ ᠯᠤᠭᠰᠠᠨ ᠨᠢ ᠠᠷᠠᠯ P ᠶ ᠢᠨ ᠪᠠᠢ ᠷᠢ ᠨᠢ ᠤ ᠮᠠᠷ ᠠ ᠠᠴᠠ ᠵᠡᠬᠦᠨᠰ ᠢ 75° ᠤᠨ ᠴᠢ ᠭᠯᠡᠯ ᠳᠦ ᠪᠠᠢ ᠵᠤ᠂ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠨ 8 ᠲ ᠠᠯᠠᠢ ᠶᠢᠨ ᠮᠢᠯ ᠠᠶᠠᠯ ᠠᠭᠠ B ᠭᠠᠵᠠ ᠷ ᠠᠴ ᠠ ᠠ ᠵᠢᠭᠯ ᠠᠬᠤ ᠳᠤ ᠠᠷᠠᠯ P ᠶᠢᠨ ᠪᠠ ᠢᠷᠢ ᠨᠢ ᠤᠮᠠᠷ ᠠ ᠠᠴ ᠠ ᠵᠡᠬᠦᠨ ᠰᠢ 60 ° ᠤ ᠨ ᠴ ᠢᠭᠯᠡᠯ ᠳᠦ ᠪᠠᠢ ᠬᠤ ᠶ ᠢ ᠬᠡᠮᠵ ᠢᠨ ᠣᠯᠪᠠ ᠂ ᠡᠨᠡ ᠦᠶ ᠡᠰ ∆AB P ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠ ᠢ ᠨᠢ_____ ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮᠢᠯ᠃ 13. ᠵᠢᠷᠤ ᠭ 9 ᠮᠡᠳ ᠤ᠂ ∆ AB C ᠳ ᠦ ᠂ AB = AC , ∠B AC = 120° , AD ⊥ AC , BC ᠲᠠᠢ ᠴ ᠡᠭ D ᠳᠦ ᠣᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ AD = 3 ᠪ ᠣᠯ᠂ BC =____ 14. ᠵᠢ ᠷᠤᠭ 10 ᠮᠡᠳ ᠤ᠂ ∆AB C ᠳᠦ ᠂ ∠A CB = 90° , ∠ A = 30° , BC = 4 ᠴ ᠡᠭ C ᠪᠠᠷ ᠲᠤ ᠭᠤᠷᠢ ᠭ ᠤᠨ ᠲᠦ᠋ ᠪ ᠪᠤ ᠯᠭᠠᠵᠤ (5) (6) D A B C (7) (8) (9) (10) CB ᠪᠠ ᠷ ᠷᠠᠳ ᠢᠤᠰ ᠪᠤᠯ ᠭᠠᠨ ᠨᠤ ᠮᠤ ᠪᠠ ᠢᠭᠤ ᠯᠪᠠᠯ AB ᠲᠠᠢ ᠴᠡᠭ D ᠳᠤ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ ᠳᠠᠬ ᠭᠭᠠ ᠲᠤᠰ ᠪᠤᠷᠢ ᠴᠡᠭ B ᠪᠠ D ᠪᠠᠷ ᠲᠣᠭᠤ ᠷᠢᠭ ᠤ ᠨ ᠲ ᠥᠪ ᠪᠣᠯᠭᠠ 12 BD ᠡᠴᠡ ᠶᠡᠬᠡ ᠤᠷᠳ ᠤ ᠪᠠ ᠷ ᠷᠠᠳ ᠢᠤᠰ ᠪᠤᠯᠭᠠᠨ ᠬᠤᠶ ᠠᠷ ᠨ ᠤᠮᠤ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ ᠲ ᠡᠳᠡᠭᠡᠷ ᠨᠢ ᠴ ᠡᠭ E ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠵᠤ᠂ ᠰᠠᠴ ᠤᠷᠠᠭ CE ᠶᠢ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ AB ᠲᠠᠢ ᠴ ᠡᠭ F ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ ᠲᠡᠭᠡᠪᠡᠯ AF ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨ ᠢ᠄ ( ) 15. ᠵᠢᠷᠤᠭ 11 ᠮᠡᠳᠤ᠂ ∆AB C ᠳ ᠦ ᠂ ∠A BC ᠶᠢᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠪᠣᠯᠤ ᠨ∠A CB ᠶᠢᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠭᠴ ᠢ ᠨᠢ ᠴᠡᠭ D ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ ᠴᠡᠭ D ᠶ ᠢ ᠲ ᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨBC ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠤ ᠭᠤᠨ ᠪᠠᠢ ᠭᠤᠯᠪᠠ ᠯ AB ᠲᠠᠢ E ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ AC ᠲᠠᠢ ᠴᠡᠭ F ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ∠B ED + ∠C FD = 240° ᠪᠤᠯ ∠B DC =______ 16. ᠵᠢ ᠷᠤᠭ 12 ᠮᠡᠳ ᠤ᠂ ∆AB C ᠳᠦ ᠂ ∠C = ∠A BC , BE ⊥ AC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ ᠴ ᠡᠭ E ᠂ ∆ BD E ᠨᠢ ᠲᠡᠴ ᠴᠡᠭᠦ ᠦ ᠲᠠᠯᠠᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠂ ᠬ ᠡᠷᠪᠡ AD = 4 ᠪ ᠣᠯ ᠬᠡᠷᠴ ᠢᠮ BE ᠶᠢᠨ ᠤ ᠷᠳᠤ ᠨᠢ_____ 17. ᠵᠢᠷᠤᠭ 13 ᠮᠡᠳ ᠦ᠂∠A OB = 30° , ᠴᠡᠭ P ᠨ ᠢ ∠A OB ᠶᠢᠨ ᠳᠣ ᠲᠤᠷ ᠠ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠂ ᠴᠡᠭ 𝑃 1 ᠪᠠ ᠴᠡᠭ P ᠨᠢ OA ᠶᠢ ᠨ ᠬ ᠤᠪᠢ ᠳᠦ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠠᠢ᠂ ᠴᠡᠭ 𝑃 2 ᠪᠠ ᠴᠡᠭ P ᠨᠢ OB ᠶᠢᠨ ᠬᠤᠪᠢ ᠳᠦ ᠲ ᠡᠭᠰᠢ ᠬᠡᠮᠳ ᠠᠢ᠂ 𝑃1 ᠪᠠ 𝑃 2 ᠶ ᠢ ᠬ ᠤᠯᠪᠤ ᠪᠠᠯ OA , OB ᠲᠠᠢ ᠲᠤ ᠰ ᠲ ᠤᠰ ᠴᠡᠭ E , F ᠳᠤ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ᠲᠡᠭᠡ ᠪᠡᠯ ∠E PF =______ 18. ᠵᠢ ᠷᠤᠭ 14 ᠮ ᠡᠳᠤ᠂ ∆ AB C ᠳᠦ ᠂ AB = AC , ∠A BC = 75° , AD ⊥ BC ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ D ᠂ ᠴ ᠡᠭ D ᠶᠢ ᠨ AB , AC ᠶᠢᠨ ᠬᠤ ᠪᠢ ᠳ ᠠᠬᠢ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠦ ᠴᠡᠭ ᠨᠢ ᠲ ᠤᠰ ᠲ ᠤᠰ ᠴ ᠡᠭ E , F ᠂ E ᠪᠠ F ᠶ ᠢ ᠬᠤ ᠯᠪᠤᠪᠠ ᠯ AB , AC ᠲᠠᠢ ᠴᠡᠭ M , N ᠳᠤ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ DM , DN ᠶᠢ ᠬ ᠤᠯᠪᠤ ᠨ ᠠ᠃ ᠬᠡᠷᠪ ᠡ AD = 6 ᠪᠣᠯ ∆D MN ᠶᠢ ᠨ ᠬᠦᠷ ᠢᠶᠡᠨ ᠤ ᠤ ᠷᠲᠦ ᠨ ᠢ____ ___ ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ (ᠨᠠᠢ ᠲᠡ 66 ᠬᠤᠪᠢ ) 19. ( 10 ᠬᠤ ᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ 1 ᠨᠢᠭᠡᠴ ᠢ ᠤ ᠷᠳᠤ ᠪᠣᠯᠬᠤ ᠵᠢᠵ ᠢᠭ ᠲ ᠡᠭᠰᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠢᠶᠡᠷ ᠪᠦᠷ ᠢᠯᠳᠦᠭᠰ ᠡᠨ ᠬ ᠠᠨᠠᠯᠵᠢ ᠵᠢᠷᠤᠭ ᠳᠡᠭᠡ ᠷ ᠡ ᠳ ᠦ᠂ ∆AB C ᠪᠠ ∆D EF ( ᠤ ᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ ᠴ ᠦ᠋ᠮ ᠵᠢᠵ ᠢᠭ ᠲ ᠡᠭᠰᠢ ᠳᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠤᠷᠤ ᠢ ᠶᠢ ᠨ ᠴ ᠡᠭ ) ᠵᠢᠴ ᠢ ᠰ ᠢᠯᠤᠭᠤ ᠨ 𝑙 ᠪᠠᠢ ᠨ ᠠ᠃ (1) ∆AB C ᠶ ᠢ ᠪᠠ ᠷᠠᠭᠤᠨᠰᠢ 2 ᠨᠢᠭᠡᠴ ᠢ ᠤ ᠷᠲᠤ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠵᠦ ᠳᠠᠬᠭ ᠭᠠ ᠳᠤᠷᠤᠭ ᠰᠢ 2 ᠨᠢᠭᠡᠴ ᠢ ᠤᠷᠲ ᠤ ᠵᠡᠷᠭᠡᠴ ᠡᠭᠡ ᠰ ᠢᠯᠵᠢ ᠬᠦᠯᠦᠭᠰᠡᠨ ᠦ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠢ ᠵ ᠢᠷᠤ᠃ (2) ∆D EF ᠶᠢᠨ ᠰᠢᠯ ᠤᠭᠤᠨ 𝑙 ᠶ ᠢᠨ ᠬᠤ ᠪᠢ ᠳ ᠠᠬᠢ ᠲᠡᠭ ᠰᠢ ᠬ ᠡᠮᠳᠠᠢ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠢ ᠵ ᠢᠷᠤ᠃ B A C D E F E FO P2 A P B P1 NM A B C FE D (11) (12) (13) (14) 20. ( 10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠦ᠂ ᠴᠡᠭ C ᠨᠢ ᠬ ᠡᠷᠴᠢ ᠮ AB ᠳᠡᠭᠡ ᠷ ᠡ ᠤᠷᠤᠰᠢ ᠨ ᠠ᠂ AD // EB , AC = BE , AD = BC , CF ᠨᠢ ∠D CE ᠶᠢ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ᠃ ᠪᠠᠳᠤ ᠯᠠᠭᠳ ᠠᠬᠤ ᠨᠢ᠄ CF ⊥ DE ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ F 21. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ ∆ AB C ᠳ ᠦ ᠂ ᠰᠢᠯᠤ ᠭᠤᠨ 𝑙 ᠨᠢ AB ᠲᠠᠢ ᠴᠡᠭ M ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ BC ᠲᠠᠢ ᠴᠡᠭ N ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ ᠴᠡᠭ B ᠶᠢ ᠨ ᠰ ᠢᠯᠤᠭᠤ ᠨ 𝑙 ᠶᠢᠨ ᠬᠤᠪᠢ ᠳᠠᠬ ᠢ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠦ ᠴᠡᠭ D ᠨᠢ ᠬᠡᠷᠴ ᠢᠮ BC ᠬᠡᠷᠴ ᠢᠮ ᠳᠡᠭᠡ ᠷ ᠡ ᠤᠷᠤᠰ ᠢᠬᠤ ᠪᠦᠭᠡ AD ⊥ DM , ∠B = 28° ᠲᠡᠭᠡᠪᠡᠯ ∠D AB ᠶᠢ ᠨ ᠶᠡᠭ ᠡ ᠪᠠ ᠭ ᠠ ᠶᠢ ᠣ ᠯ᠃ B D A E F C l N M A C BD 22. (12 ᠬ ᠤᠪᠢ ) ᠵ ᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ∆AB C ᠳᠦ ᠂ D ᠨᠢ BC ᠶᠢᠨ ᠳᠤ ᠮᠳᠠ ᠴᠡᠭ᠂ DE ⊥ AB ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ E ᠂ DF ⊥ AC ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ F , BE = CF ᠃ (1) ᠪᠠᠳ ᠤᠯᠠᠭ ᠳᠠᠬᠤ ᠨᠢ AD ᠨᠢ ∠B AC ᠶᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ᠃ (2) ᠪᠠᠳ ᠤᠯᠠ᠄ EF ᠶᠢ ᠬᠤ ᠯᠪᠤᠪᠠ ᠯ AD ᠨᠢ EF ᠶᠢ ᠡ ᠭᠴᠡ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤ ᠪᠢᠶᠠᠨ ᠠ᠃ 23. (12 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤ ᠭ ᠮ ᠡᠳᠤ᠂ ∆AB C ᠪᠣᠯ ᠲᠡᠴ ᠴᠡᠬᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠂ D ᠨᠢ BC ᠶᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠳ ᠡᠭᠡᠷᠡᠬ ᠢ ᠨ ᠢᠭᠡ ᠴᠡᠭ᠂ AD ᠶᠢ ᠬᠤ ᠯᠪᠤᠨ ᠠ᠂ AD ᠪᠠᠷ ᠨᠢᠭᠡ ᠲᠠᠯ ᠠ ᠪᠤᠯᠭᠠᠨ ᠲᠡᠴ ᠴᠡᠬᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ∆AD E ᠶ ᠢ ᠪᠠ ᠢᠭᠤ ᠯᠤᠭᠠ CE ᠶᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠃ ᠰ ᠤᠷᠤᠭ ᠰᠠᠨ ᠮᠡᠳᠡᠯ ᠭᠡ ᠪᠠ ᠨ ᠠ ᠰᠢᠭᠯ ᠠᠨ AC , CD , CE ᠶᠢᠨ ᠲᠣ ᠭ ᠠ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠬ ᠦᠨ ᠦ ᠬᠤᠯ ᠪᠤᠭᠳ ᠠᠯ ᠢ ᠰᠦ᠋ ᠪᠡᠭᠴᠢ ᠯᠡ᠃ E F D A B C 24. (1 2 ᠬ ᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ Rt ∆AB C ᠳ ᠦ ᠂ ∠C = 90° , ∠ A = 60° , AB = 10 cm ᠂ ᠬ ᠡᠷᠪᠡ ᠴᠡᠭ M ᠨᠢ ᠴᠡᠭ B ᠡᠴᠡ ᠡᠬᠭᠯ ᠡᠨ 2m /s ᠤᠨ ᠬᠤ ᠷᠳᠤᠴ ᠠ ᠪᠠ ᠷ ᠴ ᠡᠭ A ᠵᠦ᠋ᠭ ᠬᠦᠳ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬ ᠭᠨ ᠡ ᠃ ᠴ ᠡᠭ N ᠨᠢ ᠴᠡᠭ A ᠡᠴᠡ ᠡ ᠬᠭᠯᠡᠨ 1m /s ᠤᠨ ᠬᠤ ᠷᠳᠤᠴ ᠠ ᠪᠠ ᠷ ᠴᠡᠭ C ᠵᠦ᠋ᠭ ᠬᠦᠳ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬ ᠭᠨ ᠡ ᠃ ᠴ ᠡᠭ M , N ᠨᠢ ᠲᠤᠰ ᠪᠤᠷᠢ ᠴᠡᠭ B , A ᠡᠴ ᠡ ᠵ ᠡᠷᠭᠡᠪᠡᠷ ᠭᠠᠷᠴᠤ 𝑡( 𝑠) ᠬᠦᠳ ᠡᠯᠭᠡᠭ ᠡᠨ ᠬ ᠭᠨ ᠡ ᠃ (1) 𝑡 ᠠᠭᠤ ᠯᠤᠭᠰᠠᠨ ᠢᠯ ᠡᠷᠬᠡᠢ ᠯᠡᠯ ᠢ ᠶᠡᠷ ᠬᠡᠷᠴ ᠢᠮAM , AN ᠶᠢ ᠨ ᠤ ᠷᠳᠤ ᠶᠢ ᠢᠯ ᠡᠷᠬᠡᠢ ᠯᠡ᠃ (2) 𝑡 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠠ ᠪᠬᠤ ᠦᠶ ᠡᠰ ∆ AM N ᠨᠢ M N ᠪᠡᠷ ᠰ ᠠᠭᠤᠷᠢ ᠪᠤᠯᠭᠠᠭᠰᠠᠨ ᠠᠳ ᠠᠯᠢ ᠬᠠᠵᠠ ᠭᠤᠳ ᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠪᠣᠯ ᠬᠤ ᠪ ᠤᠢ ︖ (3) 𝑡 ᠶᠠᠮᠠᠷ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠠᠪᠬ ᠤ ᠦᠶ ᠡᠰ᠂ MN ∥ BC ᠪᠣᠯ ᠬᠤ ᠪᠤᠢ ︖ ᠡ ᠨᠡ ᠦᠶ ᠡᠰ ᠳᠡᠬᠢ 𝐶𝑁 ᠦ ᠤᠷᠳ ᠤ ᠶᠢ ᠣ ᠯ᠃ A C B M N