2019 ᠣᠨ ᠤ ᠤᠯᠠᠭᠠᠨᠬᠠᠳᠠ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 19 ᠣᠨ ᠤ ᠤᠯ ᠠᠭᠠᠨᠬ ᠠᠳᠠ ᠬᠤᠲᠠ ᠶᠢᠨ ᠠ ᠩᠬ ᠠ ᠳᠤᠮᠳᠠᠳ ᠤ ᠶᠢ ᠨ ᠰ ᠤᠷᠤᠭᠴ ᠢᠳ ᠤᠨ ᠰᠤᠷᠤᠯ ᠭ ᠠ ᠲᠡᠭᠦᠰ ᠬᠦ ᠂ ᠰᠤᠷ ᠭᠠᠭᠤᠯ ᠢ ᠳ ᠠᠪᠰᠢᠬ ᠤ ᠨᠢ ᠭᠡᠲᠦ ᠯᠲᠠᠢ ᠰᠢᠯ ᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠳᠦᠪ ᠲᠣᠭᠠᠨ ᠤ ᠤᠬᠠ ᠭᠠᠨ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰᠡ ᠳᠦᠪ ︵ ᠰ ᠡᠳᠦᠪ ᠪ ᠦᠷᠢ3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢᠲᠡ 42 ᠬᠤᠪᠢ ︶ 1.− 4 , − √2 , 0 , 4 ᠭᠡᠬᠦ ᠳᠦᠷ ᠪᠡᠨ ᠲᠤᠭ ᠠᠨ ᠳᠠᠬ ᠢ᠂ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠲᠣᠭ ᠠ ᠨᠢ A. 4 B. 0 C. − √2 D . − 4 2. 20 13 — 20 18 ᠣᠨ ᠪᠤᠯᠲ ᠠᠯ ᠠ᠂ ᠮᠠᠨ ᠤ ᠤᠯᠤᠰ ᠠᠴᠠ ︽ ᠨᠢ ᠭᠡᠨ ᠪᠦᠰ ᠡ ᠨᠢ ᠭᠡ ᠵᠠᠮ︾ ᠳᠠᠭ ᠠᠤ ᠶᠢᠨ ᠤᠯᠤᠰ ᠯᠤ ᠭ ᠠ ᠬᠢᠭᠰᠡᠨ ᠡᠳ᠋ ᠪᠠᠷᠠᠭ ᠠ ᠶᠢᠨ ᠠᠷᠠᠯᠵ ᠢᠶᠠᠨ ᠤ ᠶᠡᠷᠦ ᠩᠬᠡᠢ ᠬᠡᠮᠵᠢᠶ ᠡ ᠨᠢ 60 00 0 ᠳᠤᠩ ᠰᠢᠭᠤᠷ ᠳᠣᠯᠯ ᠠᠷ ᠢ ᠳᠠᠪᠠ ᠵᠠᠢ᠃ 60 00 0 ᠭᠡᠳᠡ ᠭ ᠲᠣᠭ ᠠ ᠶᠢ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤᠬᠠᠭᠠ ᠨᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡ ᠮᠳᠡᠭᠯᠡᠬᠦ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠨᠢ ︵ ︶ A. 6 B 0. 6 C. 6 D. 6 3. ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠪᠣᠳᠤ ᠯᠭ ᠠ ᠨᠤᠭ ᠤᠳ ᠠᠴᠠ ᠵᠦᠪ ᠨᠢ ︵ ︶ A. √ + √2 = √ B . = C. ( ) = D. = 4. ᠲᠤᠩᠭ ᠠᠯᠠᠭ ᠪᠤᠰᠤ ᠤᠭ ᠤᠲᠠᠨ ᠳᠤ ᠶᠡᠬᠡ ᠪᠠᠭ ᠠ ᠬᠡᠯᠪᠡ ᠷᠢ ᠳᠦᠷᠰᠦ ᠨᠢ ᠪᠦᠷᠢᠮ ᠦᠰᠦᠨ ᠠᠳᠠ ᠯᠢ ᠪᠣᠯᠬᠤ 4 ᠬᠠᠷ ᠠ ᠪᠦᠮᠪᠦᠭᠡ ᠪᠠ 2 ᠴᠠᠭᠠᠨ ᠪᠦᠮᠪᠦ ᠭᠡ ᠪᠤᠢ᠂ ᠤᠭ ᠤᠲᠠᠨ ᠳᠣᠲ ᠤᠷ ᠠ ᠠᠴ ᠠ ᠲᠠ ᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ 3 ᠪᠦᠮᠪᠦᠭᠡ ᠶᠢ ᠭᠠᠷᠭ ᠠᠨ ᠠᠪᠤᠨ ᠠ᠃ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠶᠠᠪᠤᠳᠠᠯ ᠨᠤᠭ ᠤᠳ ᠲᠠ ᠬᠢ ᠵᠠᠢᠯ ᠠᠰᠢᠭᠦᠢ ᠶᠠᠪᠤᠳᠠᠯ ᠪᠣᠯ ᠬᠤ ᠨᠢ ︵ ︶ A. 3 ᠨᠢ ᠪᠦᠷ ᠬᠠᠷ ᠠ ᠪᠦᠮᠪᠦᠭᠡ B . 2 ᠬᠠᠷ ᠠ ᠪᠦᠮᠪᠦᠭᠡ 1 ᠴᠠᠭᠠᠨ ᠪᠦᠮᠪᠦᠭᠡ C. 2 ᠴᠠᠭᠠᠨ ᠪᠦᠮᠪᠦᠭ ᠡ 1 ᠬᠠᠷ ᠠ ᠪᠦᠮᠪᠦᠭᠡ D. ᠠ ᠳᠠᠭ ᠲᠤ ᠪᠡᠨ 1 ᠬᠠᠷ ᠠ ᠪᠦᠮᠪᠦᠭᠡ 5. ᠳᠣᠣ ᠷᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠪᠣᠯ ᠨᠢᠭᠡ ᠭᠸᠥᠮ ᠸ᠋ᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠭᠤᠷᠪ ᠠᠨ ᠬᠠᠷ ᠠᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠮᠦᠨ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠡᠨᠡ ᠬᠦ ᠭ ᠸᠥᠮᠸ᠋ᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠨᠢ ︵ ︶ A. ᠭᠤᠷᠪ ᠠᠨ ᠢ ᠷᠮᠡᠭᠳᠦ ᠬᠤᠰ ᠢᠭᠤᠯᠵᠢᠨ B . ᠱᠣᠷᠤ ᠯᠵᠢᠨ C. ᠭᠤᠷᠪ ᠠᠨ ᠢ ᠷᠮᠡᠭᠳᠦ ᠪᠠᠭ ᠠᠨ ᠠ D. ᠪᠦᠭᠡᠷᠡ ᠩᠬᠡᠢ ᠪᠠᠭᠠᠨ ᠠ 6. ᠲᠡᠩ ᠴᠡᠳ ᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢ ᠨ ᠪᠦ ᠯᠦᠭ { − 2 2 ᠦ ᠨ ᠬᠠᠷᠢᠭ ᠤ ᠴᠤᠭᠯ ᠠᠷᠠᠯ ᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠡ ᠵᠦᠪ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠨᠢ ︵ ︶ 7. ᠵᠢᠷ ᠤᠭ1 ᠪᠤᠯ ᠶᠢᠰᠦ ᠳᠦᠭᠡᠷ ᠣᠨ ᠤ ᠨᠢ ᠭᠡᠨ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠨ ᠰ ᠢᠯ ᠰᠠᠪᠠ ᠳᠤ ᠲᠤᠰᠤ ᠭᠠᠭᠰᠠᠨ ᠲᠠᠢᠯ ᠪᠤᠷᠢ ᠵᠢᠷᠤᠭ ᠮᠦᠨ ︵ ᠤ ᠰᠤ ᠲᠤᠰᠤ ᠭᠠᠬᠤ ᠬᠤᠷᠳᠤᠴᠠ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ ︶᠂ ᠰᠠᠪ ᠠ ᠳᠠᠬ ᠢ ᠤᠰᠤᠨ ᠤ ᠭᠦᠨ ︵ h︶ ᠪᠤᠯ ᠤᠨ ᠬᠤᠭ ᠤᠴᠠᠭ ᠠ ︵t︶ ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠬᠤᠯ ᠪᠤᠭᠳᠠᠯ ᠢ ᠵᠦᠪ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠶᠡᠬᠡ ᠲᠦ ᠯᠦᠪ ᠵᠢᠷᠤᠮᠠᠭ ᠨᠢ ︵ ︶ 8. ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ ᠷᠤᠮᠪ A BC D ᠶᠢᠨ ᠬᠦᠷ ᠢᠶᠡᠨ ᠦ ᠤᠷᠲᠤ ᠨᠢ 20 , ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ A C, B D ᠨᠢ ᠴᠡ ᠭ O ᠳᠤ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠵᠤ᠂ E ᠨᠢ CD ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠪᠤᠯ OE ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠨ ᠢ ︵ ︶ A. 2.5 B . 3 C. 4 D. 5 9. ᠨᠢᠭᠡ ᠵᠦᠢᠯ ᠦᠨ ᠨᠡᠷᠡᠲᠦ ᠭᠠᠷ ᠤᠲᠠᠰᠤ ᠶᠢ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠰ ᠠᠷ ᠠ ᠳ ᠤ 4 00 ᠲᠦᠮᠡ ᠪᠣᠷᠤᠯ ᠠᠭᠤᠯ ᠵᠠᠢ᠃ ᠳ ᠥᠷᠪᠡ ᠲᠠᠪᠤᠳ ᠤᠭᠠᠷ ᠰᠠᠷ ᠠ ᠶᠢᠨ ᠪᠤᠷᠤᠭ ᠤᠯᠤᠯ ᠲᠠ ᠶᠢᠨ ᠬᠡᠮ ᠵᠢᠶ ᠡ ᠨᠢ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠨ ᠨᠡᠮᠡᠭ ᠳᠡᠵᠦ᠂ ᠲᠠᠪᠤᠳᠤ ᠭᠠᠷ ᠰᠠᠷ ᠠ ᠶᠢᠨ ᠪᠤᠷᠤᠭᠤᠯ ᠤᠯᠲᠠ ᠶᠢ ᠨ ᠬᠡ ᠮᠵᠢᠶ ᠡ ᠨᠢ 900 ᠲᠦᠮᠡ ᠳᠦ ᠬᠦᠷᠪᠡ᠃ ᠰᠠᠷ ᠠ ᠪᠦᠷᠢ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠨᠡ ᠮᠡᠭᠳᠡᠯ ᠦᠨ ᠨᠣᠷᠮ ᠠ ᠶᠢ ᠣᠯ᠃ ᠰᠠᠷ ᠠ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠳᠣᠮᠳᠠᠴᠢ ᠨᠡᠮᠡᠭ ᠳᠡᠯ ᠦ ᠨ ᠨᠣᠷᠮ ᠠ ᠶᠢ ᠭᠡᠪᠡᠯ᠂ ᠰᠡᠳᠦᠪ ᠦᠨ ᠤᠳᠬ ᠠ ᠶᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠨ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠰ ᠠᠨ ᠲ ᠡᠭᠰᠢᠳᠭᠡᠯ ᠡᠴ ᠡ ᠵ ᠦᠪ ᠨᠢ ︵ ︶ A. 40 0 ︵ 1+ ︶ = 90 0 B. 4 00 ︵1+ 2 ︶= 90 0 C. 90 0 ︵1− ︶ 2 = 400 D . 4 00 ︵1− ︶ 2 = 900 10 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ A B ᠨᠢ ᠶᠢᠨ ᠬᠦᠪ ᠴᠢ᠂ OC A B ᠨᠢ ᠯᠤ ᠭ ᠠ ᠴᠡᠭ C ᠳᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠴᠡ ᠭ D ᠨᠢ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡ ᠴᠡᠭ ᠂ AD C = 30 , ᠲᠡᠭᠡᠪ ᠡᠯ ᠶᠢᠨ ᠭᠷᠡᠳᠦᠰ ᠦ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ ︵ ︶ A 30 B 4 0 C 5 0 D 60 11 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ ᠴᠡ ᠭ P ᠨᠢ ᠤᠷ ᠪᠠᠭᠤ ᠬᠠᠷᠢᠴᠠᠯ ᠲᠤ ᠹ ᠦᠩᠺᠽ ( ) ᠦᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂ ᠴᠡᠭ ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠣᠷᠢ ᠨᠢ M , ᠬᠡᠷᠪᠡ PO M ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 2 ᠲᠠ ᠢ ᠲᠡ ᠩᠴᠡᠪᠡ ᠯ ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ ︵ ︶ A − B 4 C − 2 D 2 12 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ D , E ᠨᠢ ᠲᠤᠰ ᠪᠦᠷᠢ A BC ᠶᠢᠨ ᠲᠠᠯ ᠠ A B, A C ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ ᠂ A DE = AC B , ᠬᠡᠷᠪᠡ AD =2 , AB = 6 , AC = 4 ᠪᠣᠯ AE ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ ︵ ︶ A. B. 2 C. 3 D. 4 13 . ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ ᠴᠡᠭ D ᠨᠢ BC ᠶᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠤᠰ ᠢᠵᠤ᠂ D E AB ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ E,AC ᠯᠦᠭᠡ ᠴᠡᠭ F ᠲᠦ ᠣ ᠭᠲᠤᠯ ᠣᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ A = 3 5 o, D =1 5 ᠪᠣᠯ A CB ᠶᠢᠨ ᠭᠷᠡᠳᠦᠰ ᠦ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ ︵ ︶ A. 65 B. 70 C . 7 5 D. 8 5 14 .ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ ᠰ ᠡᠷᠭᠦᠯ ᠡᠩ ᠪᠠᠭᠰᠢ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 1 ᠪᠣᠯᠬᠤ ᠲᠡᠭᠰᠢ ᠳᠦᠷᠪ ᠡᠯᠵᠢᠨ ᠬᠡᠯ ᠪᠡᠷᠢᠲᠦ ᠴᠠᠭᠠᠰᠤ ᠶᠢ ᠳᠠ ᠷᠠᠭᠠ ᠬᠢ ᠠᠷᠭ ᠠ ᠪ ᠡᠷ ᠨᠤᠭ ᠤᠯᠪᠠ᠄ ① ᠲᠡ ᠭᠰᠢ ᠳᠦᠷᠪ ᠡᠯᠵᠢᠨ ᠬᠡᠯᠪᠡ ᠷᠢᠲᠦ ᠴᠠ ᠭᠠᠰᠤᠨ ᠤ ᠳᠦᠷᠪ ᠡᠨ ᠥᠨᠴᠦ ᠭ ᠳᠣᠲᠤᠭᠰ ᠢ ᠬᠠᠨ ᠳᠤᠭᠤᠯ ᠤᠨ ᠨᠤᠭ ᠤᠯᠵᠤ᠂ ᠳᠦᠷᠪ ᠡᠨ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠭᠤᠯ ᠪᠠ᠂ ᠳᠡᠯᠭᠡᠭ ᠰᠡᠨ ᠦ ᠳᠠ ᠷᠠᠭ ᠠ ᠨᠤᠭ ᠤᠯᠤᠭᠠ ᠰᠤ ᠪᠠ ᠷ ᠨᠢ ᠳᠦᠷᠪ ᠡᠨ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤᠲᠤ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠳᠦ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠢ ᠬᠠᠢ ᠴᠢᠯᠠᠵᠤ ᠣᠷ ᠬᠢᠨ ᠠ᠃ ② ᠦᠯ ᠡᠳᠡᠭᠰᠡᠨ ᠲᠡᠭ ᠰᠢ ᠳᠦᠷᠪ ᠡᠯᠵᠢᠨ ᠬᠡᠯᠪᠡᠷ ᠢᠲᠦ ᠴᠠᠭᠠᠰᠤ ᠶᠢ ᠠᠳᠠᠯᠢ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠳᠠᠬ ᠢᠨ ᠳᠠᠪᠲᠠ ᠨ ᠨᠤᠭ ᠤᠯᠵᠤ᠂ 20 19 ᠤᠳᠠ ᠭ ᠠ ᠨᠤᠭ ᠤᠯᠤᠭᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠦᠯ ᠡᠳᠡᠭᠰᠡᠨ ᠲᠡ ᠭᠰᠢ ᠳᠦᠷᠪ ᠡᠯᠵᠢ ᠨ ᠬᠡ ᠯᠪᠡᠷ ᠢᠲᠦ ᠴᠠᠭᠠᠰᠤ ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ︵ ︶ A . 2 B. C . D . ᠬᠤᠶ ᠠᠷ᠂ ᠬᠤ ᠭᠤᠰᠤ ᠨ ᠳᠤ ᠲᠠᠭ ᠯᠠ ︵ ᠰ ᠡᠳᠦᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢᠲᠡ 12 ᠬᠤᠪᠢ ︶ 15 . ᠦᠷᠡ ᠵᠢᠭᠳᠡᠭᠦᠨ ᠪᠣᠯᠭ ᠠᠨ ᠵᠠ ᠳᠠᠯ᠄ − 2 + = __ __ __ _ 16 . ᠳᠣ ᠣᠷᠠᠬ ᠢ ᠵᠢᠷᠤᠭ ᠪᠣᠯ ᠠ᠂ ᠡ ᠬᠤᠶ ᠠᠷ ᠬᠠᠷᠪᠤᠯ ᠲᠠ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠴᠢᠨ ᠤ10 ᠤᠳᠠ ᠭ ᠠ ᠶᠢᠨ ᠬᠠᠷᠪᠤᠯ ᠲᠠ ᠶᠢᠨ ᠬᠢᠷ ᠢ ᠪᠦᠳᠦ ᠮᠵᠢ ᠶᠢᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠬᠦᠰᠦ ᠨᠦᠭᠳᠦ ᠪᠠ ᠳᠠᠬᠢ ᠷ ᠱᠤᠭᠤᠮᠤᠨ ᠪᠦᠷᠢᠳ ᠭᠡᠯ ᠦ ᠨ ᠵᠢᠷ ᠤᠭ ᠮᠦᠨ᠃ ᠲᠠᠨ ᠤ ᠦᠵ ᠡᠬᠦ ᠪᠡᠷ ᠠ᠂ ᠡ ᠬᠤᠶ ᠠᠷ ᠲᠠᠮᠢᠷ ᠴᠢᠨ ᠤ__ __ __ ᠶᠢᠨ ᠬᠠᠷᠪᠤᠯ ᠲᠠ ᠶᠢᠨ ᠬᠢᠷ ᠢ ᠪ ᠦᠳᠦ ᠮᠵᠢ ᠨᠡᠩ ᠳᠣᠭ ᠲᠠᠪᠦᠷ ᠢᠳᠠᠢ᠃ ︵ ᠠ ᠪᠤᠶ ᠤ ᠡ ᠶᠢ ᠲᠠᠭᠯ ᠠ︶ 17. ᠨᠢᠭᠡᠨ ᠠᠰ ᠳᠡᠭᠡ ᠭᠰᠢ ᠪᠣᠰᠬ ᠠᠭᠰᠠ ᠨ ᠮᠣᠳᠤᠨ ᠪᠠᠭ ᠠᠨ ᠠ ᠨᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠠᠴᠠ 3 .1m ᠵᠠᠢ ᠲᠠᠢ ᠬᠡᠰᠡᠭ ᠲᠤᠩ ᠬᠤᠭᠤᠷ ᠠᠪᠠ᠃ ᠮᠣᠳᠤᠨ ᠪᠠᠭ ᠠᠨ ᠠ ᠶᠢᠨ ᠣᠷᠤᠢ ᠨᠢ ᠭᠠ ᠵᠠᠷ ᠤ ᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠳᠤ ᠤᠨᠠ ᠵᠤ᠂ ᠭᠠᠵᠠᠷ ᠤᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠲᠠᠢ 38 ᠤᠨ ᠥᠨᠴ ᠦᠭ ᠡᠭᠦᠰᠭᠡᠭ ᠰᠡᠨ ᠪᠣᠯ ᠬᠤᠭᠤᠷ ᠠᠬᠤ ᠠᠴᠠ ᠡᠮ ᠦᠨᠡᠬᠢ ᠮᠣᠳᠤᠨ ᠪᠠᠭᠠᠨ ᠠ ᠶᠢᠨ ᠥᠨᠳᠦ ᠷ ᠨᠢ ᠪᠠᠷᠤᠭ_ __ __ _m ᠃ ︵ ᠯᠠᠪᠯᠠᠬ ᠤ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢᠮᠲᠠ᠄ sin 38 0 62 co s38 0. 79 t an 38 0.7 8 ︶ 18. ᠬᠤ ᠶᠠᠳᠤᠭ ᠠᠷ ᠵᠡ ᠷᠭᠡ ᠶᠢᠨ ᠹᠦᠩᠺ ᠽ y=ax 2+b x+c ︵a 0 ︶ ᠶᠢᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠥ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠪᠡ᠂ ① b ② a – b + c = 0 ③ ᠨᠢᠭᠡ ᠥᠯ ᠦ ᠮᠡᠳᠡᠭᠳᠡᠭ ᠴᠢᠳᠡ ᠢ ᠬᠤᠶ ᠠᠳᠤᠭ ᠠᠷ ᠵᠡᠷᠭᠡ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠳᠭᠡᠯ 1 0 ︵ 0 ︶ ④ − ᠪᠣᠶ ᠤ ᠦᠶ ᠡᠰ ᠲᠦ᠂ y 0 ᠡᠮᠦᠨᠡ ᠬᠢ ᠦᠭ ᠦᠯᠡᠯ ᠨᠦᠭ ᠥᠳ ᠡᠴᠡ ᠵᠦ ᠪ ᠨᠢ __ __ __ _︵ ᠪᠣᠢ ᠪᠦᠬᠦᠢ ᠵᠦᠪ ᠦᠭ ᠦᠯᠡᠯ ᠦᠨ ᠳᠤᠭᠠ ᠷ ᠲᠡᠮᠳᠡ ᠭ ᠢ ᠲᠠᠭᠯ ᠠ︶ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠤᠳᠤᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰ ᠡᠳᠦᠪ 19 . ︵10 ᠬᠤᠪᠢ ︶ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢᠯ ᠠᠵᠤ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣᠯ᠄ – – – – + ᠡᠭᠦᠨ ᠳᠦ a = | – √ |– tan 60 + ( ) ᠠ ᠡ ᠳᠣᠮ ᠳᠠᠴᠢ ᠲᠣᠭ ᠠ 8 8 ᠳᠣᠮ ᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ 8 8 ᠮᠣᠳᠤ 8 8 20 . ︵1 0 ᠬᠤᠪᠢ ︶ ᠥ ᠭᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ᠄ AC ᠪᠣᠯ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ AB CD ᠶᠢᠨ ᠲᠠᠰᠢᠭᠤ ᠪᠴᠢ᠃ ︵1︶ ᠠᠰ ᡂᠢ ᠪᠠ ᠭᠤᠷᠲᠢᠭ ᠢᠶᠠᠷ ᠬᠡᠷ ᠴᠢᠮ AC ᠶᠢᠨ ᠡᠭᠴᠡ ᠲᠠ ᠯᠠᠯᠠᠨ ᠬᠤᠪᠢᠶ ᠠᠭᠴᠢ ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠵᠤ᠂ AD ᠯᠤ ᠭ ᠠ ᠴᠡᠭ E ᠳᠤ ᠤᠭᠲ ᠤᠯᠤᠯᠴᠠ ᠭᠤᠯᠵᠤ᠂ CE ᠶᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠃ ︵ ᠵᠢᠷ ᠤᠭᠰ ᠠᠨ ᠣᠷᠤ ᠮᠦᠷ ᠢᠶᠠ ᠨ ᠦᠯ ᠡᠳᠡᠭᠡᠨ ᠡ᠂ ᠪᠠᠢᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠠᠷᠭ ᠠ ᠪᠠᠨ ᠪᠢᠴᠢᠬᠦ ᠦᠭᠡᠢ ︶ ︵2︶ ᠡᠮ ᠦᠨᠡᠬᠢ ︵1︶ ᠶᠢᠨ ᠨᠦ ᠬᠦᠴᠡᠯ ᠳᠤᠤᠷ ᠠ᠂ ᠬᠡᠷᠪᠡ A B =3 , B C=5 ᠪᠣᠯ D CE ᠶᠢᠨ ᠬᠦᠷᠢᠶ ᠡᠨ ᠦ ᠤᠷ ᠳᠤ ᠶᠢ ᠣ ᠯ᠃ 21 .︵12 ᠬᠤᠪᠢ ︶ ᠤᠯᠠᠭ ᠠᠨᠬᠠᠳᠠ ᠬᠤᠲ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠳᠣᠮᠳᠠᠳ ᠤ ᠰ ᠤᠷᠭᠠᠭ ᠤᠯᠢ ᠨᠢ ︽ ᠳᠡᠯ ᠡᠬᠡᠢ ᠶᠢᠨ ᠨᠣᠮ ᠣᠩᠰᠢ ᠬᠤ ᠡᠳᠦ ᠷ︾ ᠢ ᠪᠡᠯ ᠭᠡᠯᠡᠬ ᠦ ᠵᠢ ᠴᠢ︽ ᠨᠣᠮ ᠤᠨ ᠦᠨᠦ ᠷ ᠠ ᠩᠬᠢᠯ ᠤᠭᠰ ᠠᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠬᠦᠷᠢᠶᠡᠯ ᠡᠩ ︾ ᠢ ᠪᠠᠢᠭ ᠤᠯᠬᠤ ᠤᠷᠢᠶ ᠠ ᠶᠢ ᠴᠤᠤᠷ ᠢᠶᠠᠳᠤᠨ ᠳᠠ ᠭᠠᠬᠤ ᠶᠢᠨ ᠲᠦ ᠯᠦᠭᠡ᠂ ︽ᠤ ᠩᠰᠢᠯ ᠭ ᠠ ᠲᠠᠢ ᠬᠠᠮᠲᠤ ᠦ ᠰᠦᠨ ᠪᠤᠢᠵ ᠢᠬᠤ︾ ᠨᠣᠮ ᠣ ᠩᠰᠢᠬ ᠤ ᠠ ᠵᠢᠯᠯ ᠠᠭᠠᠨ ᠥᠷᠨ ᠢᠭᠦᠯᠪᠡ᠃ ᠡᠨ ᠡ ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠠ ᠵᠢᠯᠯ ᠠᠭᠠᠨ ᠳᠠᠬ ᠢ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠤ ᠨ ᠨᠣᠮ ᠤᠩᠰᠢᠭᠰ ᠠᠨ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠣᠢ ᠯᠠᠭᠠᠬᠤ ᠶᠢᠨ ᠲᠦ ᠯᠦᠭᠡ ᠂ ᠪ ᠦᠬᠦ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠤᠨ ᠳᠤᠮᠳᠠ ᠠᠴᠠ ᠲᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ ᠨᠢᠭᠡ ᠪᠦᠯᠦ ᠭ ᠰ ᠤᠷᠤᠭ ᠴᠢ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠴᠤ ᠪᠠᠢᠴᠠᠭ ᠠᠯᠲᠠ ᠬᠢᠪᠡ᠂ ᠴᠤᠭ ᠯᠠᠭᠤᠯ ᠤᠨ ᠣᠯᠤᠭ ᠰᠠᠨ ᠲᠤ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠪᠠᠨ ᠡᠮᠬᠢᠳ ᠭᠡᠵᠦ ᠪᠠᠰ ᠠ ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠭᠦᠢ ᠴᠡᠳ ᠪᠣᠰᠤ ᠳᠠ ᠬᠢᠷ ᠱᠤᠭᠤᠮᠤᠨ ᠪᠦᠷᠢᠳ ᠭᠡᠯ ᠦ ᠨ ᠵᠢᠷ ᠤᠭ ᠪᠠ ᠳᠤᠭ ᠤᠷᠢᠭ ᠪᠦᠷ ᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠢ ᠵᠤᠬᠢᠶᠠᠪᠠ᠃ ︵1︶ ᠲᠠ ᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ ᠰᠤᠭᠤ ᠯᠤᠨ ᠠᠪᠤᠭᠰ ᠠᠨ ᠰᠤᠷᠤᠭ ᠴᠢ ᠨᠡ ᠢᠲᠡ __ __ __ ᠨᠡᠷ ᠡ᠂ ︽2 ᠳᠡ ᠪᠲᠡᠷ ︾ ᠭᠡᠵᠦ ᠲᠡᠮᠳᠡ ᠭᠯᠡᠵᠦ ᠪᠠᠢᠭ ᠠ ᠳᠡ ᠪᠢᠭᠦᠷ ᠬᠡᠯᠪᠡ ᠷᠢ ᠶᠢᠨ ᠲᠦ ᠪ ᠥᠨᠴᠦ ᠭ ᠦᠨ ᠭᠷᠡᠳᠦ ᠰ ᠦᠨ ᠲᠣᠭ ᠠ__ __ _ ᠭ ᠷᠡᠳᠦᠰ᠂ ᠳᠠᠬᠢᠷ ᠱ ᠤᠭᠤᠮᠤᠨ ᠪᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠢ ᠨᠦᠬᠦᠪᠦᠷ ᠢᠯᠡᠵᠦ ᠪᠦᠲ ᠦᠨ ᠪᠣᠯᠭ ᠠ᠃ ︵2︶ ᠪᠠᠢᠴᠠᠭ ᠠᠭᠰᠠᠨ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠦᠨᠳ ᠦᠰᠦᠯᠡᠨ᠂ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠠᠴᠠ 1 ᠳᠡ ᠪᠲᠡᠷ ᠨᠣᠮ ᠪᠠ 4 ᠳᠡ ᠪᠲᠡᠷ ᠨᠣᠮ ᠤᠩᠰᠢᠭᠰ ᠠᠨ ᠰ ᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠳᠣᠮ ᠳᠠ ᠠ ᠴᠠ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠬᠤ ᠶᠠᠷ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠩᠭᠤᠨ ᠠᠪᠴᠤ ᠶᠠᠷ ᠢᠯᠴᠠᠭ ᠤᠯᠬᠤ ᠪ ᠠᠷ ᠳᠣᠭᠲᠠᠪ ᠠ᠂ ᠮᠣᠳᠤᠯ ᠵᠢᠨ ᠵᠢᠷᠤᠭ ᠪᠤᠶ ᠤ ᠲᠣᠭ ᠠᠴᠢᠨ ᠵᠢᠭᠰᠠᠭ ᠠᠬᠤ ᠠᠷᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠨ ᠬᠤᠶ ᠠᠷ ᠰ ᠤᠷᠤᠭ ᠴᠢ ᠶᠢᠨ ᠨᠣᠮ ᠤᠩᠰᠢᠭᠰ ᠠᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠬᠡᠮᠵᠢ ᠶ ᠡ ᠨᠢ ᠪᠦᠷ 4 ᠪᠣᠯᠬᠤ ᠳᠣᠬᠢ ᠶᠠᠯᠳᠤᠮᠵ ᠢ ᠶᠢ ᠣᠯ᠃ 22. (1 2 ᠬᠤᠪᠢ ) ᠨᠢ ᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠨᠢ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠤᠷ ᠠᠯᠢᠭ ᠤᠨ ᠨᠠᠢᠷ ᠤ ᠨ ᠠ ᠵᠢᠯᠯ ᠠᠭᠠᠨ ᠥᠷᠨ ᠢᠭᠦᠯᠬᠦ ᠪᠡᠷ ᠪᠣᠯ ᠵᠤ᠂ ᠪᠢᠴᠢ ᠬᠠᠨ ᠨᠠᠢᠵ ᠠ ᠲᠤᠩᠭ ᠠᠯᠠᠭ ᠢ ᠲᠣᠮᠢ ᠯᠠᠵᠤ᠂ ᠰᠣᠶ ᠣᠯ ᠤᠨ ᠬᠡᠷᠡᠭᠰ ᠡᠯ ᠦ ᠨ ᠬᠡ ᠳᠦ ᠳᠡᠯᠭᠡᠭᠦᠷ ᠲᠦ ᠣᠴᠢᠵ ᠤ ᠡᠳ ᠦᠢ ᠲᠡ ᠳᠦᠢ ᠪᠢᠴᠢᠭ ᠦᠨ ᠬᠡᠷᠡᠭᠰ ᠡᠯ ᠦᠨ ᠬᠠᠢᠷ ᠴᠠᠭ ᠠᠪ ᠴᠤ ᠱᠠᠩᠨᠠ ᠯ ᠤᠨ ᠪᠣᠳᠠᠰ ᠪᠣᠯᠭ ᠠᠬᠤ ᠪᠠ ᠷ ᠳᠤᠭᠲᠠᠪ ᠠ᠃ ᠡᠨᠡ ᠲᠦᠷᠦᠯ ᠦᠨ ᠪᠢᠴᠢᠭ ᠦᠨ ᠬᠡᠷᠡᠭᠰ ᠡᠯ ᠦ ᠨ ᠬᠠᠢᠷ ᠴᠠᠭ ᠤᠨ ᠬᠠᠳᠠ ᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠬᠠᠢᠷ ᠴᠠᠭ ᠪᠦᠷᠢ 10 ᠲᠦᠭ ᠦᠷᠢᠭ ᠂ ᠳᠠ ᠩᠰᠠ ᠪᠣᠳᠤᠬ ᠤ ᠦ ᠶᠡᠰ ᠦᠨ ᠪᠢᠴᠢᠬ ᠠᠨ ᠨᠠ ᠢᠵᠠ ᠲᠤᠩᠭᠠ ᠯᠠᠭ ᠪᠣᠯᠤᠨ ᠬᠤᠳᠠᠯ ᠳᠤᠭ ᠴᠢ ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠶᠠᠷᠢᠯ ᠴᠠᠭ ᠠ ᠶᠢ ᠬᠢᠨᠠᠮᠠ ᠭᠠᠢ ᠤᠩᠰ ᠢᠭᠠᠷ ᠠᠢ᠄ ︵1︶ ᠬᠤ ᠶᠠᠷ ᠬᠦᠮᠦ ᠨ ᠦ ᠶᠠᠷᠢᠯ ᠴᠠᠭ ᠠ ᠶᠢᠨ ᠠᠭᠤᠯ ᠭ ᠠ ᠶᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠪᠡ ᠯ᠂ ᠪ ᠢᠴᠢᠬ ᠠᠨ ᠨᠠ ᠢᠵᠠ ᠲᠤᠩᠭ ᠠᠯᠠᠭ ᠣᠣᠯ ᠨᠢ ᠬᠡ ᠳᠦᠢ ᠪᠢᠴᠢᠭ ᠦ ᠨ ᠬᠡ ᠷᠡᠭᠰ ᠡᠯ ᠦᠨ ᠬᠠᠢᠷ ᠴᠠᠭ ᠬᠤᠳᠠ ᠯᠳᠤᠨ ᠠ ᠪᠬᠤ ᠪ ᠠᠷ ᠲᠦ ᠯᠦᠪᠯᠡ ᠵᠦ ᠪᠠ ᠢᠭᠰᠠᠨ ᠪᠤᠢ ︖ ︵2︶ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠠᠴᠠ ᠪᠠᠰᠠ ᠪᠣᠯᠤᠳ ᠪᠢᠷ ᠪᠠ ᠲᠣᠰᠤ ᠨ ᠪ ᠢᠷ ᠢ ᠨᠡᠢᠲ ᠡ5 0 ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠴᠤ ᠰᠠᠩᠨ ᠠᠯ ᠤ ᠨ ᠪ ᠤᠳᠠᠰ ᠢᠶᠠᠨ ᠨᠦᠬᠦ ᠪᠦᠷᠢᠯᠡᠬ ᠦ ᠪᠡᠷ ᠲᠤᠭᠲᠠᠪ ᠠ᠃ ᠬᠤᠶ ᠠᠷ ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠤᠭᠰ ᠠᠨ ᠱᠠᠩᠨᠠ ᠯ ᠤᠨ ᠪᠣᠳᠠᠰ ᠤᠨ ᠶᠡᠷ ᠦᠩᠬᠡ ᠢ ᠵᠠ ᠷᠤᠳᠠᠯ ᠢ 40 0 ᠲᠦᠭ ᠦᠷᠢᠭ ᠡᠴᠡ ᠬᠡᠳᠦ ᠷᠡᠭᠦᠯ ᠵᠦ ᠪ ᠣᠯᠬᠤ ᠦᠭᠡᠢ᠃ ᠡᠭᠦᠨ ᠳ ᠦ ᠪᠣᠯᠤᠳ ᠪᠢᠷ ᠦ ᠨ ᠬᠠᠳᠠ ᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠪᠢᠷ ᠪᠦᠷᠢ 8 ᠲᠦ ᠭᠦᠷᠢ ᠭ᠂ ᠲᠤᠰ ᠤᠨ ᠪᠢᠷ ᠦ ᠨ ᠬᠠ ᠳᠠᠭᠰᠠᠨ ᠦᠨ ᠡ ᠨᠢ ᠪᠢᠷ ᠪᠦᠷᠢ 6 ᠲᠦᠭ ᠦᠷᠢᠭ ᠂ ᠬᠡᠯᠡᠯ ᠴᠡᠭᠡ ᠪᠡᠷ ᠳᠠ ᠮᠵᠢᠵ ᠤ᠂ ᠡᠨᠡ ᠤᠳᠠ ᠭ ᠠ ᠬᠤᠳᠠ ᠯᠳᠤᠭᠴᠢ ᠠᠴᠠ 8 ᠨᠤᠭ ᠤᠯᠵᠤ ᠥ ᠭᠬᠦ ᠪᠡᠷ ᠪᠣᠯ ᠪᠠ᠃ ᠲᠡᠭᠡᠪ ᠡᠯ ᠪ ᠢᠴᠢᠬ ᠠᠨ ᠨᠠ ᠢᠵᠠ ᠲᠤᠩᠭ ᠠᠯᠠᠭ ᠬᠠᠮ ᠤᠭ ᠤᠨ ᠠᠷ ᠪᠢᠨ ᠳᠤ ᠪᠠᠨ ᠬᠡᠳᠦ ᠢ ᠪᠣ ᠯᠤᠳ ᠪᠢᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪ ᠴᠤ ᠪ ᠣᠯᠬᠤ ᠪᠣᠢ︖ 23. ︵12 ᠬᠤᠪᠢ ︶ᠵᠢᠷᠤᠭ ᠮᠡᠳᠥ ᠂ A B ᠨᠢ ᠶᠢᠨ ᠳᠢᠶᠠᠮ ᠧᠲᠷ ᠂ C , D ᠨᠢ ᠬᠠᠭ ᠠᠰ ᠳᠤᠭ ᠤᠷᠢᠭ AB ᠶᠢ ᠭᠤᠷᠪ ᠠᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠬᠡᠰᠡᠭ ᠬᠤᠪᠢᠶ ᠠᠭᠰᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠷᠢ ᠴᠡᠭ ᠂ ᠴᠡ ᠭ C ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ AD ᠶᠢᠨ ᠥᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤ ᠮ ᠯᠤ ᠭ ᠠ ᠡᠭᠴᠡ ᠰᠢᠯ ᠤᠭᠤᠨ CE ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ E ᠪᠣᠯᠤᠨ ᠠ᠃ ︵1︶ ᠪᠠᠲᠤ ᠯᠠ᠄ CE ᠪᠣᠯ ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢ ᠯᠤᠭᠣᠨ᠃ ︵2︶ ᠬᠡᠷ ᠪᠡ ᠶᠢᠨ ᠷᠠᠳᠢᠤᠰ ᠨᠢ 2 ᠪ ᠣᠯ ᠵᠢᠷᠤᠭ ᠲᠠᠬ ᠢ ᠰ ᠡᠭᠦᠳᠡ ᠷ ᠲᠠ ᠢ ᠬᠡ ᠰᠡᠭ ᠦ ᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠶᠢ ᠣᠯ᠃ 24. ︵12 ᠬᠤᠪᠢ ︶ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠮᠠ ᠲᠧᠷ ᠢᠶᠠᠯ ᠢ ᠤᠩᠰᠢᠭᠠ ᠷᠠᠢ᠄ ᠮᠠᠨ ᠤ ᠮᠡᠳᠡᠬ ᠦ ᠪᠡᠷ ᠨᠢ ᠭᠡᠳᠦᠭᠡᠷ ᠵᠡᠷᠭᠡ ᠶ ᠢᠨ ᠹᠦᠩᠺᠽ ︵ 0 , , ᠨᠢ ᠪᠠᠢᠩᠭ ᠤ ᠲᠣᠭ ᠠ︶ ᠶᠢᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠨᠢ ᠨᠢ ᠭᠡᠨ ᠰᠢᠯᠤᠭᠤᠨ ᠮᠦ ᠨ᠂ ᠳᠡᠭᠡᠳᠦ ᠳᠤ ᠮᠳᠠᠳᠤ ᠳᠤ ᠰᠤᠷᠤᠯ ᠣᠠᠬᠤ ᠦᠶ ᠡᠰ ᠲᠤ᠂ ᠰᠢᠯᠤᠭᠤᠨ ᠢ ᠶᠡᠷᠦ ᠨᠢ 𝐴 𝐵 𝐶 ︵ A 0 , B 0 , A , B, C ᠨᠢ ᠪᠠᠢᠩᠭ ᠤ ᠲᠣᠭ ᠠ ︶ ᠤᠨ ᠬᠡᠯᠪᠡ ᠷᠢ ᠪᠡ ᠷ ᠪᠢᠴ ᠢᠳᠡᠭ ᠂ ᠴᠡ ᠭ p ︵x0 , y 0 ︶ ᠡᠴᠡ ᠰᠢᠯ ᠤᠭᠤᠨ A x+B y+C =0 ᠬᠦᠷᠬᠦ ᠵᠠᠢ ᠶᠢ ᠲ ᠤᠮᠢᠶ ᠠ d = | | √ ᠪᠠᠷ ᠪᠤᠳᠤᠳᠠᠭ᠃ ᠵᠢᠱᠢᠶ ᠡᠯᠡᠪᠡ ᠯ᠄ ᠴᠡᠭ P ︵3, 4 ︶ ᠡᠴᠡ ᠰᠢᠯ ᠤᠭᠤᠨ y= −2 x+5 ᠬᠦᠷ ᠬᠦ ᠵᠠ ᠢ ᠶᠢ ᠣᠯ᠃ ᠪᠣᠳᠤ ᠬᠤ ᠨᠢ᠄ y= −2 x+ 5 ᠂ 2x +y −5 = 0 , ᠡ ᠭᠦᠨ ᠳᠦ A =2 , B= 1 , C= − ᠴᠡᠭ P︵3 ,4 ︶ ᠡ ᠴᠡ ᠰᠢᠯᠤ ᠭᠤᠨ y= −2 x+ 5 ᠬᠦ ᠷᠬᠦ ᠵᠠ ᠢ ᠨᠢ ᠄ d = | | √ = | | √ = √ = √ ᠡᠮᠦᠨᠡ ᠬᠢ ᠮᠠ ᠲᠧᠷ ᠢᠶᠠᠯ ᠢ ᠦᠨᠳ ᠦᠰᠦᠯ ᠡᠨ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠰᠠᠭᠤᠳᠠ ᠯ ᠢ ᠰᠢᠢᠳ ᠪᠦᠷᠢᠯᠡ᠄ ︵1︶ ᠴᠡᠭ Q ︵− 2, 2 ︶ ᠡᠴᠡ ᠰᠢᠯ ᠤᠭᠤᠨ 3x −y +7 =0 ᠬᠦᠷ ᠬᠦ ᠵᠠ ᠢ ᠶᠢ ᠣᠯ᠃ ︵2︶ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠥ ᠂ ᠰ ᠢᠯᠤᠭᠤ ᠨy= − x ᠢ y ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ ᠳᠠ ᠭᠠᠤ ᠳᠡᠭᠡᠭᠰ ᠢ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ 2 ᠨᠢ ᠭᠡᠴᠢ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠪᠡ ᠯ ᠥ ᠭᠡᠷ ᠡ ᠨᠢᠭᠡᠨ ᠰᠢᠯ ᠤᠭᠤᠨ ᠢ ᠣᠯᠤ ᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠡᠨ ᠡ ᠬᠤ ᠶᠠᠷ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠤᠭᠤᠨ ᠤ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠵᠠᠢ ᠶᠢ ᠣ ᠯ᠃ 25 . ︵14 ᠬᠤᠪᠢ ︶ ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠥ ᠂ ᠰ ᠢᠯᠤᠭᠤᠨ − ᠨᠢ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ᠂ y ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠯᠤᠭ ᠠ ᠲᠤᠰ ᠲᠤᠰ B , C ᠬᠤ ᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠨᠢ ᠴᠡᠭ B , C ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠵ ᠤ᠂ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠯ ᠦᠭᠡ ᠥᠭᠡᠷ ᠡ ᠨᠢ ᠭᠡ ᠴᠡ ᠭ A ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠵ ᠤ᠂ ᠣᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ D ︵1︶ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ︵2︶ ᠲᠡᠩᠭᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠡᠴᠡ ᠨᠢ ᠭᠡᠨ ᠴᠡᠭ E ᠶᠢ ᠡᠷᠢᠨ ᠣ ᠯᠵᠤ᠂ E C + ED ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠪᠣᠯ ᠭᠠᠪᠠᠯ ᠂ E C + E D ᠶᠢᠨ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠪ ᠠᠭ ᠠ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣᠯ ᠃ ︵3︶ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠲᠡ ᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ APB= O CB ᠪᠠᠢᠭ ᠤᠯᠤᠭᠳᠠᠬᠤ ᠴᠡ ᠭ P ᠤᠷᠤᠰ ᠢᠨ ᠠ ᠦᠦ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠤᠰ ᠢᠪᠠᠯ ᠴᠡᠭ P ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣ ᠯ᠃ ᠬᠡᠷᠪᠡ ᠣᠷᠤᠰ ᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠤᠷ ᠬᠠᠢᠯ ᠠ᠃ 26 .︵14 ᠬᠤᠪᠢ ︶ ᠠᠰᠠᠭᠤᠳᠠ ᠯ ᠄ ᠵᠢᠷᠤᠭ 1 ᠮᠡᠳᠥ ᠂ t ᠳᠦ᠂ A CB = 90 , AC=BC , ᠴᠡ ᠭ C ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ AB ᠶᠢᠨ ᠵᠡᠷ ᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠤᠭᠤᠨ L ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠃ E DF= 90 , ᠴᠡᠭ D ᠨᠢ ᠰ ᠢᠯᠤᠭᠤᠨ L ᠳᠡᠭᠡᠷ ᠡ ᠰ ᠢᠯᠵᠢᠬᠦ ᠪᠦᠭᠡᠳ ᠥᠨᠴᠦ ᠭ ᠦᠨ ᠨᠢᠭᠡ ᠲᠠᠯ ᠠ D E ᠬᠡ ᠵᠢᠶᠡᠳᠡ ᠴᠡᠭ B ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠵ ᠤ᠂ ᠨᠦᠭᠦᠭᠡ ᠨᠢᠭᠡ ᠲᠠᠯ ᠠ D F ᠨᠢ AC ᠯᠦ ᠭᠡ ᠴᠡᠭ P ᠳᠤ ᠣᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ D P ᠪᠠ DB ᠶᠢᠨ ᠲᠤᠭ ᠠᠨ ᠤ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠭᠦᠨ ᠦ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠢ ᠰ ᠣᠳᠤᠯ ᠤᠨ ᠠ᠃ ᠰᠦᠪᠡ ᠭᠴᠢᠯ ᠡᠨ ᠢ ᠯᠡᠷᠡᠭᠦᠯ ᠬᠦ ︵1︶ ᠵᠢᠷᠤᠭ 2 ᠮᠡ ᠳᠥ᠂ ᠨᠢᠭᠡᠨ ᠲᠤᠭᠠ ᠨ ᠤ ᠤᠬᠠᠭᠠ ᠨ ᠤ ᠳᠤᠷ ᠠ ᠪ ᠠᠬᠠᠲ ᠤ ᠪ ᠠᠭ ᠠ ᠳᠤᠭ ᠤᠢᠯ ᠠᠩ ᠤᠨᠴᠠ ᠭᠠᠢ ᠠᠴᠠ ᠶᠡᠷᠦ ᠶᠢ ᠨ ᠪ ᠣᠯᠬᠤ ᠭᠡᠳᠡᠭ ᠲᠤᠭ ᠠᠨ ᠤ ᠤᠬᠠᠭᠠ ᠨ ᠤ ᠦᠵ ᠡᠯ ᠰ ᠠᠨᠠᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠨ᠂ ᠴᠡᠭ D ᠨᠢ ᠴᠡᠭ P ᠪᠣᠯᠤᠨ ᠴᠡᠭ C ᠶᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠪ ᠣᠯᠲᠠᠯ ᠠ ᠰ ᠢᠯᠵᠢᠬ ᠦ ᠦ ᠶᠡᠰ᠂ ᠮᠦ ᠷᠳᠡᠯ ᠭᠡ ᠪᠡᠷ ᠳᠠᠮᠵᠢᠭᠤᠯ ᠤᠨ DP =D B ᠪᠣᠯᠬᠤ ᠶᠢ ᠣᠯᠵ ᠤ ᠪᠣ ᠯᠤᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲ ᠠ ᠪᠠᠲᠤ ᠯᠠᠭᠰᠠᠨ ᠶᠠᠪᠤᠴᠠ ᠪᠠᠨ ᠪᠢᠴᠢᠭᠡᠷ ᠡᠢ᠃ ᠲᠤᠭ ᠠᠨ ᠤ ᠤᠬ ᠠᠭᠠᠨ ᠤ ᠪᠣᠳᠤ ᠯᠭᠠᠰᠢᠷ ᠠᠯ ︵2︶ ᠵᠢᠷ ᠤᠭ 3 ᠮᠡᠳᠥ ᠂ ᠬᠡ ᠷᠪᠡ ᠴᠡᠭ P ᠨᠢ AC ᠲᠠ ᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢ ᠭᠡ ᠴᠡᠭ ︵ᠦᠵᠦᠭ ᠦᠷ ᠤᠨ ᠴᠡᠭ A ,C ᠶᠢ ᠪᠠᠭᠲᠠᠭ ᠠᠬᠤ ᠦᠭᠡᠢ ︶᠂ ᠡᠮᠦᠨᠡ ᠬᠢ ︵1︶ ᠶᠢᠨ ᠰᠡᠩᠬ ᠡᠷᠡᠭᠦᠯ ᠦᠯ ᠢᠶᠡ ᠷ ᠡᠨᠡᠬᠥ ᠪᠠᠭ ᠠ ᠳᠤᠭ ᠤᠢᠯ ᠠᠩ ᠨᠢ ᠴᠡᠭ D ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ DG CD ᠶᠢ ᠪᠠᠢᠭ ᠤᠯᠵᠤ᠂ BC ᠯᠦ ᠭᠡ ᠴᠡᠭ G ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠭᠤᠯᠪᠠ ᠯ DP DB ᠶᠢ ᠪᠠᠲᠤ ᠯᠠᠵᠤ ᠪᠣᠯ ᠤᠨ ᠠ ᠭᠡᠵᠦ ᠦ ᠵᠡᠪᠡ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠪ ᠠᠲᠤᠯ ᠠᠭᠰᠠᠨ ᠶᠠᠪᠤᠴᠠ ᠪᠠᠨ ᠭᠦᠢᠴ ᠡᠳᠬᠡ᠃ ᠥᠷᠭᠡᠳᠭᠡᠨ ᠭᠦᠨᠵ ᠡᠭᠡᠢᠷ ᠡᠭᠦᠯᠬᠦ ︵3 ︶ ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠥ ᠂ᠡᠮᠦ ᠨᠡᠬᠢ ︵1 ︶ ᠶᠢᠨ ᠨᠦᠬᠦᠴᠡ ᠯ ᠳᠣ ᠣᠷ ᠠ ᠂ M ᠨᠢ AB ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠡᠭ ︵ᠦᠵ ᠦᠭᠦᠷ ᠤᠨ ᠴᠡᠭ A ,C ᠶᠢ ᠪᠠᠭᠲ ᠠᠭᠠᠬᠤ ᠦᠭᠡᠢ ︶᠂ N ᠨᠢ ᠰ ᠠᠴᠤᠷᠠ ᠭ ᠱᠤᠭᠤᠮ B D ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠪᠦᠭᠡᠳ A M=B N,M N ᠢ ᠬᠤᠯ ᠪᠤᠪᠠᠯ BC ᠯᠤᠭ ᠠ ᠴᠡ ᠭ Q ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠃ ᠡᠨᠡᠬᠥ ᠲᠤᠭ ᠠᠨ ᠤ ᠤᠬ ᠠᠭᠠᠨ ᠤ ᠳᠤᠷ ᠠ ᠪ ᠠᠬᠠᠲᠤ ᠪᠠᠭ ᠠ ᠳᠤᠭ ᠤᠢᠯ ᠠᠩ ᠨᠢ ᠣ ᠯᠠᠨ ᠤᠳᠠ ᠭ ᠠM ᠴᠡᠭ ᠢ ᠠᠪᠴᠤ ᠳᠠᠬ ᠢᠨ ᠳᠠᠪᠲᠠ ᠨ ᠳᠤᠷ ᠰᠢᠯᠳᠠ ᠬᠢᠬᠦ ᠳᠦ᠂ ᠴᠡᠭ M ᠨᠢ ᠶᠠᠮᠠ ᠷ ᠨᠢ ᠭᠡᠨ ᠪᠠᠢᠷ ᠢᠨ ᠳ ᠤ ᠪᠠᠢᠬᠤ ᠦᠶ ᠡᠰ ᠲᠦ BQ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠶᠡᠬᠡ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠲᠠ ᠢ ᠪᠣᠯᠬᠤ ᠶᠢ ᠢᠯᠡᠷ ᠡᠭᠦᠯᠦ ᠨ ᠮᠡ ᠳᠡᠵᠠᠢ᠃ ᠬᠡᠷ ᠪᠡ AC = BC = 4 ᠪ ᠣᠯ ᠲᠠ BQ ᠶᠢᠨ ᠬᠠᠮ ᠤᠭ ᠶᠡᠬᠡ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠢ ᠰ ᠢᠭᠤᠳ ᠪᠢᠴᠢᠵ ᠤ ᠭᠠᠷ ᠭ ᠠ᠃