ᠤᠷᠢᠳᠤᠰ ᠬᠤᠲᠠ ᠶᠢᠨ 2018 ᠣᠨ ᠤ ᠠᠩᠬ᠎ᠠ ᠳᠤᠮᠳᠠᠳᠤ ᠶᠢᠨ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠲᠡᠭᠦᠰᠴᠦ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠳᠠᠪᠰᠢᠬᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠳᠦᠪ 2018鄂尔多斯.docx

ᠨᠢᠭᠡ᠂ ᠭᠠᠭᠴᠠ ᠬᠠᠷ ᠢᠭᠤᠯ ᠲᠠ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰ᠊ ᠡᠲᠤᠪ᠃ ︵ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤᠪ ᠲᠤ 10 ᠪᠠᠭ ᠠ ᠰ᠊ ᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ︶ 1. ᠳᠠᠷᠠᠭ ᠠᠬᠢ 227 , − 2018 , √ 4 , π ᠲᠥᠷᠪ ᠠ ᠲᠤᠭ ᠠ ᠠᠴ ᠠ ᠶᠤ ᠰᠤᠬᠦ ᠢ ᠲᠤᠭ ᠠ ᠨᠢ ︵︶ A. 227 B. −20 18 C. √4 D. π 2. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠵᠥ ᠤ ᠪᠤ ᠯᠬᠤ ᠨᠢ ︵︶ A. 3𝑥 − 𝑥= 3 B. 𝑎 2 ÷ 𝑎 4 = 1𝑎 C. ︵𝑥 − 1︶ 2= 𝑥 2 − 2𝑥 − 1 D. ︵− 2𝑎 2︶ 3= −6 𝑎 6 3. ᠳ᠋᠊ ᠠᠷᠠᠭ ᠠᠬᠢ ᠹᠦᠩᠺᠼ ᠦ ᠳ᠋᠊ ᠠᠴᠠ ᠬᠤᠪᠢᠰᠬ ᠤᠨ 𝑥 ᠤ ᠨ᠊ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ᠨᠢ 𝑥 1 ᠪᠣᠯᠬᠤ ᠨᠢ ︵︶ A. y= √𝑥 − 1 B. y= 1 √𝑥︔ 1 C. y= 1𝑥︔ 1 D. y = ︵𝑥 − 1︶ 0 4. O ᠪᠠᠷ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠥᠨ ᠴᠥᠭ ᠬᠡᠮᠵᠢᠭ ᠦᠷ ᠤ ᠣᠯᠣᠨ ᠲᠡᠭ ᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤᠤᠢ AB C ᠶᠢ ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠤ ᠲᠠᠯᠪᠢᠪ ᠠᠯ᠂ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭ ᠦᠨ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡ ᠬ᠋᠊ B ᠨᠢ ᠲᠡᠭ ᠭᠷ ᠠᠳᠦ᠋ᠰ ᠤᠨ ᠣ ᠷᠣᠰᠢ ᠵᠤ ᠪᠠᠢᠭ ᠠ ᠰ᠊ ᠢᠯᠤᠭᠤᠨ DE ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠣᠰᠢᠨ ᠠ᠃ ᠥᠨᠴ ᠥᠭ ᠬᠡᠮᠵᠢᠬᠦᠷ ᠪᠣᠯ ᠣᠨ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤᠤᠢ ᠨᠢ ᠵᠥᠪᠬ ᠠ ᠨᠢ ᠭᠡ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠᠯ ᠤᠨ ᠴᠡᠭ ᠲᠠ ᠢ ᠪᠦ ᠭᠡᠳ ᠣ ᠭᠳᠣᠯ ᠣᠯᠴᠠᠯ ᠤ ᠨ᠊ ᠴᠡ ᠬ᠋᠊ ᠨᠢ P ᠪᠣᠯᠣᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ∠EO P= 13 4°5 0′ ᠪᠣᠯ ∠CB D ᠶᠢᠨ ᠭᠷᠠᠳᠦ᠋ᠰ ᠤᠨ ᠲᠤᠭ ᠠ ᠨᠢ ︵︶ A. 45 °10 ′ B. 44°50′ C. 46 °1 0′ D. ᠳᠣᠭᠳᠠ ᠭᠠᠵᠤ ᠲᠠᠢ ᠯᠬᠦ ᠦᠭᠡᠢ 5. ᠨᠢᠭᠡᠨ ᠬᠦᠴ ᠢᠷᠳᠡᠯᠳᠡᠢ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠲᠡᠳᠬᠦᠬ ᠦ ᠶᠢᠨ ᠲᠤ ᠯᠥᠭᠡ ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠬᠢᠨ ᠵᠣᠭᠤᠰ ᠲᠤᠰᠠ ᠯᠠᠬᠤ ᠠᠵᠢᠯ ᠯᠠᠭᠠᠨ ᠢ ᠥᠷᠨ ᠢᠬᠦᠯᠪᠡ᠃ ᠣ ᠲᠤ ᠤ ᠥᠬᠦ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢ ᠨ᠊ ᠰ᠊ ᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠨ᠊ ᠲᠤ ᠰᠠᠯᠠᠭᠰᠠ ᠨ᠊ ᠵᠣᠭᠤᠰ ᠠᠴᠠ 10 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢᠨ ᠳᠤᠰ ᠠᠯᠠᠭᠰᠠᠨ ᠵᠣᠭᠤᠰ ᠢ ᠲᠠᠰᠢᠷᠠ ᠮ᠊ ᠢᠶᠠᠷ ᠰᠤᠩᠭᠤᠨ ᠠᠪᠤᠭᠠ ᠳ᠋᠊᠂ ᠲᠤᠭ ᠠ ᠪᠠᠷᠢᠮᠳᠠ ᠶᠢ ᠤ ᠦᠷᠢᠳᠭᠡᠭ ᠰᠡᠨ ᠨᠢ ᠳ᠋᠊ ᠣᠣᠷᠠᠬ ᠢ ᠬᠦ ᠰᠥᠨᠥ ᠭᠳᠦ ᠮᠡᠳᠤ᠄ ᠳᠣᠣᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠠᠴᠠ ᠵᠥᠪ ᠪᠣᠯ ᠬᠤ ᠨ᠊ ᠢ A. 10 ᠰᠤᠷᠤᠭᠴᠢ ᠪᠣᠯ ᠪᠦᠭᠦᠳᠡ ᠶ᠊ ᠶᠢᠨ ᠨᠢ ᠭᠡ ᠶᠠᠩ ᠵᠤ B. ᠳᠤᠮ ᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠤᠭ ᠠ ᠨᠢ 40 C. ᠮᠤᠳ᠋ ᠨᠢ 90 D. ᠰᠠᠷ ᠨᠢᠴᠠ ᠨᠢ 40 0 6. ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ ᠷᠤᠮᠪ AB CD ᠲᠤ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠯᠬ ᠤᠮ ᠳᠠᠷᠠᠭ ᠠ ᠪᠠᠷ ᠵᠢᠷᠤᠭ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠃ ① ᠴᠡᠭ C ᠪᠠ ᠴᠡᠭ D ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤ ᠨ᠊ ᠲᠥᠪ ᠪᠤ ᠯᠭᠠᠨ᠂ 12 𝐶𝐷 ᠠᠴᠠ ᠶᠡᠭᠡ ᠤ ᠷᠳᠤ ᠪᠠ ᠷ᠊ ᠷᠠ ᠳᠢᠤᠰ ᠪᠣᠯᠭ ᠠᠨ ᠨᠤᠮᠤ DE O P A B C 0 30 60 90 120 150 180 ᠳᠤᠰᠠ ᠯᠠᠭᠰᠠ ᠨ᠊ ᠵᠣᠭᠤᠰ ︵ ᠨᠢ ᠭᠡᠴᠢ᠄ ᠲᠤ ᠭᠥᠷᠢᠭ︶ ᠬᠦᠮᠦᠨ ᠦ ᠲᠤᠭ ᠠ ︵ᠨᠢᠭᠡᠴ ᠢ᠄ ᠬ᠊ ᠦᠮᠦᠨ︶ 20 2 30 4 50 3 90 1 A B C D EM N ᠤᠷᠳᠤ ᠰ᠊ ᠬᠤᠳᠠ ᠢᠢᠨ 20 18 ᠣᠨ ᠤ ᠠᠩᠬ ᠠ ᠲᠤ ᠮᠳᠠᠳᠤ ᠢᠢ ᠨ᠊ ᠰᠤᠷ ᠤᠯᠭ ᠠ ᠲᠡᠭ ᠦᠰᠴ ᠤ ᠰᠤᠷ ᠭᠠᠭ ᠤᠯᠢ ᠲᠠᠪᠰ ᠢᠬᠤ ᠰᠢᠯ ᠭᠠᠯᠳᠠ ᠢᠢᠨ ᠰᠡᠳᠤ ᠤ A B C D O x y ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊᠂ ᠬᠤᠶ ᠠᠷ ᠨᠤᠮ ᠤ ᠨᠢ ᠴᠡ ᠬ᠋᠊ ᠲᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠨ᠊ ᠠ᠃ ② ᠰᠢᠯ ᠤᠭᠤᠨ 𝑀𝑁 ᠢ ᠤ ᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠊ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠴᠡᠭ A ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠵ ᠤ᠂ 𝐶𝐷 ᠲᠠᠢ ᠴᠡᠭ E ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠂ 𝐵𝐸 ᠶᠢ ᠬᠤᠯ ᠪᠣᠨ ᠠ᠃ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠭᠡᠯᠡᠯᠭᠡ ᠠᠴᠠ ᠪᠤᠷᠤᠭ ᠤ ᠪᠤ ᠯᠬᠤ ᠨᠢ ︵︶ A. ∠AB C= 60° B. 𝑆 ∆𝐴𝐵𝐸 = 2𝑆 ∆𝐴𝐷 𝐸 C. ᠬᠡᠷ ᠪᠡ AB = 4 ᠤ ᠣᠯ BE = 4√ 7 D. sin ∠CB E= √2114 7. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ 𝑦 1᠂ 𝑦2 ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠲᠤᠰᠤ ᠨ᠊ ᠬᠦᠳᠡᠯᠭᠡ ᠭᠦᠷᠳ ᠤ ᠮᠠ ᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠪᠠ ᠴᠠᠬᠢᠯ ᠭᠠᠨ ᠬᠦᠳᠡ ᠯᠭᠡᠭᠦᠷ ᠲᠤ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡ ᠶᠢᠨ ᠶᠠᠪᠤᠭᠰᠠᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ S ︵ᠨᠢᠭᠡᠴᠢ᠄ ᠺᠢᠯ ᠣᠮᠧᠳ᠋ᠷ︶ ᠪᠣᠯ ᠣᠨ ᠵᠠᠷᠤᠭᠰ ᠠᠨ ᠵᠠᠷᠤᠳᠠᠯ y ︵ᠨᠢᠭᠡ ᠴᠢ᠄ ᠲᠤ ᠬᠥᠷᠢᠭ︶ ᠶᠢᠨ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠬᠣ ᠯᠪᠣᠭᠳᠠᠯ ᠢ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠠ ᠡ᠃ ᠴᠠᠬᠢᠯ ᠭᠠᠨ ᠬᠦᠳᠡᠯ ᠭᠡᠭᠦᠷ ᠲᠤ ᠮᠠᠰᠢᠨ ᠲᠡ ᠷᠭᠡᠨ ᠦ ᠺᠢ ᠯᠣᠮᠧᠳ᠋ᠷ ᠪᠦᠷᠢ ᠲᠤ ᠤ ᠠ ᠵᠠ ᠷᠤᠭᠰ ᠠᠨ ᠵᠠ ᠷᠤᠳᠠᠯ ᠨᠢ ᠲᠤᠰᠤ ᠨ᠊ ᠬᠦ ᠳᠡᠯᠭᠡᠭᠦᠷ ᠲᠤ ᠮᠠᠰᠢ ᠨ᠊ ᠲᠡ ᠷᠭᠡ ᠶᠢᠨ ᠺᠢᠯᠣᠮᠧᠳ᠋ᠷ ᠪᠦᠷᠢ ᠲᠤ ᠤ ᠠ ᠵᠠ ᠷᠤᠬᠤ ᠵᠠᠷᠤᠳ ᠠᠯ ᠢ 𝑥 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠭᠡᠵᠤ ᠵᠢᠱᠢ ᠪᠡᠯ᠂ ᠵᠥ ᠤ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠰ ᠠᠨ ᠲᠡ ᠭᠰᠢᠳ ᠭᠡᠯ ᠨᠢ ︵︶ A. 36𝑥 = 9 𝑥︔ 0.54 B. 36 𝑥︔ 0.54 = 9𝑥 C. 36 𝑥: 0.54 = 9𝑥 D. 36𝑥 = 9 𝑥: 0.54 8. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠳᠢᠶ ᠠᠮᠧᠳ᠋ᠷ ᠨᠢ 2 ᠪᠤ ᠯᠬᠤ ᠲᠤᠭ ᠣᠷᠢᠭ ᠬᠡᠯᠪᠡ ᠷᠢᠳᠤ ᠲᠡ ᠮᠦᠷ ᠬᠠᠪᠳᠠᠰᠤ ᠠᠴᠠ ᠲᠤ ᠤ ᠥᠨᠴᠥ ᠬ᠋᠊ ᠨᠢ 90° ᠪᠣᠯᠬᠤ ᠨᠢᠭᠡᠨ ᠲᠡ ᠪᠢᠬᠦᠷ ᠬᠡᠯ ᠪᠡᠷᠢᠳᠤ CA B ᠶᠢ ᠬᠠᠢᠴᠢ ᠯᠠᠨ ᠠᠪᠬᠤ ᠪᠥᠭᠡᠳ ᠴᠡᠭ C , A , B ᠨᠢ ᠴᠥᠮ ⨀ O ᠳᠡ ᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠬᠠᠢᠴᠢᠯ ᠠᠵᠤ ᠠᠪᠤᠭ ᠰᠠᠨ ᠳᠡᠪᠢᠬ ᠦᠷ ᠬᠡᠯᠪᠡ ᠷᠢᠳᠤ ᠪᠠᠷ ᠨᠢᠭᠡᠨ ᠱᠣᠷᠣᠯ ᠶᠢᠨ ᠬᠦᠷᠢᠶ ᠡᠯᠡᠪᠡ ᠯ᠊᠂ ᠲᠤᠰ ᠱᠣᠷᠣᠯ ᠶᠢᠨ ᠤ ᠰᠠᠭ ᠤᠷᠢ ᠲᠣᠭᠤᠷ ᠢᠭ ᠤ ᠨ᠊ ᠷᠠ ᠳᠢᠤᠰ ᠨᠢ ︵︶ A. 12 B. √2 C. √ 22 D. √ 24 9. ᠵᠢᠷᠤᠭ ᠮᠡ ᠲᠤ᠂ ᠴᠡᠭ A ︵-2, 0︶᠂ ᠴᠡᠭ B ︵0 ,1︶᠂ ᠬᠡ ᠷᠴᠢᠮ AB ᠪᠠ ᠷ᠊ ᠲᠠ ᠯ᠊ ᠠ ᠪᠠᠨ ᠪᠣᠯᠭ ᠠᠭᠰᠠ ᠨ᠊ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ A BCD ᠶᠢ ᠬᠣ ᠶᠠᠳᠤᠭ ᠠᠷ ᠮᠥ ᠴᠡ ᠳ᠋᠊ ᠦ ᠤ ᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠃ ᠬᠤᠣᠰ ᠮᠤᠷᠤᠢ ᠱᠤᠭᠤᠮ y = 𝑘𝑥 ︵𝑘 𝑏c ᠪᠣᠯ᠂ 𝑎 𝑏 ᠶᠢᠨ ᠤᠷᠪᠠ ᠭᠤ ᠦᠭ ᠦᠯᠡᠯ ᠨᠢ ᠵᠢᠩᠬ ᠢᠨᠢ ᠦᠭ ᠦᠯᠡᠯ ᠪᠣᠯ ᠤᠨ ᠠ᠃ ③ ᠬᠡ ᠷᠪᠡ 𝑀 ︵𝑎, 2︶ , 𝑁 ︵1, 𝑏︶ ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ᠊ ᠬᠤᠪᠢ ᠲᠤ ᠲᠡᠭᠰᠢ ᠬᠡ ᠮᠳᠤ ᠴᠡᠭ ᠪᠣᠯ 𝑎 + 𝑏= −1᠃ ④ ᠨᠢ ᠭᠡᠨ ᠣᠯᠠ ᠨ᠊ ᠥᠨᠴᠥ ᠭᠳᠥ ᠶᠢ ᠨ᠊ ᠲᠠ ᠯ᠊ ᠠ ᠶᠢᠨ ᠪᠣᠳᠠᠭ ᠠᠯᠢ ᠲᠤᠭ ᠠ ᠨᠢ 1 ᠨᠡᠮᠡᠭ ᠳᠡᠬᠦ ᠪᠦᠷᠢ ᠲᠣᠳᠣ ᠭᠠᠳᠤ ᠥᠨᠴᠥ ᠬ᠋᠊ ᠦᠨ ᠨᠠᠢᠯ ᠡᠪᠤᠷᠢ ᠨᠢ 180° ᠨᠡᠮᠡ ᠭᠳᠡᠵᠦ᠂ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠥᠨᠴᠥᠭ ᠶᠢᠨ ᠨᠠᠢᠯ ᠡᠪᠤᠷᠢ ᠨᠢ ᠬᠤᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠃ ⑤ √11 ᠶᠢᠨ ᠪᠦ ᠬᠥᠯᠢ ᠲᠣᠭᠠ ᠨ᠊ ᠤ ᠭᠡᠰᠡᠭ ᠨᠢ 𝑎᠂ ᠪᠤᠳᠠᠷ ᠬᠠᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠭᠡᠰᠡ ᠬ᠋᠊ ᠨᠢ b ᠪᠣᠯ 𝑎𝑏 = 3√ 11 − 3᠃ 14. ᠬᠠᠪ ᠳᠠᠭᠠᠢ ᠶ᠊ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠤ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ᠊ ᠰ᠊ ᠢᠰᠲᠧᠮ ᠲᠤ᠂ ᠴᠡᠭ P ︵𝑎 ,𝑏︶ ᠶᠢᠨ ᠬᠤᠪᠢ ᠳ᠋᠊ ᠤ᠂ ᠪᠢᠳᠠ Q ︵− 𝑏+ 1, 𝑎+ 1︶ ᠶᠢ ᠴᠡᠭ P ᠶᠢᠨ ᠳᠠᠭ ᠠᠯᠳᠠ ᠴᠡᠭ ᠭᠡᠵ ᠤ ᠲᠤᠭ ᠲᠠᠭᠠᠪ ᠠ᠃ 𝐴1 ᠶᠢᠨ ᠳᠠᠭ ᠠᠯᠲᠠ ᠴᠡᠭ ᠨᠢ 𝐴 2᠂ 𝐴 2 ᠳᠠᠭ ᠠᠯᠲᠠ ᠴᠡᠭ ᠨᠢ 𝐴 3 ᠁ ᠡᠨᠡ ᠬᠦ ᠵᠢᠮ ᠡ ᠶᠤ ᠰᠤ ᠪᠠ ᠷ᠊ ᠣᠯᠬ ᠤ ᠨᠢ 𝐴1 , 𝐴2 , 𝐴 3 ᠁ 𝐴𝑛 ᠬᠡᠷ ᠪᠡ 𝐴 1 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ︵3,1︶ ᠤ ᠣᠯ ᠴᠡᠭ 𝐴2018 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ︵︶᠃ 15. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭ ᠰᠢ ᠲ᠊ ᠥᠷᠪᠡ ᠯᠵᠢᠨ AB CD ᠶᠢᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 4᠂ ᠤᠷᠳᠤ ᠨᠢ 4 ᠪᠣᠯᠬ ᠤ ᠬᠡ ᠷᠴᠢᠮ PQ ᠶᠢᠨ ᠬᠤᠶ ᠠᠷ ᠦᠵ ᠦᠬᠦᠷ ᠦᠨ ᠴᠡᠭ ᠨᠢ ᠲᠡ ᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ AB CD ᠶᠢᠨ ᠵᠡᠷᠭᠡᠯ ᠡᠳᠡᠭᠡ ᠬᠤᠶ ᠠᠷ ᠲᠠ ᠯ᠊ ᠠ ᠳᠡᠭᠡᠷ ᠡ ᠭᠤᠯ ᠭᠤᠨ ᠰᠢᠯᠵ ᠢᠬᠦ ᠪᠥᠭᠡᠳ ᠴᠡ ᠬ᠋᠊ P ᠨᠢ A→ B→ C→ Q ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠭᠤᠯ ᠭᠤᠨ ᠰ᠊ ᠢᠯᠵᠢᠵ ᠤ᠂ ᠴᠡᠭ D ᠬᠦᠷᠪᠡ ᠯ᠊ ᠭᠤᠯ ᠭᠤᠬ ᠤ ᠪᠠᠨ ᠵᠣᠭᠰᠣ ᠨ᠊ ᠠ᠃ ᠪᠥᠬᠦ ᠭᠤᠯ ᠭᠤᠨ ᠰᠢᠯ ᠵᠢᠬᠦ ᠶᠠᠪᠤᠴᠠ ᠲᠤ PQ ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ R ᠶᠢᠨ ᠶᠠᠪ ᠤᠵᠤ ᠦᠩᠭᠡᠷᠡ ᠭᠰᠡᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ ᠨᠢ ︵︶᠃ 16. ᠵᠢᠷ ᠤᠭ ① ᠮᠡ ᠲᠤ᠂ AF, BE ᠪᠣᠯ ∆AB C ᠶᠢᠨ ᠳᠤᠮᠲᠠ ᠱᠤᠭᠤ ᠮ᠊᠂ AF ⊥ BE ᠡᠭᠴᠡ ᠶᠢᠨ ᠰ᠊ ᠠᠭᠤᠷᠢ ᠨᠢ P ᠬᠡ ᠷᠪᠡ BC = 𝑎 , AC = b , AB = c ᠭᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ 𝑎 2 + 𝑏 2 = 5𝑐 2 ᠪᠤ ᠯᠣᠨ ᠠ᠃ ᠳᠡ ᠭᠡᠷᠠᠬ ᠢ ᠰ᠊ ᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠢ ᠠ ᠰᠢᠭᠯ ᠠᠨ ᠪᠣᠳᠣ᠃ ᠵᠢᠷᠤᠭ ② ᠮᠡᠲᠤ᠂ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ AB CD ᠲᠤ᠂ E᠂ F᠂ G ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ AD , BC , CD ᠶᠢᠨ ᠳᠤᠮ ᠳᠠ ᠴᠡᠭ᠂ BE ⊥ GE ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ E , AD = 8 , AB = 2√ 5 ᠪᠣᠯ AF = ︵︶᠃ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬ᠊ ᠠᠷᠢᠭ ᠤᠯᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ︵ᠨᠠᠢᠲᠡ 72 ᠬᠤᠪᠢ︶ R P A B C D Q 图 1 P A B C E F 图 2 G F E D CB A ① ② 17. ︵ᠲᠤᠰ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ① ᠤᠷᠢᠳᠠ ᠪᠠᠷ ᠬᠢᠯᠪᠠ ᠷᠴᠢᠯ ᠠᠵᠤ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣ ᠯ᠊᠃ 2𝑥𝑥: 1 + 4︔ 2𝑥 𝑥 2︔ 1 ÷ 𝑥︔ 2 𝑥 2︔ 2 𝑥: 1᠂ ᠡᠭᠦᠨ ᠲᠤ 𝑥= −2 2+ 2sin 45° + |− 3|᠃ ② ᠲᠡ ᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢ ᠨ᠊ ᠪᠥᠯᠥ ᠬ᠋᠊ { 𝑥− 3 ︵𝑥− 2︶ ≤ 8① 2𝑥︔ 15 𝑥: 12 − 1② ᠶᠢ ᠪᠣᠳᠣᠭᠠ ᠳ᠋᠊ ᠲᠡᠭᠦᠨ ᠦ ᠰᠥᠭᠡᠷᠬ ᠦ ᠪᠤᠰᠤ ᠪᠥᠬᠦ ᠯᠢ ᠲᠤᠭ ᠠᠨ ᠤ ᠬᠠᠷᠢᠭ ᠤ ᠶᠢ ᠣᠯ᠃ 18. ︵ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ︽ᠠᠯᠳᠠ ᠮᠥᠩᠬᠦᠨ ᠠᠭᠤᠯᠠ ᠪᠣᠯ ᠪᠠᠴᠤ ᠬᠥᠭᠡᠮ ᠳᠦᠭ ᠨᠤᠭ ᠣᠭᠠᠨ ᠠᠭᠤᠯ ᠠ ᠤᠰᠤ ᠶᠢ ᠡᠰᠡ ᠬᠦᠢᠴ ᠡᠮᠤᠢ︾ ᠭᠡᠳᠡ ᠭᠴᠢᠯ ᠠ ᠣᠷᠳᠤᠰ ᠬᠤᠲ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠬᠤᠰ ᠢᠭᠤᠨ ᠨᠤᠭ ᠣᠭᠠᠷ ᠠᠭᠤᠯ ᠤᠯᠲᠠ ᠶᠢᠨ ᠬᠤᠲ ᠠ ᠶᠢᠨ ᠪᠦ ᠳᠥᠭᠡᠨ ᠪᠠᠢᠭ ᠤᠯᠤᠯᠲᠠ ᠶᠢ ᠲᠠ ᠰᠤᠷᠠᠯᠲ ᠠ ᠦᠭᠡᠢ ᠲᠦ᠋ ᠯᠬᠢᠨ ᠠᠬ ᠢᠭᠤᠯ ᠵᠤ ᠪᠣᠢ᠃ ᠠ ᠡ ᠬᠠᠪ ᠤᠷ ᠲ᠊ ᠥᠷᠪᠡ ᠨ᠊ ᠲᠥᠷ ᠥᠯ ᠦ ᠨ᠊ ᠮᠤᠳᠤᠨ ᠰᠣᠶ ᠣᠭ ᠠ ᠶᠢ ᠲᠠᠷ ᠢᠵᠠᠢ᠃ ᠣᠢ ᠵᠢᠭᠤ ᠯᠤᠯᠲᠠ ᠶᠢᠨ ᠰᠠᠯᠠᠭ ᠠ ᠮᠥᠴᠢᠷ ᠦ ᠨ᠊ ᠬᠢᠨ ᠡᠳᠡᠭᠡᠷ ᠲᠠᠷ ᠢᠭᠰᠠᠨ ᠮᠤᠳᠤᠨ ᠰᠣᠶᠤᠭ ᠠ ᠠᠴ ᠠ 4 00 0 ᠰ᠊ ᠠᠭᠤᠷᠢ ᠶ᠊ ᠢ ᠲᠠ ᠰᠢᠷᠠ ᠮ᠊ ᠢᠶᠠᠷ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠴᠤ᠂ ᠡᠯ ᠡ ᠲ᠊ ᠥᠷᠥ ᠯ᠊ ᠦᠨ ᠲᠠᠷᠢᠭᠰᠠ ᠨ᠊ ᠮᠤᠳᠤᠨ ᠰᠣᠶᠤᠭ ᠠ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠲᠤᠭ ᠠ ᠶᠢ ᠰᠠᠪᠠᠭ ᠠᠨ ᠤ ᠦᠷᠢᠳᠭᠡᠯ ᠦ ᠨ᠊ ᠵᠢᠷᠤᠭ ᠢᠶᠠᠷ ᠲᠦ᠋ᠷᠰᠤ ᠯᠡᠵᠤ᠂ ᠪᠠᠰᠠ ᠡᠯ ᠡ ᠲᠥᠷᠥᠯ ᠦᠨ ᠮᠤᠳᠤᠨ ᠰᠣ ᠶᠤᠭ ᠠ ᠶᠢᠨ ᠤᠨᠳᠣᠷᠣᠭᠰᠠᠨ ᠰᠠᠭ ᠤᠷᠢ ᠲᠤᠭ ᠠ ᠶᠢ ᠰᠠᠪᠠᠭ ᠠᠨ ᠪᠦᠷ ᠢᠳᠭᠡᠯ ᠦ ᠨ᠊ ᠵᠢᠷᠤᠭ ᠢᠶᠠᠷ ᠲᠦ᠋ᠷᠰᠤᠯ ᠡᠪᠡ᠃ ᠪᠦᠷᠢ ᠲᠭᠡᠯ ᠢᠶᠡᠷ ᠲᠠ ᠮᠵᠢᠨ ᠨᠠ ᠷᠠᠰᠤᠨ ᠮᠤᠳᠤ ᠪᠠ ᠤᠯᠢᠶ ᠠᠰᠤ ᠶᠢ ᠨ᠊ ᠣ ᠨᠳᠣᠷ ᠣᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠨᠡᠯ ᠢᠶᠡᠳ ᠦᠨᠳᠤᠷ ᠪᠦ ᠭᠡᠳ ᠤᠯ ᠢᠶᠠᠰᠤ ᠶᠢ ᠨ᠊ ᠣ ᠨᠳᠣᠷ ᠣᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ 97 % ᠪᠣᠯ ᠬᠤ ᠶᠢ ᠮᠡᠳᠡᠪ ᠠᠯ᠃ ᠵᠢᠷᠤᠭ ᠲᠤ ᠦᠭ ᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠮᠡᠳᠡᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠤ ᠰᠤᠡᠯ ᠵᠤ᠂ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠠᠰᠠᠭᠤᠯ ᠲᠠ ᠶᠢ ᠬᠠᠷᠢᠭ ᠤᠯ᠃ ︵1︶ ᠲᠤᠭ ᠣᠷᠢᠭ ᠪᠦᠷᠢᠳ ᠭᠡᠯ ᠦ ᠨ᠊ ᠵᠢᠷ ᠤᠭ ᠲᠠᠬᠢ ᠨᠠᠷᠠᠰᠤᠨ ᠮᠤᠳᠤ ᠲᠠ ᠢ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠭᠡᠰᠡᠭ ᠦᠨ ᠲᠥᠪ ᠥᠨᠴᠥ ᠬ᠋᠊ ᠨᠢ ︵︶ ᠭᠷ ᠠᠳᠦ᠋ᠰ᠃ ᠪᠠᠰᠠ ᠰᠠᠪᠠᠭ ᠠᠨ ᠤ ᠦᠷᠢᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤᠭ ᠢ ᠪᠦᠷᠢᠨ ᠪᠣᠯᠭ ᠠ᠃ ︵2︶ ᠲᠤᠰ ᠬᠤᠰ ᠢᠭᠤᠨ ᠡᠨᠡ ᠵᠢᠯ ᠨᠠᠢᠲᠡ 32 ᠲᠤ ᠮᠡᠨ ᠰᠠᠭ ᠤᠷᠢ ᠮᠤᠳᠤ ᠲᠠ ᠷᠢᠭᠰᠠᠨ ᠪᠣᠯ ᠪᠠᠷᠤᠭ ᠬᠡᠳᠤᠢ ᠰᠠᠭ ᠤᠷᠢ ᠮᠤᠳᠤ ᠣᠨᠳᠣᠷ ᠣᠭᠰ ᠠᠨ ᠪᠣᠢ︖ ︵3︶ ᠣ ᠢᠵᠢᠭ ᠤᠯᠤᠯ ᠲᠠ ᠶᠢᠨ ᠰᠠᠯᠠᠭ ᠠ ᠮᠥᠴᠢᠷ ᠡ ᠴᠡ ᠢᠷᠠᠬ ᠦ ᠵᠢᠯ ᠲᠤ ᠡᠨᠡ ᠲᠥᠷᠪ ᠠ ᠲᠥᠷᠥᠯ ᠦᠨ ᠮᠤᠳᠤᠨ ᠰᠣᠶ ᠣᠭ ᠠ ᠠᠴ ᠠ ᠬᠣ ᠶᠠᠷ ᠲᠥᠷᠥᠯ ᠢ ᠰ᠊ ᠤᠩᠭᠤᠨ ᠲᠠᠷᠢᠬ ᠤ ᠪᠠᠷ ᠲᠣᠭᠲᠠᠭ ᠠᠪᠠ᠃ ᠮᠤᠳᠤᠯ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ︵ ᠤ ᠤᠶᠤ ᠬᠥᠰᠥ ᠨᠥᠭᠵ ᠢᠬᠦᠯᠬᠦ ᠠᠷᠭ ᠠ︶ ᠪᠠ ᠷ᠊ ᠣ ᠨᠳᠣᠷ ᠣᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠨᠠᠯ ᠢᠶᠡᠳ ᠦᠨᠳ ᠤᠷ ᠤ ᠣᠯᠬᠤ ᠬᠤᠶ ᠠᠷ ᠲ᠊ ᠥᠷᠥ ᠯ᠊ ᠦᠨ ᠮᠤᠳᠤᠨ ᠰᠣᠶᠤᠭ ᠠ ᠶᠢ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ ᠰᠤᠩᠭᠤᠵ ᠤ ᠠ ᠪᠬᠤ ᠶᠢᠨ ᠲᠤᠬ ᠢᠶᠠᠯᠳ ᠤᠮᠵᠢ ᠶᠢ ᠣᠯ᠃ ︵ᠨᠠᠷ ᠠᠰᠤᠨ ᠮᠤᠳᠤ᠂ ᠤ ᠯᠢᠶᠠᠰᠤ᠂ ᠬᠠᠢ ᠯᠠᠰᠤ᠂ ᠤᠳᠠ ᠮᠤᠳᠤ ᠶᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑎, 𝑏, 𝑐,𝑑 ᠢᠶᠠᠷ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡ︶ 19. ︵ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤ ᠤ ᠨᠠ ᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ∆AB C ᠲᠤ ∠B AC = 45°᠂ AD ⊥BC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ D , BD = 6 , DC = 4 , AD ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣ ᠯ᠊᠃ ᠰᠢᠶᠥ ᠥ ᠮᠢ ᠩᠭ ᠠ ᠨᠤᠭ ᠤᠯᠬᠤ ᠠᠷ ᠬ᠋᠊ ᠠ ᠶᠢ ᠠᠰᠢᠭ ᠯᠠᠨ ᠤᠷᠠᠨ ᠠᠷᠭ ᠠ ᠪᠠᠷ ᠲᠤᠰ ᠰᠡᠲᠤᠪ ᠢ ᠬᠠᠷ ᠢᠭᠤᠯ ᠵᠠᠢ᠃ ᠰᠢᠶᠥ ᠥ ᠮᠢ ᠩᠭ ᠠ ᠶᠢᠨ ᠰᠡᠳᠬᠢᠨ ᠰᠥᠪᠡ ᠭᠴᠢᠯ ᠡᠭᠰᠡᠨ ᠶᠠᠪᠤᠴᠠ ᠶᠢ ᠦᠨᠳᠤᠰ ᠤᠯᠡᠵ ᠤ᠂ ᠳᠠ ᠷᠠᠭᠠ ᠬᠢ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷᠢᠭ ᠤᠯ᠃ ︵1︶ AB , AC ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠪᠣᠯᠭ ᠠᠨ ∆ AB D ᠪᠠ ∆A CD ᠶᠢᠨ ᠲᠡ ᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠵᠢᠷ ᠤᠭ ᠲᠥᠷᠰᠤ ᠶᠢ ᠵᠢᠷ ᠤᠪᠠᠯ᠂ ᠴᠡ ᠬ᠋᠊ D ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠬᠡ ᠮᠳᠤ ᠴᠡᠭ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ E , F ᠪᠣᠯ ᠣᠨ ᠠ᠃ EB ᠪᠠ FC ᠶᠢ ᠦ ᠷᠬᠦᠯᠵ ᠢᠯᠡᠬᠦ ᠯᠪᠡᠯ ᠴᠡᠭ G ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠃ ᠪᠠᠲᠤᠯ ᠠ᠄ ᠲᠤ ᠷᠪᠡᠯ ᠶᠢᠨ AE FG ᠨᠢ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ᠃ ︵2︶ AD = 𝑥 ᠭᠡᠵ ᠤ ᠵᠢᠱ ᠢᠪᠡᠯ᠂ 𝑥 ᠦᠨ ᠲᠤᠬᠠ ᠢ ᠲᠡ ᠭᠰᠢᠳ ᠭᠡᠯ ᠪᠠᠢᠭ ᠤᠯᠵᠤ᠂ AD ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣ ᠯ᠊᠃ 20. ︵ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠸᠠᠩ ᠠᠨᠢᠶ ᠠ ᠶᠢᠨ ᠭᠡᠷ ᠦ ᠨ᠊ ᠨᠠ ᠷᠠᠪᠴ ᠶᠢᠨ ᠳ᠋᠊ ᠤ ᠨᠢ ᠭᠡᠨ ᠬᠤᠪ ᠴᠠᠰᠤ ᠬᠠᠲᠠᠭ ᠠᠬᠤ ᠵᠢᠶᠠᠰᠠ ᠶᠢ ᠲᠤᠭ ᠳᠠᠪᠤᠷ ᠢᠵᠢᠭ ᠤᠯᠤᠨ ᠲᠡᠯᠭᠡ ᠵᠤ ᠲᠠ ᠯᠪᠢᠭ ᠰᠠᠨ ᠨᠢ ᠵᠢᠷ ᠤᠭ ① ᠮᠡᠲᠤ᠃ ᠵᠢᠷ ᠤᠭ ②᠂ ③ ᠪᠣᠯ ᠬᠤ ᠪᠴᠠᠰᠤ ᠬᠠᠳᠠ ᠭᠠᠭ᠋ᠤ ᠵᠢᠶᠠᠰᠠ ᠶᠢ ᠳᠡᠯᠭᠡᠭ ᠰᠡᠨ ᠦ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠬᠠᠵᠠᠭᠤ ᠲᠠ ᠯ᠊ ᠠ ᠶᠢᠨ ᠣᠭᠳᠣᠯ ᠣᠭᠠᠰᠤ ᠵᠢᠷᠤᠭ ᠮᠥᠨ᠃ ∆ AO B ᠤ ᠣᠯ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠨᠢ 13 0c m ᠪᠣᠯᠬᠤ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ᠃ ᠬᠤᠪᠴᠠᠰᠤ ᠬᠠᠳ ᠠᠭᠠᠬᠤ ᠵᠢᠶᠠᠰ ᠠᠨ ᠤ ᠬᠦ ᠨᠳᠡᠯᠡ ᠨ᠊ ᠰ᠊ ᠠᠪᠠᠭ ᠠ OE , OF ᠨᠢ O ᠤ ᠠᠷ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ ᠡᠷᠬᠢᠯ ᠳᠤᠵᠤ ᠪᠣᠯᠣ ᠨ᠊ ᠠ᠂ ᠪᠠᠰᠠ OE = OF = 13 0c m᠂ OA᠂ OB ᠳᠡᠭᠡᠷ ᠡᠬᠢ C᠂ D ᠴᠡᠭ ᠲᠤ ᠲᠤᠰ ᠲᠤᠰ CN᠂ DM ᠲᠤ ᠯᠬᠤ ᠰᠠᠪᠠᠭ ᠠ ᠤ ᠠᠢᠵᠤ᠂ ᠲᠡᠷ ᠡ ᠨ᠊ ᠢ ᠴᠡ ᠬ᠋᠊ C᠂ D ᠢᠶᠡᠷ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤ ᠨ᠊ ᠲ᠊ ᠥᠪ ᠪᠣᠯᠭ ᠠᠨ ᠡᠷ ᠬᠢᠯᠳ ᠥᠵᠤ ᠪᠣᠯᠣᠨ ᠠ᠃ ︵1︶ ᠵᠢᠷ ᠤᠭ ② ᠮᠡ ᠲᠤ᠂ ᠬᠡᠷᠪᠡ EF ᠨᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ᠂ ᠸᠠ ᠩ᠊ E F G B CD A ᠠᠨᠢᠶ ᠠ ᠶᠢᠨ ᠬᠤᠪ ᠴᠠᠰᠤ ᠶᠢ ᠬᠤᠪ ᠴᠠᠰᠤ ᠨ᠊ ᠡᠯᠬᠦᠬ ᠦᠷ ᠲᠤ ᠡᠯᠭᠦ ᠭᠰᠡᠨ ᠦ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠶᠡᠷ ᠤᠩᠬᠡᠢ ᠤᠷᠳᠤ ᠨᠢ 11 0c m ᠪᠣᠯ ᠲᠡᠭᠦᠨ ᠢ ᠬᠤ ᠪᠴᠠᠰᠤ ᠬᠠᠳᠠ ᠭᠠᠭ᠋ᠤ ᠵᠢᠶᠠᠰ ᠠᠨ ᠤ ᠬᠦ ᠨᠳᠡᠯᠡᠨ ᠰᠠᠪᠠᠭ ᠠ OE ᠳᠡ ᠭᠡᠷ ᠡ ᠡᠯᠭᠦᠪᠡᠯ ᠭᠠᠵᠠᠷ ᠲᠤ ᠴᠢᠷᠥ ᠭᠳᠡᠨ ᠡ ᠤᠤ︖ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲ᠊ ᠣᠳᠣᠷᠬᠠᠢᠯ ᠠ᠃ ︵0. 1cm ᠬᠦᠷ ᠲᠡᠯ ᠡ ᠨᠠᠷᠢᠪ ᠴᠢᠯᠠ᠃︶ ︵2︶ ᠵᠢᠷᠤᠭ ③ ᠮᠡᠳᠤ᠂ ᠲᠤᠯ ᠬᠤ ᠰᠠᠪᠠᠭ ᠠ DM ᠨᠢ ᠴᠡ ᠬ᠋᠊ 𝑀 ′ ᠳᠡ ᠭᠡᠷ ᠡ ᠲᠤᠯ ᠬᠤ ᠦᠶ ᠡᠰ ∠EO B= 78° ᠴᠡᠭ E ᠨᠢ ᠭᠠᠵᠠᠷ ᠤ ᠨ᠊ ᠭᠠᠳᠠ ᠷᠭᠤ ᠠᠴᠠ ᠬᠠᠮᠤᠭ ᠦᠨᠳᠤᠷ ᠵᠶᠢᠳᠠᠢ᠃ ᠬᠡᠷᠪᠡ ᠬᠤᠪ ᠴᠠᠰᠤᠨ ᠡᠯᠭ ᠦᠭᠦᠷ ᠲᠤ ᠡᠯ ᠭᠦᠭᠰᠡᠨ ᠬᠤᠪ ᠴᠠᠰᠤ ᠶᠢ ᠭᠠᠵᠠᠷ ᠲᠤ ᠴᠢᠷᠤᠭ ᠳᠡᠬᠦᠯ ᠬᠦ ᠦᠭᠢ ᠡ ᠭᠡᠪᠡ ᠯ᠊ ᠬ᠊ ᠤᠪᠴᠠ ᠰᠤᠨ ᠡᠯᠭᠦᠭᠦᠷ ᠪᠠ ᠬᠤᠪ ᠴᠠᠰᠤ ᠨ᠊ ᠤ ᠶᠡᠷᠤᠩᠬᠡᠢ ᠤᠷᠳᠤ ᠶᠢ ᠬᠠᠮᠤᠭᠤᠯ ᠤᠷᠳᠤ ᠲᠤ ᠪᠠ ᠨ᠊ ᠤ ᠠᠷᠤᠭ ᠬᠡᠳᠤᠢ ᠰᠡᠨᠲᠢᠮᠧᠳ᠋ᠷ ᠪᠠᠢᠯᠭᠠᠪ ᠠᠯ ᠵᠣᠬᠢᠬᠤ ᠪᠣᠢ︖ ︵ᠳᠦ᠋ᠩ ᠲᠤ ᠨᠢ ᠤ ᠥᠬᠦᠯ ᠢ ᠲᠤᠭ ᠠ ᠦ ᠯᠡᠳᠡᠭᠡ︶ ︵ᠯᠠᠪᠯᠠᠬᠤ ᠲᠤᠭ ᠠ ᠤ ᠠᠷᠢᠮᠳᠠ᠄ √3 ≈ 1710 , sin 78° ≈ 4950 , cos 78° ≈ 15 , sin 18° ≈ 310 , cos 18° ≈ 1920︶ 21. ︵ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰ᠊ ᠡᠲᠤᠪ ᠨᠠᠢ ᠲᠡ 8 ᠬᠤᠪᠢ︶ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ☉O ᠪᠣᠯ ∆AB C ᠢ ᠪᠠᠭᠳᠠ ᠭᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠣᠷᠢᠭ᠂ AC ᠨᠢ ᠳᠢᠶᠠᠮ ᠧᠳ᠋ᠷ᠂ ᠬᠥᠪ ᠴᠢ BD = BA᠂ BE ⊥ DC᠂ ᠪᠥᠭᠡᠳ DC ᠶᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤ ᠮ᠊ ᠲᠠ ᠢ ᠴᠡ ᠬ᠋᠊ E ᠲᠤ ᠣ ᠭᠳᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠃ ⑴ ᠪᠠᠳᠤᠯ ᠠ᠄ BE ᠪᠣᠯ ᠶᠢᠨ ☉O ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨ᠊᠃ ⑵ ᠬᠡᠷᠪᠡ sin ∠𝐵𝐶 𝐸 = 34 , AB = 3 ᠤ ᠣᠯ᠂ AD ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠶᠢ ᠣ ᠯ᠊᠃ A B C D EO 图 1 图 2 A B C D EF MN O 图 3 A B C D ON′ M′ EF 22. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲ ᠡ 9 ᠬᠤᠪᠢ︶ ᠮᠠᠯᠴᠢᠨ ᠪᠠᠭᠠᠳᠤᠷ ᠨᠢ ᠭᠡᠨ ᠲᠥᠷᠥᠯ ᠦᠨ ᠴᠠᠭᠠᠨ ᠢᠳᠡᠭᠡ ᠨ᠊ ᠢ ᠦᠢ ᠯᠡᠳᠪᠤᠷᠢᠯ ᠡᠵᠤ᠂ ᠰ᠊ ᠦᠯᠵ ᠢᠶ ᠡ ᠶᠢ ᠠᠰᠢᠭ ᠯᠠᠨ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯ ᠬᠤ ᠤ ᠠᠷ ᠤ ᠣᠯᠪᠠ᠃ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠤ ᠭᠰᠠᠨ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ᠊ ᠰ᠊ ᠦᠯᠵ ᠢᠶᠡᠨ ᠳᠡ ᠭᠡᠷ ᠡ ᠵᠠᠬᠢᠶ ᠠᠯᠠᠵᠤ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠪᠠᠭ ᠠᠳᠤᠷ᠂ ᠠ ᠪᠤᠶᠤ ᠡ ᠲᠥᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠶ᠊ ᠢ ᠰ᠊ ᠤᠩᠭᠤᠵ ᠤ ᠭᠤᠪᠢᠭᠤ ᠯᠬᠤ ᠰᠦᠳᠭᠡᠯ ᠢᠶᠡᠨ ᠲᠤᠰᠢ ᠶᠠᠨ ᠪᠠᠷᠠᠭ ᠠ ᠪᠠᠨ ᠭᠤᠪᠢᠭᠤ ᠯᠤᠨ ᠬᠦᠷᠭᠡᠨ ᠡ᠃ ᠠ ᠲᠥᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠪᠠ ᠷ᠊ 2 ᠺᠢᠯ ᠣᠭᠷᠠᠮ ᠢ ᠭᠤᠪᠢᠭᠤᠯ ᠵᠤ᠂ ᠡ ᠲᠥᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤ ᠯᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠪᠠ ᠷ᠊ 3 ᠺᠢᠯᠥ ᠭᠷᠠᠮ ᠢ ᠭᠤᠪᠢᠭᠤᠯ ᠬᠤ ᠲᠤ ᠨᠠᠢᠲᠡ 42 ᠲᠤ ᠬᠥᠷᠢᠭ ᠦᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦᠳᠭᠡᠯ ᠬᠡ ᠷᠡᠭᠰ ᠠ ᠡ᠃ ᠠ ᠲ᠊ ᠥᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠪᠠ ᠷ᠊ 5 ᠺᠢᠯ ᠣᠭᠷᠠᠮ ᠢ ᠭᠤᠪᠢᠭᠤ ᠯᠵᠤ᠂ ᠡ ᠲᠤ ᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠪᠠ ᠷ᠊ 4 ᠺᠢᠯᠥ ᠭᠷᠠᠮ ᠢ ᠭᠤᠪᠢᠭᠤ ᠯᠬᠤ ᠲᠤ ᠨᠠᠢᠲᠡ 70 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠦ ᠨ᠊ ᠭᠤᠪᠢᠭᠤ ᠯᠬᠤ ᠰᠦ᠋ᠢ ᠲᠭᠡᠯ ᠬᠡᠷᠡᠭᠰ ᠠ ᠡ᠃ ︵1︶ ᠠ᠂ ᠡ ᠬᠣ ᠶᠠᠷ ᠲᠥᠷᠭᠡᠨ ᠭᠤᠪ ᠢᠭᠤᠯ ᠬ᠋᠊ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠶᠢᠨ ᠭᠤᠪᠢᠭᠤ ᠯᠬᠤ ᠰᠦ᠋ᠢ ᠲᠭᠡᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠺᠢ ᠯᠣᠭᠷ ᠠᠮ ᠤ ᠦᠷᠢ ᠳᠡᠭᠡ ᠨ᠊ ᠬᠡ ᠳᠤᠢ ᠲᠤ ᠬᠥᠷᠢᠭ ᠪᠣᠢ︖ ︵2︶ ᠪᠠᠭ ᠠᠳᠤᠷ ᠨᠢ ᠭᠡ ᠡᠳᠤᠷ ᠦᠢᠯ ᠡᠳᠪᠤᠷᠢᠯᠡᠭ ᠰᠨ ᠡ ᠴᠠᠭᠠᠨ ᠢᠳᠡᠭᠡ ᠨ᠊ ᠢᠶᠡᠨ ᠲᠤᠰ ᠡᠳᠤᠷ ᠲ᠊ ᠤᠨᠢ ᠪᠥᠬᠦ ᠶᠢᠨ ᠢ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠵᠤ ᠪᠣᠯᠬ ᠤ ᠪᠦ ᠭᠡᠳ ᠭᠤᠪᠢᠭᠤ ᠯᠬᠤ ᠰᠦ᠋ᠢ ᠲᠭᠡᠯ ᠪᠠᠭ ᠠ ᠪᠤ ᠯᠬᠤ ᠲᠤ ᠷᠭᠡᠨ ᠭᠤᠪᠢᠭᠤ ᠯᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫ ᠠᠨᠢ ᠶᠢ ᠣᠩᠭᠤᠯ ᠵᠤ ᠪᠠᠷᠠᠭ ᠠ ᠪᠠᠨ ᠭᠤᠪᠢᠭᠤᠯ ᠨ᠊ ᠬ᠊ ᠦᠷᠭᠡᠨ ᠡ ᠭᠡᠵ ᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ ᠲᠤᠰ ᠴᠠ ᠭᠠᠨ ᠢᠳᠡᠭᠡ ᠨ᠊ ᠢ ᠦᠢᠯ ᠡᠳᠪᠤᠷᠢᠯ ᠡᠬᠦ ᠬᠡᠮᠵᠢᠶ ᠡ ᠺᠢ ᠯᠥᠭᠷ ᠠᠮ ᠲᠤ ᠬᠡᠷᠡᠭᠯᠠᠬ ᠦ ᠥᠪᠡᠷ ᠦᠨ ᠥᠷᠳᠡ ᠬ᠋᠊ ᠨᠢ ᠺᠢᠯᠥ ᠭᠷᠠᠮ ᠪᠦᠷᠢ ᠲᠡᠭᠡᠨ 𝑦1 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ︵ᠭᠤᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦᠳ ᠭᠡᠯ ᠢ ᠪᠠᠭᠲᠠᠭ ᠠᠬᠤ ᠦᠭᠡᠢ︶ ᠪᠣᠯ ᠣᠨ ᠦᠢᠯ ᠡᠳᠪᠤᠷᠢᠯ ᠡᠬᠦ ᠬᠡ ᠮᠵᠢᠶ ᠡ x ᠺᠢᠯᠥ ᠭᠷᠠᠮ ᠤᠨ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦ ᠨ᠊ ᠬᠤᠯᠪᠣᠭᠳᠠᠯ ᠤ ᠨ᠊ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠨᠢ 𝑦1 = { − 2𝑥 + 58 ︵0 𝑥 8︶ 42 ︵𝑥 ≥ 8︶᠂ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠬᠤ ᠥᠨ ᠡ ᠨᠢ ᠺᠢᠯᠥ ᠭᠷᠠᠮ ᠪᠦᠷᠢ ᠳᠠᠬ ᠠ 𝑦 2 ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠪᠣᠯᠣ ᠨ᠊ ᠦᠢ ᠯᠡᠳᠪᠤᠷᠢᠯ ᠡᠬᠦ ᠬᠡ ᠮᠵᠢᠶ ᠡ x ᠺᠢᠯᠥᠭ ᠷᠠᠮ ᠤ ᠨ᠊ ᠬᠤᠭᠤᠷ ᠣᠨᠳᠤᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦ ᠨ᠊ ᠬᠣ ᠯᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠨᠢ 𝑦 2 = −6 𝑥+ 120 ︵0 𝑥 13︶ ᠪᠣᠯ᠂ ᠪᠠᠭ ᠠᠳᠤᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠬᠡᠳᠤᠢ ᠺᠢᠯᠤᠭ ᠷᠠᠮ ᠴᠠᠭᠠᠨ ᠢᠳᠡᠭᠡ ᠨ᠊ ᠢ ᠦᠢᠯ ᠡᠳᠪᠤ ᠷᠢᠯᠠᠬ ᠦ ᠦᠶ ᠡᠰ ᠣ ᠯᠬᠤ ᠠᠰᠢᠭ ᠬᠤᠨ ᠵᠢᠪᠤ ᠷᠢ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠪᠣᠯ ᠬᠤ ᠪᠣᠢ︖ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠠᠰᠢᠭ ᠬᠤᠨ ᠵᠢᠪᠤᠷᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠲᠥᠬᠦ‍ ᠷᠢᠭ ᠪᠣᠢ︖ 23. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 11 ᠬᠤᠪᠢ︶ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠰᠢᠯᠤᠭᠤᠨ y = 12 𝑥 − 3 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢᠭ᠂ y ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ B , C ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠬ᠋᠊ ᠲᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ᠊ ᠨᠢ B , C ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠬ᠋᠊ ᠢ ᠳᠠᠭ ᠠᠷᠢᠬ ᠤ ᠪᠦ ᠭᠡᠳ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠ ᠢ ᠣ ᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠬᠤ ᠨᠥᠬᠦᠭᠡ ᠨᠢᠭᠡ ᠣᠭᠳ ᠣᠯᠣᠯᠴ ᠠᠯ ᠤ ᠨ᠊ ᠴᠡ ᠬ᠋᠊ ᠨᠢ A᠃ ︵1︶ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ᠊ ᠤᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ ︵2︶ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ 𝑆 ∆𝐷 𝐵𝐶 = 𝑆∆ 𝐴𝐵𝐶 ᠪᠣᠯ ᠭᠠᠭ᠋ᠤ ᠴᠡᠭ D ︵ᠴᠡ ᠬ᠋᠊ A ᠲᠠᠢ ᠲᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ︶ ᠣᠷᠣᠰ ᠶᠢᠨ ᠠ ᠤᠤ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣᠰ ᠢᠪᠠᠯ ᠴᠡᠭ D ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ ᠬᠡᠷᠪᠡ ᠣ ᠷᠣᠰᠢᠬ ᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ ︵3︶ ᠥᠷᠭᠡᠨ ᠨᠢ 2᠂ ᠤᠷᠳᠤ ᠨᠢ ᠬᠦᠷᠢᠶᠡᠨ ᠤᠷᠳᠤ ᠪᠣᠯ ᠬᠤ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ︵ᠰᠡᠭᠦᠳᠡ ᠷ᠊ ᠲᠠ ᠢ ᠭᠡᠰᠡᠭ︶ ᠶᠢ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠦ ᠨ᠊ ᠴᠢᠭᠯᠡᠯ ᠲᠠ ᠭᠠᠤ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠪᠡ ᠯ᠊᠂ y ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠨᠢᠭᠡ ᠬᠣ ᠣᠰ ᠡᠰᠡᠷ ᠬᠦ ᠲᠠ ᠯ᠊ ᠠ ᠨᠢ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢ ᠨ᠊ ᠱᠤᠭᠤ ᠮ᠊ ᠲᠠ ᠢ ᠴᠡ ᠬ᠋᠊ P ᠪᠠ Q ᠲᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠ ᠵᠤ᠂ ᠰᠢᠯᠤᠭᠤᠨ CB ᠲᠠ ᠢ ᠴᠡ ᠬ᠋᠊ M ᠪᠠ N ᠲᠤ ᠣᠭᠳᠣᠯ ᠣᠯᠴᠠᠨ ᠠ᠃ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥ ᠭᠳᠥ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠬᠦ ᠶᠠᠪᠤᠴᠠ ᠳ᠋᠊ ᠤ ᠴᠡ ᠬ᠋᠊ P᠂ Q᠂ M᠂ N ᠢᠶᠡᠷ ᠣᠷ ᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠢᠶᠡᠨ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠲᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠨᠢ ᠵᠢᠰᠢᠬᠦ ᠯᠵᠢᠨ ᠪᠣᠯᠬᠤ ᠦᠶ ᠡᠰ ᠦ ᠨ᠊ ᠴᠡ ᠬ᠋᠊ M ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ 24. ︵ᠲᠤᠰ ᠪᠠ ᠬ᠋᠊ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 12 ᠬᠤᠪᠢ︶ ︵1︶ 【 ᠣᠷ ᠣᠯᠳᠣ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠠ ᠵᠢᠭᠯ ᠠᠵᠤ ᠣ ᠯᠬᠤ 】 ᠵᠢᠷᠤᠭ ① ᠮᠡ ᠲᠤ᠂ ∆AB C ᠶᠢ ᠴᠡᠭ A ᠶᠢ ᠲᠤᠭ ᠣᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤᠨ ᠵᠡᠬᠦᠦ ᠶᠢ ᠨ᠊ ᠳᠠᠭ ᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 60° ᠡᠷᠬᠢᠭ ᠦᠯᠪᠡ ᠯ᠊ ∆ ADE ᠶᠢ ᠣᠯᠣᠨ ᠠ᠂ BD ᠶᠢ ᠬᠤᠯ ᠪᠣᠪᠠ ᠯ᠊ ∠AB D = ︵︶ ᠭᠷᠠᠳᠦ᠋ᠰ᠃ ⑵ 【 ᠠᠳᠠ ᠯᠢᠳᠬ ᠠᠨ ᠰ᠊ ᠦᠪᠡ ᠭᠴᠢᠯ ᠡᠬᠦ 】 ᠵᠢᠷᠤᠭ ② ᠮᠡᠲᠤ᠂ ᠲᠡ ᠩᠴᠡᠬᠦ ᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ABC ᠶᠢᠨ ᠲᠣᠳᠣᠷ ᠠ ᠲᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠡᠭ P ᠶᠢ ᠠ ᠪᠴᠤ᠂ PA᠂ PB᠂ PC ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠ᠄ PA᠂ PB᠂ PC ᠶᠢᠨ ᠤᠷᠳᠤ ᠪᠠᠷ ᠭᠤᠷᠪ ᠠᠨ ᠲᠠᠯ ᠠ ᠪᠠᠨ ᠪᠣᠯᠭ ᠠᠭᠰᠠᠨ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠢ ᠯᠠᠪᠳᠠ ᠢ ᠪᠦᠷᠢ ᠯᠳᠥᠬᠦ ᠯᠵᠤ ᠲᠠᠢ ᠯᠦᠨ ᠡ᠃ x y OA B C M N P QC BA O y x 图 1 A B C D E ① ② 图 2 A B C P ⑶ 【 ᠠᠰᠠᠭᠤᠳᠠ ᠯ᠊ ᠢ ᠰᠢᠢᠳᠪ ᠤᠷᠢᠯ ᠡᠬᠦ 】 ᠵᠢᠷᠤᠭ ③ ᠮᠡ ᠲᠤ᠂ ᠴᠡ ᠬ᠋᠊ P ᠨᠢ ᠲᠠ ᠯ᠊ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ √7 ᠪᠣᠯ ᠬᠤ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠲᠣᠳᠣᠷ ᠠ ᠣ ᠷᠣᠰᠢᠬ ᠤ ᠲᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠡᠭ᠂ ∠APC = 90°᠂ ∠B PC = 120°᠃ ∆ APC ᠶᠢᠨ ᠲᠠᠯ ᠠᠪᠠᠢ ᠶᠢ ᠣᠯ᠃ ⑷ 【 ᠥᠷᠭᠡᠳᠭᠡᠨ ᠬᠡᠷ ᠡᠭᠯᠠᠬ ᠦ 】 ᠵᠢᠷᠤᠭ ④ ᠪᠣᠯ᠂ A᠂ B᠂ C ᠭᠤ ᠷᠪᠠᠨ ᠲᠤᠰᠬ ᠤᠨ ᠤ ᠪᠠᠢᠷ ᠢᠯᠠᠯ ᠤ ᠨ᠊ ᠬᠠᠪ ᠳᠠᠭᠠᠢ ᠵᠢᠷᠤᠭ ᠪᠣᠯᠣᠨ ᠠ᠂ ᠬᠡᠮᠵᠢᠯ ᠲᠡ ᠪᠠᠷ ᠲᠠ ᠮᠵᠢᠨ AC = 4 , BC = 5 , ∠A CB = 30° ᠶᠢ ᠣᠯᠵ ᠤ ᠮᠡ ᠳᠡᠪᠡ᠃ ᠴᠡᠭ P ᠨᠢ ∆AB C ᠶᠢᠨ ᠳᠣᠲᠣᠷ ᠠ ᠣ ᠷᠣᠰᠢᠬ ᠤ ᠨᠢ ᠭᠡ ᠬᠦᠳᠡᠯ ᠬᠦ ᠴᠡ ᠬ᠋᠊᠂ PA᠂ PB᠂ PC ᠶᠢ ᠬᠤᠯ ᠪᠣᠨ ᠠ᠃ PA + PB + PC ᠶᠢᠨ ᠬᠠᠮ ᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠣᠯ᠃ ③ 图 3 A B C P 图 4 A P B C ④