2018 ᠣᠨ ᠤ ᠣᠷᠳᠣᠰ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

ᠨᠢᠭᠡ ᠂ ᠭᠠ ᠭᠴᠠ ᠬᠠᠷᠢ ᠭᠤᠯᠲ ᠠ ᠰᠤᠩ ᠭᠤᠬ ᠤ ᠰᠡᠲᠤᠪ ᠃ ( ᠲᠤᠰ ᠶᠡᠭᠡ ᠰᠡᠲᠤᠪ ᠲᠤ 10 ᠪᠠᠭ ᠠ ᠰ ᠡᠲᠤᠪ ᠲᠠᠢ᠂ ᠪᠠᠭ ᠠ ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢ ᠲᠡ 30 ᠬᠤᠪᠢ) 1. ᠳᠠᠷᠠᠭᠠᠬᠢ 227 , − 2018 , √ 4 , π ᠲᠥᠷᠪᠡᠨ ᠲᠣᠭ ᠠ ᠠᠴᠠ ᠶᠤᠰᠤ ᠬᠦᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 227 B. −2018 C. √4 D. π 2. ᠳᠠᠷᠠᠭᠠᠬᠢ ᠪᠣᠳ ᠣᠯᠭ ᠠ ᠠᠴᠠ ᠵᠥ ᠪ ᠪᠣᠯᠬᠤ 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.ᠳᠠᠷᠠᠭᠠᠬᠢ ᠹᠦᠩ ᠺᠼ ᠦᠳ ᠡᠴᠡ ᠬᠤᠪᠢᠰᠬᠤ ᠨ 𝑥 ᠤᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠠᠪᠬ ᠤ ᠬᠡᠪᠴᠢ ᠶ 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 ᠪ ᠡᠷ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠥᠨᠴᠥᠭ ᠬᠡᠮᠵᠢᠭᠦᠷ ᠪᠣᠯᠣ ᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭ ᠳᠥ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤᠤᠢ ABC ᠶᠢ ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ ᠲᠠᠯᠪᠢᠪᠠᠯ᠂ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭ ᠦᠨ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ B ᠨᠢ ᠲᠡᠭ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠤᠨ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠭ ᠠ ᠰᠢᠯᠤᠭᠤ ᠨ DE ᠳᠡᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠨ ᠠ᠃ ᠥᠨᠴᠥᠭ ᠬᠡᠮᠵᠢᠬᠦᠷ ᠪᠣᠯᠣ ᠨ ᠭᠤᠷᠪᠠᠯᠵ ᠢᠨ ᠤᠤ ᠢ ᠨᠢ ᠵᠥᠪᠬᠡᠨ ᠨᠢᠭᠡ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠯ ᠤᠨ ᠴᠡᠭ ᠲᠠᠢ ᠪᠥᠭᠡᠳ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠯ ᠤᠨ ᠴᠡᠭ ᠨᠢ P ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ∠EOP = 134°50 ′ ᠪᠣᠯ ∠ CBD ᠶᠢᠨ ᠭᠷ ᠠᠳᠦ᠋ ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 45°10 ′ B. 44°50 ′ C. 46°10 ′ D. ᠳᠣᠭ ᠳᠠᠭᠠᠵᠤ ᠲᠠᠢ ᠯᠬᠦ ᠦᠭᠡᠢ 5. ᠨᠢᠭᠡᠨ ᠬᠦᠴᠢᠷ ᠳᠡᠯᠳᠡᠢ ᠰᠤᠷᠤ ᠭᠴᠢ ᠶᠢ ᠲᠡᠳᠬ ᠦᠬᠦ ᠶᠢᠨ ᠲᠥᠯᠥ ᠭᠡ ᠨᠢᠭᠡᠨ ᠰᠤ ᠷᠭᠠᠭ ᠤᠯᠢ ᠶᠢ ᠨ ᠬᠢᠨ ᠵᠣᠭᠣ ᠰ ᠲᠤᠰᠠᠯᠠᠬᠤ ᠠᠵᠢ ᠯᠯᠠᠭᠠᠨ ᠢ ᠥᠷᠨᠢᠬᠦᠯᠪᠡ᠃ ᠣ ᠳᠥ ᠪᠥᠬᠥ ᠰᠤᠷᠭᠠᠭᠤᠯᠢ ᠶᠢᠨ ᠰᠤᠷ ᠤᠭᠴᠢ ᠶᠢ ᠨ ᠲᠤᠰᠠᠯᠠᠭᠰᠠ ᠨ ᠵᠣᠭᠣ ᠰ ᠠᠴᠠ 10 ᠰᠤᠷᠤ ᠭᠴᠢ ᠶᠢᠨ ᠳᠤᠰᠠᠯᠠᠭᠰᠠ ᠨ ᠵᠣᠭᠣ ᠰ ᠢ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠰᠤᠩ ᠭᠤᠨ ᠠᠪᠤᠭᠠᠳ᠂ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠳᠠ ᠶᠢ ᠪ ᠦᠷᠢᠳᠭ ᠡᠭᠰᠡᠨ ᠨᠢ ᠳᠣᠣ ᠷᠠᠬᠢ ᠬᠦᠰᠥᠨᠥ ᠭᠳᠦ ᠮᠡᠳ ᠤ᠄ ᠳᠣᠣ ᠷᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠥ ᠪ ᠪᠣᠯᠬᠤ ᠨᠢ A. 10 ᠰᠤᠷᠤ ᠭᠴᠢ ᠪᠣᠯ ᠪᠦᠭᠦᠳᠡ ᠶᠢᠨ ᠨᠢᠭᠡ ᠶᠠᠩ ᠵᠤ B. ᠳᠤᠮᠳᠠ ᠪ ᠠᠢᠷᠢ ᠶᠢᠨ ᠲᠣ ᠭ ᠠ ᠨᠢ 40 C. ᠮᠤᠳ᠋ ᠨᠢ 90 D. ᠰᠠᠷ ᠨᠢᠴᠠ ᠨᠢ 400 6. ᠵᠢᠷᠤ ᠭ ᠮᠡᠳᠤ᠂ ᠷᠤᠮᠪ ABC D ᠳ ᠤ᠂ ᠳᠠᠷᠠᠭᠠ ᠬᠢ ᠠᠯᠬᠤ ᠮ ᠳᠠᠷᠠᠭ ᠠ ᠪᠠᠷ ᠵᠢᠷᠤ ᠭ ᠪᠠᠢᠭᠤᠯᠤ ᠨ ᠠ᠃ ① ᠴᠡᠭ C ᠪᠠ ᠴᠡᠭ D ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠨ᠂ 12 𝐶𝐷 ᠡᠴᠡ ᠶᠡᠭᠡ ᠤᠷᠳᠤ ᠪ ᠠᠷ ᠷᠠᠳᠢᠤᠰ ᠪᠣᠯᠭᠠᠨ ᠨᠤᠮᠤ ᠪᠠᠢ ᠭᠤᠯᠪ ᠠᠯ᠂ ᠬᠣᠶᠠᠷ ᠨᠤᠮᠤ ᠨᠢ ᠴ ᠡᠭ ᠳᠤ ᠣᠭᠳ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ② ᠰᠢᠯᠤᠭᠤ ᠨ𝑀𝑁 ᠢ ᠪᠠᠢᠭᠤᠯᠪᠠᠯ ᠶᠠᠭ DE O P A B C 0 30 60 90 120 150 180 ᠳᠤᠰᠠ ᠯᠠᠭᠰᠠ ᠨ ᠵᠣ ᠭᠣᠰ ( ᠨᠢᠭᠡ ᠴᠢ᠄ ᠲᠥᠭ ᠥᠷᠢᠭ ︶ ᠬᠦᠮᠦᠨ ᠦ ᠲᠣᠭ ᠠ ( ᠨᠢᠭᠡ ᠴᠢ᠄ ᠬᠦᠮᠦᠨ ) 20 2 30 4 50 3 90 1 A B C D EM N ᠤᠷᠳ᠋ ᠤᠰ ᠬᠤᠳ᠋ ᠠ ᠢᠢᠨ 20 18 ᠣᠨ ᠤ ᠠᠩᠬ ᠠ ᠲᠤᠮᠳ᠋ ᠠᠳ᠋ᠤ ᠢᠢᠨ ᠰᠤᠷᠤᠯᠭ ᠠ ᠲᠡᠭᠦᠰᠴᠤ ᠰᠤᠷᠭᠠᠭᠤᠯ ᠢ ᠲᠠᠪᠰᠢᠬᠤ ᠰᠢᠯᠭᠠᠯᠳ᠋ ᠠ ᠢᠢᠨ ᠰᠡᠳ᠋ ᠤᠪ A B C D O x y ᠰᠠᠢᠬᠠᠨ ᠴᠡᠭ A ᠶᠢ ᠳᠠᠭᠠᠷᠢᠵᠤ ᠂ 𝐶𝐷 ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠂ 𝐵𝐸 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠃ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠭᠡ ᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠪ ᠤᠷᠤᠭᠤ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. ∠ ABC = 60° B. 𝑆 ∆𝐴𝐵𝐸 = 2𝑆 ∆𝐴𝐷𝐸 C. ᠬᠡᠷᠪᠡ AB = 4 ᠪᠣᠯ BE = 4√ 7 D. sin ∠CBE = √2114 7. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ ᠂ 𝑦 1 ᠂ 𝑦2 ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠲᠤᠰᠤ ᠨ ᠬᠦᠳᠡᠯᠭᠡᠭᠦᠷᠳᠤ ᠮᠠᠰᠢᠨ ᠲᠡᠷᠭᠡ ᠪᠠ ᠴᠠᠬᠢᠯ ᠭᠠᠨ ᠬᠦᠳ ᠡᠯᠭᠡᠭᠦᠷᠳᠤ ᠮᠠᠰᠢ ᠨ ᠲᠡᠷ ᠭᠡ ᠶᠢᠨ ᠶᠠᠪᠤᠭ ᠰᠠᠨ ᠵᠠᠮ ᠤ ᠨ ᠤᠷ ᠳᠤ S (ᠨᠢᠭᠡᠴᠢ᠄ ᠺᠢᠯ ᠣᠮᠧᠲᠷ) ᠪᠣᠯᠣ ᠨ ᠵᠠᠷᠤᠭ ᠰᠠᠨ ᠵᠠᠷᠤᠳ ᠠᠯ y (ᠨᠢᠭᠡᠴᠢ᠄ ᠲᠥᠬᠥᠷ ᠢᠭ) ᠶᠢᠨ ᠬᠣᠭᠣᠷᠣᠨᠳᠣ ᠬᠢ ᠬᠣᠯᠪᠣᠭᠳᠠ ᠯ ᠢ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠨ ᠡ᠃ ᠴᠠᠬᠢᠯ ᠭᠠᠨ ᠬᠦᠳ ᠡᠯᠭᠡᠭᠦᠷᠳᠤ ᠮᠠᠰᠢ ᠨ ᠲᠡᠷᠭᠡᠨ ᠦ ᠺᠢᠯᠣ ᠮᠧᠲᠷ ᠪᠦᠷᠢ ᠳᠦ ᠪᠡᠨ ᠵᠠᠷᠤᠭ ᠰᠠᠨ ᠵᠠᠷᠤᠳ ᠠᠯ ᠨᠢ ᠲᠤᠰᠤ ᠨ ᠬᠦᠳᠡᠯᠭᠡ ᠭᠦᠷᠳᠤ ᠮᠠᠰᠢᠨ ᠲᠡᠷᠭᠡ ᠶᠢᠨ ᠺᠢᠯᠣ ᠮᠧᠲᠷ ᠪᠦᠷᠢ ᠳᠦ ᠪᠡᠨ ᠵᠠᠷᠤᠬ ᠤ ᠵᠠᠷᠤᠳᠠᠯ ᠢ 𝑥 ᠲᠥᠬᠥᠷᠢᠭ ᠭᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ ᠵᠥᠪ ᠪᠠᠢᠭᠤᠯᠤ ᠭᠰᠠᠨ ᠲᠡᠭ ᠰᠢᠳᠭᠡᠯ 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. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠳᠢᠶᠠᠮᠧᠲᠷ ᠨᠢ 2 ᠪᠣᠯᠬᠤ ᠲᠣᠭᠣ ᠷᠢᠭ ᠬᠡᠯᠪᠡᠷᠢᠳᠤ ᠲᠡᠮᠦ ᠷ ᠬᠠᠪᠳᠠᠰᠤ ᠠᠴᠠ ᠲᠥᠪ ᠥᠨᠴᠥᠭ ᠨᠢ 90° ᠪᠣᠯᠬᠤ ᠨᠢᠭᠡᠨ ᠲᠡᠪᠢ ᠬᠦᠷ ᠬᠡᠯᠪᠡᠷᠢᠳᠤ CA B ᠶᠢ ᠬᠠᠢᠴᠢᠯ ᠠᠨ ᠠᠪᠬᠤ ᠪᠥᠭᠡᠳ ᠴᠡᠭ C , A , B ᠨᠢ ᠴᠥᠮ ⨀O ᠳᠡᠭ ᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠨ ᠠ᠃ ᠬᠡᠷᠪᠡ ᠬᠠᠢᠴᠢᠯᠠᠵ ᠤ ᠠᠪᠤᠭ ᠰᠠᠨ ᠳ ᠡᠪᠢᠬᠦᠷ ᠬᠡᠯᠪᠡᠷᠢᠳ ᠤ ᠪᠠᠷ ᠨᠢᠭᠡᠨ ᠱᠣᠷ ᠣᠯᠵᠢᠨ ᠬᠦᠷᠢᠶᠡᠯᠡᠪᠡᠯ᠂ ᠲᠤᠰ ᠱᠣᠷ ᠣᠯᠵᠢᠨ ᠤ ᠰᠠ ᠭᠤᠷᠢ ᠲᠣ ᠭᠣᠷᠢ ᠭ ᠤᠨ ᠷᠠ ᠳᠢᠤᠰ ᠨᠢ ( ) A. 12 B. √ 2 C. √ 22 D. √ 24 9. ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠴᠡᠭ A( -2, 0)᠂ ᠴᠡᠭ B(0,1 )᠂ ᠬᠡᠷᠴᠢᠮ AB ᠪ ᠠᠷ ᠲ ᠠᠯ ᠠ ᠪᠠᠨ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳ ᠥ A BCD ᠶᠢ ᠬᠣᠶᠠᠳᠤ ᠭᠠᠷ ᠮᠥᠴᠡ ᠳ ᠦ ᠪᠠᠢ ᠭᠤᠯᠤ ᠨ ᠠ᠃ ᠬᠣᠣ ᠰ ᠮᠤᠷᠤ ᠢ ᠱᠤᠭᠤᠮ y= 𝑘𝑥 (𝑘 𝑏c ᠪᠣᠯ᠂ 𝑎 𝑏 ᠶᠢᠨ ᠤᠷ ᠪᠠᠭᠤ ᠦᠭᠦᠯᠡᠯ ᠨᠢ ᠵᠢᠩᠬ ᠢᠨᠢ ᠦᠭᠦᠯᠡᠯ ᠪᠣᠯᠤ ᠨ ᠠ᠃ ③ ᠬᠡᠷᠪᠡ 𝑀 (𝑎 ,2 ), 𝑁( 1, 𝑏) ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠦᠨ ᠬᠤᠪᠢ ᠳ ᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠴ ᠡᠭ ᠪᠣᠯ 𝑎+ 𝑏= −1 ᠃ ④ ᠨᠢᠭᠡᠨ ᠣᠯᠠᠨ ᠥᠨᠴᠥᠭᠳᠥ ᠶᠢ ᠨ ᠲ ᠠᠯ ᠠ ᠶᠢᠨ ᠪᠣᠳᠠᠭᠠᠯᠢ ᠲᠣᠭ ᠠ ᠨᠢ 1 ᠨᠡ ᠮᠡᠭᠳᠡᠬᠦ ᠪᠦᠷᠢ ᠲᠣᠳ ᠣᠭᠠᠳᠤ ᠥᠨᠴᠥᠭ ᠦ ᠨ ᠨᠠᠢᠯᠡᠪᠤ ᠷᠢ ᠨᠢ 180° ᠨᠡᠮ ᠡᠭᠳᠡᠵᠦ᠂ ᠭᠠ ᠳᠠᠭᠠ ᠳᠤ ᠥᠨ ᠴᠥᠭ ᠶᠢᠨ ᠨᠠᠢᠯᠡᠪᠤ ᠷᠢ ᠨᠢ ᠬᠣᠪᠢᠷ ᠠᠬᠤ ᠦᠭᠡᠢ᠃ ⑤ √11 ᠶᠢᠨ ᠪᠥᠬᠥᠯᠢ ᠲᠣᠭᠠᠨ ᠤ ᠭᠡᠰᠡᠭ ᠨᠢ 𝑎᠂ ᠪᠤᠳᠠᠷᠬᠠᠢ ᠲᠣᠭᠠᠨ ᠤ ᠭᠡᠰᠡᠭ ᠨᠢ b ᠪᠣᠯ 𝑎𝑏 = 3√ 11 − 3᠃ 14 . ᠬᠠ ᠪᠳᠠᠭᠠᠢ ᠶ ᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭ ᠳᠤ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ ᠧᠮ ᠳᠦ᠂ ᠴᠡᠭ P (𝑎 ,𝑏 ) ᠶᠢᠨ ᠬᠤᠪᠢ ᠳ ᠤ᠂ ᠪᠢᠳᠠ Q (− 𝑏+ 1, 𝑎+ 1) ᠶᠢ ᠴᠡᠭ P ᠶᠢ ᠨ ᠳᠠᠭᠠᠯᠳᠠ ᠴᠡᠭ ᠭᠡᠵᠤ ᠲᠣᠭ ᠲᠠᠭᠠᠪᠠ ᠃ 𝐴1 ᠶᠢᠨ ᠳᠠᠭᠠᠯᠲᠠ ᠴᠡᠭ ᠨᠢ 𝐴 2 ᠂ 𝐴 2 ᠳᠠᠭᠠᠯᠲᠠ ᠴᠡᠭ ᠨᠢ 𝐴3 ᠁ ᠡᠨᠡ ᠬᠦ ᠵᠢᠮ ᠡ ᠶᠤᠰᠤ ᠪ ᠠᠷ ᠣᠯᠬᠤ ᠨᠢ 𝐴1 , 𝐴2 , 𝐴 3 ᠁ 𝐴𝑛 ᠬᠡᠷᠪᠡ 𝐴1 ᠶᠢᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ( 3,1 ) ᠪᠣᠯ ᠴᠡᠭ 𝐴20 18 ᠶᠢᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ( )᠃ 15. ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠂ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨ ABC D ᠶᠢᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 4 ᠂ ᠤᠷ ᠳᠤ ᠨᠢ 4 ᠪᠣᠯᠬᠤ ᠬᠡᠷᠴᠢ ᠮ PQ ᠶᠢᠨ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠬᠦᠷ ᠦ ᠨ ᠴᠡᠭ ᠨᠢ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨABC D ᠶᠢᠨ ᠵᠡᠷᠭᠡᠯᠡᠳᠡᠭᠡ ᠬᠣᠶᠠᠷ ᠲᠠᠯ ᠠ ᠳᠡᠭᠡᠷ ᠡ ᠭᠣᠯᠭᠤᠨ ᠰᠢᠯᠵᠢᠬᠦ ᠪᠥᠭᠡᠳ ᠴᠡᠭ P ᠨᠢ A → B→ C→ Q ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠭᠤᠯᠭᠤᠨ ᠰᠢᠯᠵᠢᠵ ᠤ᠂ ᠴᠡᠭ D ᠬᠦᠷᠪᠡ ᠯ ᠭᠤᠯᠭᠤᠬ ᠤ ᠪᠠᠨ ᠵᠣᠭᠰᠣᠨ ᠠ᠃ ᠪᠥᠬᠥ ᠭᠤ ᠯᠭᠤᠨ ᠰᠢᠯᠵᠢᠬᠦ ᠶᠠᠪᠤᠴᠠ ᠳᠤ PQ ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ R ᠶᠢᠨ ᠶᠠᠪᠤᠵ ᠤ ᠦᠩ ᠭᠡᠷᠡᠭᠰᠡ ᠨ ᠵᠠᠮ ᠤ ᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( )᠃ 16. ᠵᠢᠷᠤᠭ ① ᠮᠡᠲᠤ᠂ AF, BE ᠪᠣᠯ ∆ABC ᠶᠢᠨ ᠳᠤᠮᠲ ᠠ ᠱᠤᠭᠤᠮ ᠂ AF ⊥ BE ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ P ᠬᠡᠷ ᠪᠡ BC = 𝑎 , AC = b , AB = c ᠭᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ 𝑎 2 + 𝑏 2 = 5𝑐 2 ᠪᠣᠯᠣ ᠨ ᠠ᠃ ᠳᠡᠭᠡᠷᠡ ᠬᠢ ᠰᠢᠨᠵᠢ ᠴᠢᠨᠠᠷ ᠢ ᠠᠰᠢᠭᠯ ᠠᠨ ᠪᠣᠳᠣ᠃ ᠵᠢᠷᠤ ᠭ ② ᠮᠡᠲᠤ᠂ ᠵᠢᠰᠢᠭᠦᠯᠵᠢᠨ ABCD ᠳᠦ᠂ E ᠂ F ᠂ G ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ AD , BC , CD ᠶᠢᠨ ᠳᠤᠮᠳ ᠠ ᠴᠡᠭ᠂ BE ⊥ GE ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ E , AD = 8 , AB = 2√ 5 ᠪᠣᠯ AF = ( )᠃ ᠭᠤᠷᠪᠠ ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬ ᠠᠷᠢᠭᠤᠯᠬ ᠤ ᠰᠡᠲ ᠤᠪ᠃ ( ᠨᠠᠢ ᠲᠡ 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. ( ᠲᠤ ᠰ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ) ① ᠤᠷ ᠢᠳᠠᠪᠠᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢᠯᠠᠵᠤ᠂ ᠳᠠᠷ ᠠᠭ ᠠ ᠨᠢ ᠬᠡᠮ ᠵᠢᠭᠳᠡᠯ ᠢ ᠨᠢ ᠣᠯ᠃ 2𝑥𝑥+ 1 + 4− 2𝑥 𝑥 2 −1 ÷ 𝑥− 2 𝑥 2 −2 𝑥+ 1 ᠂ ᠡᠭᠦᠨ ᠳᠦ 𝑥 = −2 2+ 2sin 45° + |− 3|᠃ ② ᠲᠡᠩᠴᠡᠳᠭᠡᠯ ᠪᠤᠰᠤ ᠶᠢᠨ ᠪᠥᠯᠥᠭ { 𝑥 − 3( 𝑥− 2) ≤ 8 ① 2𝑥 −15 𝑥+ 12 − 1② ᠶᠢ ᠪᠣᠳᠣᠭᠠᠳ ᠲᠡᠭᠦᠨ ᠦ ᠰᠥᠭᠡᠷᠬᠦ ᠪᠤᠰᠤ ᠪᠥᠬᠥᠯᠢ ᠲᠣᠭ ᠠᠨ ᠤ ᠬᠠ ᠷᠢᠭᠤ ᠶᠢ ᠣᠯ᠃ 18. (ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ︽ᠠᠯᠳᠠ ᠮᠥᠩᠬᠦᠨ ᠠᠭᠤᠯᠠ ᠪᠣᠯᠪᠠᠴᠤ ᠬᠥᠭᠡᠮ ᠳᠦᠭ ᠨᠣᠭᠣᠭᠠᠨ ᠠᠭᠤᠯᠠ ᠤᠰᠤ ᠶᠢ ᠡᠰᠡ ᠬᠦᠢᠴᠡᠮᠤᠢ ︾ ᠭᠡᠳᠡᠭᠴᠢᠯᠡᠨ ᠣᠷᠳ ᠤᠰ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡᠨ ᠬᠤᠰᠢᠭᠤᠨ ᠨᠣᠭᠣᠭᠠᠷᠠ ᠭᠤᠯᠤ ᠯᠲᠠ ᠶᠢᠨ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠪᠥᠳᠥᠭᠡᠨ ᠪᠠᠢᠭ ᠤᠯᠤᠯ ᠲᠠ ᠶᠢ ᠲᠠᠰᠤᠷᠠᠯᠲᠠ ᠦᠭᠡᠢ ᠲᠦ᠋ᠯᠬ ᠢᠨ ᠠᠬᠢᠭᠤ ᠯᠵᠤ ᠪᠣᠢ᠃ ᠡᠨᠡ ᠬᠠᠪᠤᠷ ᠲᠥᠷᠪᠡᠨ ᠲᠥᠷᠥ ᠯ ᠦᠨ ᠮᠤᠳᠤ ᠨ ᠰᠣᠶᠣᠭ ᠠ ᠶᠢ ᠲᠠᠷᠢᠵᠠᠢ᠃ ᠣᠢᠵᠢᠭᠤᠯᠤ ᠯᠲᠠ ᠶᠢᠨ ᠰᠠᠯᠠ ᠭ ᠠ ᠮᠥᠴᠢᠷ ᠦᠨ ᠬᠢᠨ ᠡᠳᠡᠭᠡᠷ ᠲᠠᠷᠢᠭᠰᠠᠨ ᠮᠤᠳᠤ ᠨ ᠰᠣᠶᠣ ᠭ ᠠ ᠠᠴᠠ 4000 ᠰᠠᠭ ᠤᠷᠢ ᠶᠢ ᠲᠠᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠴᠤ᠂ ᠡᠯ ᠡ ᠲ ᠥᠷᠥᠯ ᠦᠨ ᠲᠠᠷᠢᠭᠰᠠᠨ ᠮᠤᠳᠤ ᠨ ᠰᠣᠶᠣ ᠭ ᠠ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠰᠠᠪᠠ ᠭᠠᠨ ᠪᠦᠷᠢᠳᠭ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠢᠶᠠᠷ ᠲᠦ᠋ᠷᠰᠤ ᠯᠡᠵᠤ᠂ ᠪᠠᠰᠠ ᠡᠯ ᠡ ᠲᠥᠷᠥ ᠯ ᠦᠨ ᠮᠤᠳᠤᠨ ᠰᠣᠶᠣ ᠭ ᠠ ᠶᠢᠨ ᠤᠨᠳᠣ ᠷᠣᠭᠰᠠᠨ ᠰᠠᠭᠤᠷ ᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠰᠠᠪᠠᠭᠠᠨ ᠪᠦᠷᠢᠳᠭ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠢᠶᠠᠷ ᠲᠦ᠋ᠷᠰᠤ ᠯᠡᠪᠡ᠃ ᠪᠦᠷᠢᠳᠭ ᠡᠯ ᠢᠶᠡᠷ ᠲᠠᠮᠵᠢᠨ ᠨᠠᠷᠠᠰᠤ ᠨ ᠮᠤᠳᠤ ᠪᠠ ᠤᠯᠢ ᠶᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠳᠣ ᠷᠣᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠨᠡᠯᠢᠶᠡᠳ ᠦᠨᠳᠤᠷ ᠪᠥᠭᠡᠳ ᠤᠯᠢᠶᠠᠰᠤ ᠶᠢᠨ ᠣᠨᠳᠣ ᠷᠣᠯᠲᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ 97 % ᠪᠣᠯᠬᠤ ᠶᠢ ᠮᠡᠳᠡᠪᠠᠯ᠃ ᠵᠢᠷᠤ ᠭ ᠲ ᠤ ᠦᠭᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠮᠡᠳᠡ ᠭᠡ ᠵᠠᠩᠭᠢ ᠶᠢ ᠦᠨᠳᠤᠰᠤ ᠡᠯᠵᠤ᠂ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠠᠰᠠ ᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠ ᠷᠢᠭᠤ ᠯ᠃ (1) ᠲᠣᠭᠣ ᠷᠢᠭ ᠪᠦᠷᠢᠳᠭ ᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠲᠠᠬᠢ ᠨᠠᠷᠠᠰ ᠤᠨ ᠮᠤᠳᠤ ᠲᠠᠢ ᠬᠠᠷᠠ ᠭᠠᠯᠵᠠᠭᠰᠠᠨ ᠭᠡᠰᠡᠭ ᠦᠨ ᠲᠥᠪ ᠥᠨᠴᠥᠭ ᠨᠢ ( ) ᠭᠷᠠᠳᠦ᠋ ᠰ᠃ ᠪ ᠠᠰᠠ ᠰᠠᠪᠠ ᠭᠠᠨ ᠪᠦᠷᠢ ᠳᠭᠡᠯ ᠦᠨ ᠵᠢᠷᠤ ᠭ ᠢ ᠪᠦᠷ ᠢᠨ ᠪ ᠣᠯᠭ ᠠ᠃ (2) ᠲᠤᠰ ᠬᠤᠰᠢ ᠭᠤᠨ ᠡᠨᠡ ᠵᠢᠯ ᠨᠠᠢᠲᠡ 32 ᠲᠥᠮᠡᠨ ᠰᠠᠭᠤᠷ ᠢ ᠮᠤᠳᠤ ᠲᠠᠷᠢᠭᠰᠠᠨ ᠪᠣᠯ ᠪᠠᠷᠤ ᠭ ᠬᠡᠳᠤᠢ ᠰᠠᠭᠤᠷᠢ ᠮᠤᠳᠤ ᠣᠨᠳᠣᠷ ᠣᠭᠰᠠᠨ ᠪᠣᠢ ︖ (3) ᠣᠢᠵᠢᠭᠤᠯᠤᠯᠲ ᠠ ᠶᠢᠨ ᠰᠠᠯᠠ ᠭ ᠠ ᠮᠥᠴᠢᠷ ᠡᠴᠡ ᠢᠷᠡᠬ ᠦ ᠵᠢᠯ ᠳ ᠦ ᠡᠨᠡ ᠲᠥᠷᠪᠡᠨ ᠲᠥᠷᠥ ᠯ ᠦᠨ ᠮᠤᠳᠤ ᠨ ᠰᠣᠶᠣ ᠭ ᠠ ᠠᠴᠠ ᠬᠣᠶᠠᠷ ᠲᠥᠷᠥ ᠯ ᠢ ᠰᠤᠩ ᠭᠤᠨ ᠲᠠᠷᠢᠬᠤ ᠪ ᠠᠷ ᠲᠣᠭ ᠲᠠᠭᠠᠪ ᠠ᠃ ᠮᠤᠳᠤ ᠯᠵᠢᠨ ᠵᠢᠷᠤ ᠭ ( ᠪᠤᠶᠤ ᠬᠥᠰᠥᠨᠥᠭᠵᠢ ᠬᠦᠯᠬᠦ ᠠᠷᠭ ᠠ) ᠪ ᠠᠷ ᠣᠨᠳᠣ ᠷᠣᠯᠲ ᠠ ᠶᠢᠨ ᠨᠣᠷᠮ ᠠ ᠨᠢ ᠨᠠᠯᠢᠶᠡᠳ ᠦᠨᠳᠤᠷ ᠪᠣᠯᠬᠤ ᠬᠣᠶᠠᠷ ᠲᠥᠷᠥ ᠯ ᠦᠨ ᠮᠤᠳᠤ ᠨ ᠰᠣᠶᠣ ᠭ ᠠ ᠶᠢ ᠶᠠᠭ ᠰᠠᠢᠬᠠ ᠨ ᠰᠤ ᠩᠭᠤ ᠵᠤ ᠠᠪᠬᠤ ᠶᠢᠨ ᠲᠣ ᠬᠢᠶᠠᠯᠳᠤ ᠮᠵᠢ ᠶᠢ ᠣᠯ᠃ (ᠨᠠᠷᠠᠰ ᠤᠨ ᠮᠤᠳᠤ᠂ ᠤᠯᠢ ᠶᠠᠰᠤ᠂ ᠬᠠᠢᠯᠠᠰᠤ᠂ ᠤᠳᠠ ᠮᠤᠳᠤ ᠶᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑎, 𝑏, 𝑐,𝑑 ᠢᠶᠠᠷ ᠢᠯᠡᠷᠭᠡᠢᠯᠡ ) 19. (ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢ ᠲᠡ 8 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ∆ABC ᠳᠦ ∠B AC = 45° ᠂ AD ⊥BC ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤᠷ ᠢ ᠨᠢ D , BD = 6 , DC = 4 , AD ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣ ᠯ᠃ ᠰᠢᠶᠥᠥ ᠮᠢᠩᠭ ᠠ ᠨᠤᠭᠤᠯ ᠬᠤ ᠠᠷᠭ ᠠ ᠶᠢ ᠠᠰᠢᠭᠯ ᠠᠨ ᠤᠷ ᠠᠨ ᠠᠷ ᠭ ᠠ ᠪᠠᠷ ᠲᠤᠰ ᠰᠡᠲᠤᠪ ᠢ ᠬᠠ ᠷᠢᠭᠤ ᠯᠵᠠᠢ ᠃ ᠰᠢᠶᠥᠥ ᠮᠢᠩᠭ ᠠ ᠶᠢᠨ ᠰᠡᠳᠬᠢᠨ ᠰᠥᠪᠡᠭ ᠴᠢᠯᠡᠭᠰᠡᠨ ᠶᠠᠪᠤᠴ ᠠ ᠶᠢ ᠦᠨᠳᠤᠰᠤ ᠯᠡᠵᠤ᠂ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠠᠰᠠᠭᠤᠯᠲ ᠠ ᠶᠢ ᠬᠠᠷᠢᠭᠤ ᠯ᠃ ⑴ AB , AC ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠪᠣᠯᠭᠠᠨ ∆ABD ᠪᠠ ∆A CD ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠵᠢᠷᠤ ᠭ ᠲᠥᠷᠰᠤ ᠶᠢ ᠵᠢᠷᠤ ᠪᠠᠯ᠂ ᠴᠡᠭ D ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠴᠡᠭ ᠨᠢ ᠲ ᠤᠰ ᠲᠤᠰ E , F ᠪᠣᠯᠣ ᠨ ᠠ᠃ EB ᠪᠠ FC ᠶᠢ ᠦ ᠷᠬᠦᠯᠵᠢᠯ ᠡᠬᠦᠯᠪᠡᠯ ᠴᠡᠭ G ᠳᠤ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠪᠠᠲ ᠤᠯᠠ᠄ ᠲᠥ ᠷᠪᠡᠯᠵᠢᠨ AEFG ᠨᠢ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨ᠃ ⑵ AD = 𝑥 ᠭᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ 𝑥 ᠦᠨ ᠲᠤᠬ ᠠᠢ ᠲᠡᠭᠰᠢᠳᠭᠡᠯ ᠪᠠᠢᠭᠤᠯᠵᠤ᠂ AD ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢ ᠣ ᠯ᠃ 20. (ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢ ᠲᠡ8 ᠬᠤᠪᠢ ) ᠸᠠᠩ ᠠᠨᠢᠶ ᠠ ᠶᠢᠨ ᠭᠡᠷ ᠦ ᠨ ᠨᠠᠷᠠᠪᠴᠢᠨ ᠳᠤ ᠨᠢᠭᠡᠨ ᠬᠤᠪᠴᠠᠰ ᠤ ᠬᠠᠲ ᠠᠭᠠᠬ ᠤ ᠵᠢᠶᠠᠰᠠ ᠶᠢ ᠲᠣᠭᠳᠠᠪᠤᠷᠢᠵᠢ ᠭᠤᠯᠤ ᠨ ᠲᠡᠯᠭᠡᠵᠤ ᠲᠠᠯᠪᠢᠭᠰᠠᠨ ᠨᠢ ᠵᠢᠷᠤ ᠭ ①ᠮᠡᠲᠤ᠃ ᠵᠢᠷᠤ ᠭ ②᠂ ③ ᠪᠣᠯ ᠬᠤᠪᠴᠠᠰᠤ ᠬᠠᠳ ᠠᠭᠠᠬ ᠤ ᠵᠢᠶᠠᠰᠠ ᠶᠢ ᠳᠡᠯᠭᠡᠭᠰᠡᠨ ᠦ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠬᠠᠵᠠᠭᠤ ᠲᠠᠯ ᠠ ᠶᠢ ᠨ ᠣᠭᠳ ᠣᠯᠣᠭᠠᠰᠤ ᠵᠢᠷᠤ ᠭ ᠮᠥᠨ᠃ ∆A OB ᠪᠣᠯ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 13 0cm ᠪᠣᠯ ᠬᠤ ᠲᠡᠩᠴᠡᠬ ᠦᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ᠃ ᠬᠤᠪᠴᠠ ᠰᠤ ᠬᠠᠳᠠᠭᠠᠬᠤ ᠵᠢᠶᠠᠰᠠᠨ ᠤ ᠬᠦᠨ ᠳᠡᠯᠡᠨ ᠰᠠᠪᠠᠭ ᠠ OE , OF ᠨᠢ O ᠪᠠᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯᠭᠠᠨ ᠡᠷᠬ ᠢᠯᠳᠤ ᠵᠤ ᠪᠣᠯᠣ ᠨ ᠠ᠂ ᠪᠠᠰᠠ OE = OF = 130cm ᠂ OA ᠂ OB ᠳᠡᠭᠡᠷᠡᠬᠢ C ᠂ D ᠴᠡᠭ ᠲᠦ ᠲᠤᠰ ᠲᠤᠰ CN ᠂ DM ᠲᠤᠯ ᠬᠤ ᠰᠠᠪᠠᠭ ᠠ ᠪᠠᠢᠵ ᠤ᠂ ᠲ ᠡᠷᠡ ᠨᠢ ᠴᠡᠭ C᠂ D ᠢᠶᠡᠷ ᠲᠣᠭᠣ ᠷᠢᠭ ᠤ ᠨ ᠲᠥᠪ ᠪ ᠣᠯᠭᠠᠨ ᠡᠷ ᠬᠢᠯᠳᠥᠵ ᠤ ᠪᠣᠯᠣᠨ ᠠ᠃ ⑴ ᠵᠢᠷᠤ ᠭ ② ᠮᠡᠳᠤ ᠂ ᠬᠡᠷᠪᠡ EF ᠨᠢ ᠭᠠᠵᠠ ᠷ ᠤᠨ ᠭᠠᠳᠠᠷᠭ ᠤ ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ᠂ ᠸᠠᠩ ᠠᠨᠢᠶ ᠠ ᠶᠢᠨ ᠬᠤᠪᠴ ᠠᠰᠤ ᠶᠢ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠡᠯᠬᠦᠬᠦᠷ ᠲᠦ ᠡᠯᠭᠦᠭ ᠰᠡᠨ ᠦ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠶᠡᠷᠤᠩ ᠬᠡᠢ ᠤᠷ ᠳᠤ ᠨᠢ 11 0cm ᠪᠣᠯ ᠲᠡᠭᠦᠨ ᠢ ᠬᠤᠪᠴᠠᠰᠤ ᠬᠠᠳ ᠠᠭᠠᠬᠤ ᠵᠢᠶᠠᠰᠠᠨ ᠤ ᠬᠦᠨᠳᠡ ᠯᠡᠨ ᠰᠠᠪᠠᠭ ᠠ OE ᠳᠡᠭᠡᠷ ᠡ ᠡᠯᠭᠦᠪᠡᠯ ᠭᠠᠵᠠᠷ ᠲ ᠤ ᠴᠢᠷᠥᠭᠳᠡᠨ ᠡ ᠤᠤ ︖ ᠰᠢᠯᠳᠠᠭᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢ ᠯᠠ᠃ ( 0.1 cm ᠬᠦᠷᠲᠡ ᠯ ᠡ ᠨᠠ ᠷᠢᠪᠴᠢᠯᠠ᠃ ) ⑵ ᠵᠢᠷᠤ ᠭ ③ ᠮᠡᠳᠤ ᠂ ᠲᠤᠯ ᠬᠤ ᠰᠠᠪᠠᠭ ᠠ DM ᠨᠢ ᠴᠡᠭ 𝑀 ′ ᠳᠡᠭᠡᠷ ᠡ ᠲᠤᠯ ᠬᠤ ᠦᠶᠡᠰ ∠EOB = 78° ᠴᠡᠭ E ᠨᠢ ᠭᠠᠵᠠᠷ ᠤ ᠨ ᠭᠠᠳᠠᠷᠭ ᠤ ᠠᠴᠠ ᠬᠠᠮᠤᠭ ᠦᠨᠳᠤᠷ ᠵᠠᠢᠳᠠᠢ ᠃ ᠬᠡᠷᠪᠡ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠡᠯᠭᠦᠭᠦᠷ ᠲᠦ ᠡᠯᠭᠦᠭᠰᠡᠨ ᠬᠤᠪᠴᠠᠰᠤ ᠶᠢ ᠭᠠᠵᠠᠷ ᠲᠤ ᠴᠢᠷᠤ ᠭᠳᠡᠬᠦᠯᠬᠦ ᠦᠭᠢ ᠡ ᠭᠡᠪᠡᠯ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠡᠯᠭᠦᠭᠦᠷ ᠪᠠ ᠬᠤᠪᠴᠠᠰᠤᠨ ᠤ ᠶᠡᠷᠤᠩ ᠬᠡᠢ ᠤᠷᠳᠤ ᠶᠢ ᠬᠠᠮᠤᠭᠤᠯ ᠤᠷᠳᠤ ᠳ ᠤ ᠪᠠ ᠨ ᠪᠠᠷᠤ ᠭ ᠬᠡᠳᠤᠢ ᠰᠡᠨᠲ ᠢᠮᠧᠲᠷ ᠪᠠᠢᠯᠭᠠᠪᠠᠯ ᠵᠣᠬᠢᠬᠤ ᠪᠣᠢ︖ ( ᠳᠦ᠋ᠩ ᠳᠦ ᠨᠢ ᠪᠥᠬᠥ ᠯᠢ ᠲᠣᠭ ᠠ ᠦᠯᠡᠳᠡᠭᠡ ) ( ᠯᠠᠪ ᠯᠠᠬᠤ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢ ᠮᠳᠠ᠄ √3 ≈ 1710 , sin 78° ≈ 4950 , cos 78° ≈ 15 , sin 18° ≈ 310 , cos 18° ≈ 1920 ) E F G B CD A 21. ( ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠨᠠᠢᠲᠡ 8 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ⨀o ᠪᠣᠯ ∆ABC ᠢ ᠪᠠᠭᠳ ᠠᠭᠠᠭᠰᠠ ᠨ ᠲᠣ ᠭᠣᠷᠢ ᠭ᠂ AC ᠨᠢ ᠳᠢᠶᠠᠮᠧᠲᠷ᠂ ᠬᠥᠪᠴ ᠢ BD = BA ᠂ BE ⊥ DC ᠂ ᠪᠥᠭᠡᠳ DC ᠶᠢᠨ ᠦᠷ ᠬᠦᠯᠵᠢᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲᠠᠢ ᠴᠡᠭ E ᠲᠦ ᠣᠭ ᠳᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠃ ⑴ ᠪᠠᠳᠤ ᠯᠠ᠄ BE ᠪᠣᠯ ᠶᠢᠨ ☉O ᠰᠢᠬᠠᠭ ᠴᠢ ᠰᠢᠯᠤᠭᠤ ᠨ᠃ ⑵ ᠬᠡᠷᠪᠡ sin ∠𝐵𝐶𝐸 = 34 , AB = 3 ᠪᠣᠯ ᠂ AD ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠶᠢ ᠣ ᠯ᠃ 22. ︵ ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡᠲ ᠤᠪ ᠨᠠᠢᠲ ᠡ9 ᠬᠤᠪᠢ ︶ ᠮᠠᠯᠴ ᠢᠨ ᠪᠠᠭᠠᠳᠤᠷ ᠨᠢᠭᠡᠨ ᠲᠥᠷᠥ ᠯ ᠦᠨ ᠴᠠᠭᠠᠨ ᠢᠳᠡ ᠭᠡᠨ ᠢ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠵᠤ᠂ ᠰᠦ᠋ᠯᠵᠢᠶ ᠡ ᠶᠢ ᠠᠰᠢᠭᠯ ᠠᠨ ᠪᠣᠷᠣ ᠯᠠᠭᠤ ᠯᠬᠤ ᠪ ᠠᠷ ᠪᠣᠯᠪ ᠠ᠃ ᠬᠤᠳᠠᠯᠳᠤ ᠨ ᠠᠪᠤᠭ ᠰᠠᠨ ᠤᠷ ᠢᠳᠠᠪᠠᠷ ᠰᠦ᠋ᠯᠵᠢᠶ ᠡᠨ ᠳᠡᠭᠡ ᠷ ᠡ ᠵᠠᠬᠢᠶ ᠠᠯᠠᠵᠤ᠂ ᠳᠠᠷᠠᠭ ᠠ ᠨᠢ ᠪᠠᠭᠠᠳᠤ ᠷ᠂ ᠠ ᠪᠤᠶ ᠤ ᠡ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪᠢᠭᠤᠯᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠ ᠨᠢ ᠶᠢ ᠰᠤᠩ ᠭᠤᠵᠤ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢᠳᠭᠡᠯ ᠢᠶᠡᠨ ᠲᠤᠰᠢᠶ ᠠᠨ ᠪᠠᠷᠠᠭ ᠠ ᠪᠠᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠤᠨ ᠬᠦᠷᠭᠡᠨ ᠡ᠃ ᠠ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠪᠠᠷ 2 ᠺᠢᠯᠣᠭ ᠷᠠᠮ ᠢ ᠭᠣᠪᠢᠭᠤᠯᠵᠤ᠂ ᠡ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠪᠠᠷ 3 ᠺᠢᠯᠥᠭᠷ ᠠᠮ ᠢ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠳ ᠤ ᠨᠠᠢ ᠲᠡ 42 ᠲᠥᠬᠥᠷᠢᠭ ᠦᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢᠳᠭᠡᠯ ᠬᠡᠷᠡᠭᠰᠡ ᠨ ᠡ᠃ ᠠ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨᠢ ᠪᠠᠷ 5 ᠺᠢ ᠯᠣᠭᠷᠠᠮ ᠢ ᠭᠣᠪᠢᠭᠤᠯ ᠵᠤ᠂ ᠡ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣ ᠪᠢᠭᠤᠯᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠪᠠᠷ 4 ᠺᠢᠯᠥᠭᠷ ᠠᠮ ᠢ ᠭᠣᠪᠢᠭ ᠤᠯᠬᠤ ᠳ ᠤ ᠨᠠ ᠢᠲᠡ 70 ᠲᠥᠬᠥᠷᠢᠭ ᠦᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢ ᠳᠭᠡᠯ ᠬᠡᠷᠡᠭᠰᠡᠨ ᠡ᠃ (1) ᠠ᠂ ᠡ ᠬᠣᠶᠠᠷ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠶᠢᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢᠳᠭᠡᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠺᠢᠯᠣ ᠭᠷᠠᠮ ᠪ ᠦᠷᠢ ᠳ ᠠᠬᠡᠨ ᠬᠡᠳᠤᠢ ᠲᠥ ᠬᠥᠷᠢᠭ ᠪᠣᠢ ︖ (2) ᠪᠠᠭᠠᠳᠤᠷ ᠨᠢᠭᠡ ᠡᠳᠤ ᠷ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠭᠰᠨ ᠡ ᠴ ᠠᠭᠠᠨ ᠢᠳᠡᠭᠡᠨ ᠢᠶᠡᠨ ᠲᠤᠰ ᠡᠳᠤ ᠷ ᠲᠤᠨᠢ ᠪᠥᠬᠥᠢᠨ ᠢ ᠪᠣᠷᠣ ᠯᠠᠭᠤ ᠯᠵᠤ ᠪᠣᠯᠬᠤ ᠪᠥᠭᠡ ᠳ ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢᠳᠭᠡᠯ ᠪᠠᠭ ᠠ ᠪᠣᠯᠬᠤ ᠲᠥᠷᠭ ᠡᠨ ᠭᠣᠪ ᠢᠭᠤᠯᠭ ᠠ ᠶᠢᠨ ᠺᠣᠮᠫᠠᠨ ᠢ ᠶᠢ ᠣᠩ ᠭᠤᠯᠵ ᠤ ᠪᠠᠷᠠᠭ ᠠ ᠪᠠᠨ ᠭᠣᠪᠢᠭᠤᠯ ᠨ ᠬᠦᠷᠭᠡᠨ ᠡ ᠭᠡᠵᠤ ᠵᠢᠱᠢᠪᠡᠯ᠂ ᠲᠤᠰ ᠴᠠᠭᠠᠨ ᠢᠳᠡᠭᠡᠨ ᠢ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠬᠦ ᠬᠡᠮᠵᠢᠶ ᠡ ᠺᠢ ᠯᠥᠭᠷ ᠠᠮ ᠳ ᠤ ᠬᠡᠷᠡᠭᠯᠡᠬ ᠦ ᠥ ᠪᠡᠷ ᠦᠨ ᠥᠷᠳ ᠡᠭ ᠨᠢ ᠺᠢ ᠯᠥᠭᠷ ᠠᠮ ᠪᠦᠷᠢ ᠳᠡᠬᠡᠨ 𝑦1 ᠲᠥᠬᠥᠷᠢᠭ ︵ᠭᠣᠪᠢᠭᠤᠯ ᠬᠤ ᠰᠦ᠋ᠢᠳᠭᠡᠯ ᠢ ᠪᠠ ᠭᠲᠠᠭ ᠠᠬᠤ ᠦᠭᠡᠢ ︶ ᠪᠣᠯᠣ ᠨ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠬᠦ ᠬᠡᠮᠵᠢ ᠶ ᠡ 𝑥 ᠺᠢᠯᠥᠭᠷ ᠠᠮ ᠤᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣ ᠬᠢ ᠹᠦᠩᠺ ᠼ ᠦᠨ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠢᠯᠡᠷᠭᠡᠢᠯᠡᠯ ᠨᠢ 𝑦1 = { − 2𝑥 + 58 (0 𝑥 8) 42 (𝑥 ≥ 8) ᠂ ᠪᠣᠷᠣ ᠯᠠᠭᠤᠯᠬᠤ ᠥᠨ ᠡ ᠨᠢ ᠺᠢᠯᠥᠭᠷ ᠠᠮ ᠪᠦᠷᠢ ᠳᠡᠬᠡ ᠨ𝑦 2 ᠲᠥᠬᠥᠷᠢᠭ ᠪᠣᠯᠣ ᠨ ᠦᠢᠯᠡᠳᠪᠤᠷ ᠢᠯᠡᠬᠦ ᠬᠡᠮᠵᠢ ᠶ ᠡ 𝑥 ᠺᠢ ᠯᠥᠭᠷ ᠠᠮ ᠤᠨ ᠬᠣᠭ ᠣᠷᠣᠨᠳᠤ ᠬᠢ ᠹᠦᠩᠺ ᠼ ᠦᠨ ᠬᠣᠯᠪᠣᠭᠳᠠᠯ ᠤ ᠨ 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 ᠢᠯᠡᠷᠬᠡᠢᠯᠡᠯ ᠨᠢ 𝑦 2 = −6 𝑥+ 120 (0 𝑥 13 ) ᠪᠣᠯ᠂ ᠪᠠᠭᠠᠳᠤᠷ ᠡᠳᠤ ᠷ ᠪᠦᠷᠢ ᠬᠡᠳᠤᠢ ᠺᠢᠯᠣ ᠭᠷᠠᠮ ᠴᠠᠭᠠᠨ ᠢᠳᠡᠭᠡᠨ ᠢ ᠦᠢᠯ ᠡᠳᠪᠤᠷᠢ ᠯᠡᠬᠦ ᠦᠶᠡᠰ ᠣᠯᠬᠤ ᠠ ᠰᠢᠭ ᠬᠣᠨᠵᠢᠪᠤᠷ ᠢ ᠨᠢ ᠬᠠ ᠮᠤᠭ ᠤ ᠨ ᠶᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠪᠣᠢ︖ ᠬᠠᠮᠤᠭ ᠤᠨ ᠶᠡᠭᠡ ᠠᠰᠢᠭ ᠬᠣᠨᠵᠢᠪᠤᠷ ᠢ ᠨᠢ ᠬᠡᠳᠤᠢ ᠲᠥᠬᠥᠷᠢᠭ ᠪᠣᠢ︖ 23. ( ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡ ᠲᠤᠪ ᠨᠠᠢᠲᠡ 11 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤ ᠭ ᠮᠡᠲᠤ᠂ ᠰᠢᠯᠤ ᠭᠤᠨ y = 12 𝑥 − 3 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ᠂ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ B , C ᠬᠣᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠳ ᠣᠯᠣᠯ ᠴᠠᠨ ᠠ᠃ ᠬᠠᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠨᠢ B , C ᠬᠣᠶᠠᠷ ᠴᠡᠭ ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠪᠥᠭᠡᠳ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠣᠭᠳᠣ ᠯᠣᠯᠴᠠᠬᠤ ᠨ ᠥᠬᠥᠭᠡ ᠨᠢᠭᠡ ᠣᠭ ᠳᠣᠯᠣᠯ ᠴᠠᠯ ᠤᠨ ᠴᠡᠭ ᠨᠢ A ᠃ ⑴ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠤᠨ ᠠᠨᠠᠯᠢᠰ ᠢᠯ ᠡᠷᠭᠡᠢᠯᠡᠯ ᠢ ᠣᠯ᠃ ⑵ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡ 𝑆 ∆𝐷𝐵𝐶 = 𝑆∆ 𝐴𝐵𝐶 ᠪᠣᠯᠭᠠᠬᠤ ᠴᠡᠭ D ( ᠴᠡᠭ A ᠲᠠᠢ ᠲᠠᠪᠬᠤ ᠴᠠᠬᠤ ᠦᠭᠡᠢ ) ᠣᠷᠣ ᠰᠢᠨ ᠠ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ ᠴᠡᠭ D ᠶᠢᠨ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ ᠬᠡᠷᠪᠡ ᠣ ᠷᠣᠰᠢᠬᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠳᠠ ᠭᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠣ ᠷᠬᠠᠢᠯᠠ᠃ ⑶ ᠥᠷ ᠭᠡᠨ ᠨᠢ 2 ᠂ ᠤᠷ ᠳᠤ ᠨᠢ ᠬ ᠦᠷᠢᠶ ᠡᠨ ᠤᠷᠳᠤ ᠪᠣᠯᠬᠤ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭᠳᠥ ( ᠰᠡᠭ ᠦᠳᠡᠷ ᠲᠠᠢ ᠭᠡᠰᠡᠭ ) ᠶᠢ 𝑥 ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠦᠨ ᠴᠢᠭᠯᠡᠯ ᠲᠠᠭᠠᠤ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵᠢᠬᠦᠯᠪᠡ ᠯ᠂ y ᠲᠡᠩᠭᠡᠯᠢ ᠭ ᠲᠠᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠪᠣᠯᠬᠤ ᠨᠢᠭᠡ ᠬᠣᠣ ᠰ ᠡᠰᠡᠷᠬᠦ ᠲᠠᠯ ᠠ ᠨᠢ ᠬᠠ ᠶᠠᠯᠲ ᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠲᠠᠢ ᠴᠡᠭ P ᠪᠠ Q ᠳ ᠤ ᠣ ᠭᠳᠣ ᠯᠣᠯᠴᠠᠵᠤ᠂ ᠰᠢᠯᠤᠭᠤ ᠨ CB ᠲᠠᠢ ᠴᠡᠭ M ᠪᠠ N ᠳᠤ ᠣᠭᠳ ᠣᠯᠣᠯᠴᠠᠨ ᠠ᠃ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳ ᠥ ᠵᠡᠷᠭᠡ ᠴᠡᠭᠡ ᠰᠢᠯᠵᠢᠬᠦ ᠶᠠᠪᠤᠴᠠ ᠳᠤ ᠴᠡᠭ P᠂ Q ᠂ M ᠂ N ᠢᠶᠡᠷ ᠣᠷᠣ ᠢ ᠶᠢᠨ ᠴᠡᠭ ᠢᠶᠡᠨ ᠪᠣᠯᠭ ᠠᠭᠰᠠᠨ ᠲᠥᠷᠪᠡᠯᠵ ᠢᠨ ᠨᠢ ᠵᠢᠰᠢᠬᠦᠯᠵᠢᠨ ᠪᠣᠯᠬᠤ ᠦᠶᠡᠰ ᠦᠨ ᠴᠡᠭ M ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ x y OA B C M N P QC BA O y x 24. ( ᠲᠤᠰ ᠪᠠᠭ ᠠ ᠰᠡ ᠲᠤᠪ ᠨᠠᠢ ᠲᠡ12 ᠬᠤᠪᠢ ) ⑴ 【ᠣᠷᠣ ᠯᠳᠣᠯ ᠭ ᠠ ᠠᠴᠠ ᠠ ᠵᠢᠭᠯᠠᠵᠤ ᠣᠯ ᠬᠤ 】 ᠵᠢᠷᠤ ᠭ ① ᠮᠡᠲᠤ᠂ ∆ABC ᠶᠢ ᠴᠡᠭ A ᠶᠢ ᠲᠣᠭᠣ ᠷᠢᠭᠤ ᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠬᠦᠦ ᠶᠢᠨ ᠳᠠᠭᠠ ᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 60° ᠡᠷᠬᠢᠭ ᠦᠯᠪᠡᠯ ∆ADE ᠶᠢ ᠣᠯᠣ ᠨ ᠠ᠂ BD ᠶᠢ ᠬᠣᠯᠪᠣ ᠪᠠᠯ ∠ABD = ( ) ᠭᠷᠠᠳᠦ᠋ ᠰ᠃ ⑵ 【 ᠠᠳᠠᠯᠢᠳ ᠬᠠᠨ ᠰᠦ᠋ᠪᠡᠭᠴᠢᠯ ᠡᠬᠦ 】 ᠵᠢᠷᠤ ᠭ ② ᠮᠡᠲᠤ᠂ ᠲᠡᠩᠴᠡᠬᠦᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ABC ᠶᠢᠨ ᠲᠣᠳᠣ ᠷᠠ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠡᠭ P ᠶᠢ ᠠ ᠪᠴᠤ᠂ PA ᠂ PB ᠂ PC ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠃ ᠪᠠᠳᠤ ᠯᠠ᠄ PA ᠂ PB ᠂ PC ᠶᠢᠨ ᠤᠷᠳᠤ ᠪ ᠠᠷ ᠭᠤᠷ ᠪᠠᠨ ᠲᠠᠯ ᠠ ᠪᠠᠨ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠭᠤᠷᠪᠠᠯᠵ ᠢᠨ ᠢ ᠯᠠᠪᠳᠠᠢ ᠪᠦᠷᠢ ᠯᠳᠥᠬᠥᠯ ᠵᠤ ᠲᠠᠢ ᠯᠦᠨ ᠡ᠃ ⑶ 【ᠠᠰᠠ ᠭᠤᠳᠠᠯ ᠢ ᠰᠢᠢᠳᠪᠤᠷᠢ ᠯᠡᠬᠦ 】 ᠵᠢᠷᠤ ᠭ ③ ᠮᠡᠳᠤ ᠂ ᠴᠡᠭ P ᠨᠢ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ √ 7 ᠪᠣᠯᠬᠤ ᠲᠡᠩ ᠴᠡᠬᠦ ᠦ ᠲᠠᠯᠠᠳ ᠤ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠲᠣ ᠳᠣᠷᠠ ᠣᠷᠣ ᠰᠢᠬᠤ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭ ᠡ ᠴᠡᠭ᠂ ∠APC = 90° ᠂ ∠BPC = 120° ᠃ ∆ APC ᠶᠢᠨ ᠲᠠᠯᠠᠪᠠᠢ ᠶᠢ ᠣ ᠯ᠃ ⑷ 【ᠥᠷᠭ ᠡᠳᠭᠡᠨ ᠬᠡᠷᠡᠭᠯᠡᠬᠦ 】 ᠵᠢᠷᠤ ᠭ ④ ᠪᠣᠯ᠂ A᠂ B ᠂ C ᠭᠤ ᠷᠪᠠᠨ ᠲᠤᠰᠬᠤ ᠨ ᠤ ᠪᠠᠢᠷᠢᠯᠠᠯ ᠤ ᠨ ᠬᠠ ᠪᠳᠠᠭᠠ ᠢ ᠵᠢᠷᠤ ᠭ ᠪᠣᠯᠣ ᠨ ᠠ᠂ ᠬᠡᠮᠵᠢᠯᠲᠡ ᠪᠡᠷ ᠲᠠᠮᠵᠢᠨ AC = 4 , BC = 5 , ∠ ACB = 30° ᠶᠢ ᠣᠯᠵᠤ ᠮᠡᠳᠡ ᠪᠡ᠃ ᠴᠡᠭ P ᠨᠢ ∆ABC ᠶᠢᠨ ᠳᠣᠲ ᠣᠷ ᠠ ᠣᠷᠣ ᠰᠢᠬᠤ ᠨᠢᠭᠡ ᠬᠦᠳᠡᠯᠬ ᠦ ᠴᠡᠭ ᠂ PA ᠂ PB ᠂ PC ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠃ PA + PB + PC ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠪᠠᠭ ᠠ ᠬᠡᠮ ᠵᠢᠭᠳᠡ ᠯ ᠢ ᠣᠯ᠃ 图 1 A B C D E ① ② 图 2 A B C P ③ 图 3 A B C P 图 4 A P B C ④