ᠳᠣᠯᠣᠳᠤᠭᠠᠷ ᠪᠥᠯᠥᠭ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠪᠠᠢᠷᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ᠧᠮ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠲᠤᠯᠤ ᠳ᠋ᠤᠭᠠ ᠷ ᠪᠤᠯᠦᠭ ᠦᠨ ᠪ ᠠᠶᠢᠴ ᠠᠭᠠᠯᠳ᠋ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵90 ᠮᠢᠨᠦ᠋ ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ ᠤᠪᠢ ︶ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰᠡ ᠲᠤᠪ ᠃ ( ᠰ ᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬ ᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪ ᠢ) 1. ᠭᠡᠷᠪᠠ ᠴᠡ ᠭ A (m ,n ) ᠨᠢ ᠬᠣ ᠶᠠᠳᠤᠭ ᠠᠷ ᠮᠥᠴᠡ ᠳᠦ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠴᠡᠭ B( −m ,|n |) ᠨᠢ ( ) ᠮᠥ ᠴᠡ ᠳ ᠦ ᠣ ᠷᠣᠰᠢᠨ ᠠ᠃ A. ᠨᠢᠭᠡᠳᠤᠭᠡᠷ B. ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ C. ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ D. ᠲᠥ ᠷᠪᠡᠳᠤᠭᠡᠷ 2. ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠴᠡ ᠭ P ᠨᠢ ᠬᠣᠶ ᠠᠳᠤᠭᠠ ᠷ ᠮᠥᠴᠡ ᠳᠦ ᠣᠷᠣᠰ ᠢᠨ ᠠ᠂ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠡᠴᠡ 4 ᠨᠢᠭᠡᠴᠢ ᠵᠠᠢ ᠲᠠᠢ᠂ y ᠲᠠ ᠩᠭᠡᠯ ᠢᠭ ᠠ ᠴᠠ 3 ᠨᠢ ᠭᠡᠴᠢ ᠵᠠᠢ ᠲᠠᠢ᠂ ᠲᠡᠭᠡ ᠪᠡᠯ ᠴᠡᠭ P ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (− 3,4 ) B. (3 ,4) C. ( −4 ,3) D. (4 ,− 3) 3. ᠴᠡᠭ A( −2 ,− 2) ᠶᠢ ᠲᠡᠭᠡᠭᠡᠰᠢ 1 ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷ ᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠬᠦ ᠯᠵᠤ᠂ ᠳᠠ ᠷᠠᠭ ᠠ ᠨᠢ ᠵᠡᠭᠦᠨᠰᠢ 5 ᠨᠢᠭᠡᠴᠢ ᠵᠡᠷ ᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠬᠦ ᠯᠦᠭᠰᠡᠨ ᠦ ᠳᠠ ᠷᠠᠭᠠᠬ ᠢ ᠴᠡᠭ ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (1 ,− 8) B. (1 ,− 2) C. (− 7, −1 ) D. (0 ,− 1) 4. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠳᠤ ᠂ ᠬᠡ ᠷᠪᠡ ᠨᠢᠭᠡ ᠰᠢᠳᠡ ᠷ ᠠ ᠳᠤ ᠬᠠᠭ ᠠᠨ ᠪ ᠪ ᠠᠢᠷᠢ ᠨᠢ (− 1 , −2 ) ᠂ ᠵᠠᠭ ᠠᠨ ᠤ ᠪᠠᠢᠷ ᠢ ᠨᠢ ( 1 ,− 2) ᠂ ᠲᠡ ᠭᠡᠪᠠᠯ ᠫᠤᠤ ᠶᠢᠨ ᠪᠠᠢᠷ ᠢ ᠨᠢ ( ) A. (1 ,4) B. (4 ,1) C. (− 4,1 ) D. (1 ,− 2) 5.ᠲᠡ ᠭᠰᠢ ᠦᠠ ᠴᠦᠭᠳᠦ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠤ ᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ 𝑥Oy ᠳᠦ᠂ ᠭᠡᠷ ᠴᠢᠮ AB ᠶᠢᠨ ᠬᠤᠶ ᠠᠷ ᠦ ᠵᠡᠬᠦᠷ ᠤᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ (− 1 , −1 ) ᠂ (1 ,2 )᠃ ᠭᠡᠷᠴᠢᠮ AB ᠶᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰ ᠢᠯᠵᠢᠬᠦ ᠯᠵᠤ ᠴᠡᠭ A ᠤᠨ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠠ ᠴᠡᠭ ᠤ ᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (3 ,− 1) ᠪᠣᠯ ᠴᠡᠭ B ᠶᠢᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠠ ᠴᠡ ᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. ( 4,2 ) B. ( 5,2 ) C. ( 6,2 ) D. ( 5,3 ) 6. ᠴᠡᠭ A (− 2, −3 ) ᠶᠢ ᠪᠠᠷᠠᠭ ᠤᠨᠰᠢ 3 ᠨᠢ ᠭᠡᠴᠢ ᠰᠢᠯ ᠵᠢᠬᠦ ᠯᠪᠠᠯ ᠴᠡᠭ B ᠳᠤ ᠬᠦᠷᠦᠨ ᠠ᠃ ᠲᠡᠭᠡᠪ ᠠᠯ ᠴᠡ ᠭ B ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (1 ,− 3) B. (− 2,0 ) C. (− 5, −3 ) D. (− 2, −6 ) 7. ᠦᠭ ᠬᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ᠴᠡᠭ A( 1, 0) ᠂ B (0 ,2 ) ᠴᠡᠭ P ᠨᠢ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠡᠭᠡᠷ ᠠ ᠪᠠᠢᠬᠤ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ ∆P AB ᠶᠢᠨ ᠲᠠᠯ ᠠᠪᠠᠢ ᠨᠢ 5᠂ ᠲᠡᠭᠡᠪ ᠠᠯ ᠴᠡ ᠭ P ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (− 4,0 ) B. (6 ,0) C. (− 4,0 ) ᠪᠤᠶ ᠤ ( 6,0 ) D. (0 ,12 ) ᠪᠤᠶ ᠤ ( 0, −8 ) 8.ᠲᠡ ᠭᠰᠢ ᠦᠠᠴ ᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠳᠤ᠂ ᠦ ᠭᠬᠦᠭᠳᠡᠭᠰᠡ ᠨ ᠨ ᠢ ᠭᠡᠷ ᠴᠢᠮ AB ᠶᠢᠨ ᠬᠤᠶ ᠠᠷ ᠦᠵ ᠡᠬᠦᠷ ᠤᠨ ᠴᠡᠭ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ A (4 ,− 1) ᠂ B (4 ,3) ᠃ ᠴᠡᠭ C ᠨᠢ ᠭᠡᠷᠴᠢᠮ AB ᠶᠢᠨ ᠲᠣᠮᠳ ᠡ ᠴᠡᠭ ᠂ ᠭᠡᠷᠴᠢᠮ AB ᠶᠢ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰ ᠢᠯᠵᠢᠬᠦ ᠯᠪᠠᠯ ᠴᠡᠭ A ᠤᠨ ᠬᠠᠷᠠᠭ ᠠᠯᠵᠠᠭᠰᠠᠠ ᠴᠡ ᠭ ᠤ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ( 3,5 )᠂ ᠪᠣᠯ ᠴᠡᠭ C ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (− 3, −7 ) B. (3 ,9) C. (3 ,7) D. (− 3, −9 ) 9. ᠲᠡᠭᠰᠢ ᠦᠠ ᠴᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ ᠳᠤ ᠴᠡᠭ M ᠨᠢ ᠲᠦ᠋ ᠷᠪᠡᠳᠦ ᠭᠡᠷ ᠮᠦ᠋ᠴᠠ ᠳᠤ ᠪᠠᠢᠬᠤ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ x ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠬᠦᠷᠬ ᠦ ᠵᠡ ᠢ ᠨᠢ 1᠂ y ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠬᠦᠷᠬ ᠦ ᠵ ᠡᠢ ᠨᠢ 2 ᠂ ᠲᠡ ᠭᠡᠪᠠᠯ ᠴᠡᠭ M ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (2 ,1) B. (− 2,1 ) C. (2 ,− 1) D. (1 ,− 2) (1) 10 . ᠵᠢᠷ ᠤᠭ 2 ᠮᠡ ᠳᠤ᠂ ᠲᠡᠭᠰᠢ ᠦᠠ ᠴᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠦ ᠴᠡᠭ P ᠦᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( −2 ,3) ᠂ ᠴᠡᠭ O ᠪᠠᠷ ᠲᠤᠭ ᠤᠷᠢᠭ ᠦᠨ ᠲᠦ᠋ᠪ ᠪᠤᠯᠭ ᠠᠨ ᠂ OP ᠤᠷ ᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠦ᠋ ᠰ ᠪ ᠤᠯᠭᠠᠨ ᠨᠤᠮᠤ ᠵᠢᠷᠤᠪᠠᠯ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠦᠨ ᠰᠦ᠋ᠭᠡᠷ ᠬᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠡᠢ ᠴᠡᠭ A ᠲᠦ ᠤᠭᠳ ᠤᠯᠤᠴ ᠠᠨ ᠠ ᠃ ᠲᠡᠭᠡ ᠪᠠᠯ ᠴᠡᠭ A ᠦᠨ ᠬᠦᠠᠳ ᠠᠯᠠᠠ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) ᠦᠨ ᠬᠤᠭᠤᠷ ᠤᠠᠳᠤ ᠪ ᠤᠢ ᠃ A. −4 ~ − 3 B. 3 ~ 4 C. −5 ~ − 4 D. 4 ~ 5 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳᠤ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠲᠤᠪ ᠃ ︵ᠰᠡᠲᠤᠪ ᠪᠦ ᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲ ᠡ 2 4 ᠬᠤᠪᠢ ︶ 11. ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ P (2 ,− 3) ᠪᠠ ᠴᠡᠭ Q( 2 , 𝑥) ᠦᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳᠤᠬ ᠢ ᠵᠡ ᠢ ᠨᠢ 4 ᠪᠣᠯ 𝑥 ᠦᠨ ᠭᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠨᠢ __ __ __ _ 12. ᠵᠢᠷ ᠤᠭ 3 ᠮᠡᠲᠤ᠂ ᠴᠡ ᠭ A ᠂ B ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ (− 3,3 )᠂ (3 ,3) ᠪᠣᠯ ᠴᠡᠭ C ᠶᠢᠨ ᠪᠡᠢᠷ ᠢᠰᠢᠯ ᠨᠢ __ __ __ 13. ᠨᠢᠭᠡ ᠠᠩᠭ ᠯᠢ ᠦᠭᠡ ᠶᠢᠨ ᠠᠪᠢᠶ ᠠ ᠶᠢᠨ ᠲᠠᠷ ᠠᠭᠠᠯᠠᠯ ᠨᠢ ᠲᠤ ᠤᠷᠠᠬ ᠢ ᠵᠢᠷ ᠤᠭ 4 ᠮᠡᠳᠤ (5 ,3) ᠂ (6 ,3) ᠂ (7 ,3) ᠂ (4 ,1) ᠂ (4 ,4) ᠪᠣᠯ ᠡᠨᠡ ᠠᠩᠭ ᠯᠢ ᠦᠭᠡ ᠶᠢ ᠤᠷ ᠴᠢᠭᠤᠯ ᠵᠤ ᠪᠢᠴ ᠢᠪᠠᠯ __ __ __ __ 14. ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ M (𝑎 + 5, 𝑎− 3) ᠨᠢ y ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠡᠭᠡᠷ ᠠ ᠪᠠᠢᠪ ᠠᠯ ᠴᠡ ᠭ m ᠡᠴᠡ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠬ ᠦᠷᠬᠦ ᠵᠡᠢ ᠨᠢ_ __ __ __ 15. ᠰᠢᠲᠠᠷ ᠠ ᠢᠢᠨ ᠤᠷᠤᠯ ᠳᠤᠭᠠ ᠨ ᠳ ᠤ ᠠ ᠳᠠᠯᠢ ᠦᠩ ᠭᠡ ᠢᠢᠨ ᠰᠢᠲᠡᠷ ᠠ ᠨᠢ ᠭᠡ ᠮᠦ᠋ᠷ ᠲᠡᠭᠡᠷ ᠠ 5 ᠰᠢᠲᠠᠷ ᠡ ᠪᠤᠯᠪᠠ ᠯ ᠢ ᠯᠠᠨ ᠠ ᠃ ᠭᠡᠷᠪᠡ ᠴᠠ ᠭᠠᠨ ① ᠢᠢᠨ (1 ,− 5) ᠪᠠᠶᠢᠷ ᠢ ᠨᠢ ᠪᠤᠯ ᠪᠠᠯ ᠬᠠᠷ ᠠ ᠦᠨ ᠪᠠᠶᠢᠷ ᠢ ᠨᠢ (2 ,− 4) ᠪᠤᠯ ᠪᠠᠯ ᠣᠳᠤ ᠬᠠᠷ ᠠ ᠰ ᠢᠲᠡᠷ ᠠ ᠢᠢ __ __ __ __ ᠲᠠᠯ ᠪᠢᠪᠠᠯ ᠢᠯ ᠠᠨ ᠠ ᠃ 16 . ᠵᠢᠷ ᠤᠭ 5 ᠮᠡᠳᠤ ᠂ ᠵᠢᠰᠢᠬ ᠦᠯᠵᠢᠨ ᠢᠢ ᠲᠡᠭᠰᠢ ᠦᠠᠴ ᠦᠭᠳᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦᠨ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠡ ᠭᠡᠷ ᠠ ᠲᠠᠯᠪᠢᠨ ᠠ᠃ ᠴᠡᠭ A᠂ C ᠤ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ (6 ,0) ᠂ (1 ,4) ᠪᠣᠯ ᠴᠡᠭ B ᠶᠢᠨ ᠪ ᠡᠢᠷᠢᠰᠢᠯ ᠨᠢ __ __ __ _ 17 .ᠲᠡᠭ ᠰᠢ ᠦᠠᠴ ᠦᠭᠳᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠲᠤᠳ ᠤᠷ ᠠ ᠳᠤᠷ ᠠ ᠢᠢᠨ ᠨᠢᠭᠡᠨ ᠴᠡᠭ (m ,n ) ᠨᠢ ᠲᠤᠤᠷᠠ ᠬᠢ ᠮᠡᠳᠤ ᠬᠤᠶᠠ ᠷ ᠵᠤᠶᠢᠯ ᠦᠨ ᠵᠢᠮ ᠡ ᠶᠤᠰᠤ ᠪᠠᠷ ᠬᠤᠪᠢᠷ ᠠᠯᠳᠠ ᠬᠢᠨ ᠡ ᠃ (1) f( m, n) = (m ,− n) ᠵᠢᠱᠢᠶ ᠠᠯᠠᠪᠠ ᠯ f(2 ,1) = (2 ,− 1) (2) g( m, n) = (− m, −n ) ᠵᠢᠱᠢᠶ ᠠᠯᠠᠪᠠ ᠯ g ( 2,1 )= (− 2, −1 ) ᠲᠡᠭᠡᠷ ᠠᠬᠢ ᠬᠤᠪᠢᠷ ᠭᠠᠯᠳ ᠠ ᠢᠢ ᠦᠠᠳ ᠤᠰᠤᠯ ᠠᠪᠠᠯ f[g (3 ,4) ]= f(− 3, −4 )= (− 3,4 ) ᠲᠡ ᠭᠡᠪᠠᠯ g[ f(− 3,2 )] = __ __ 18 . ᠵᠢᠷ ᠤᠭ 6 ᠮᠡ ᠳᠤ ᠂ ᠴᠡᠭ P ,Q ᠨᠢ ᠴᠡᠭ A( 2,0 ) ᠡᠴᠠ ᠵᠡᠷᠭᠡ ᠪᠠᠷ ᠭᠠᠷ ᠴᠤ ᠂ ᠲᠡᠭᠰᠢ ᠲᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ BCDE ᠢᠢᠨ ᠲᠠᠯ ᠠ ᠲᠠᠭ ᠠᠤ ᠵᠢᠭᠳ ᠡ ᠬᠤᠷᠳ ᠤᠴᠠ ᠪᠠᠷ ᠬᠦᠳ ᠡᠯᠭᠡᠭᠡ ᠨ ᠬᠢᠨ ᠡ ᠃ ᠴᠡᠭ P ᠨᠢ ᠰᠧᠺᠦ ᠨᠳ᠋ ᠪᠦᠷᠢ 1 ᠨᠢᠭᠡᠴᠢ ᠢᠢᠨ ᠬᠤᠷᠳ ᠤᠴᠠ ᠪᠠᠷ ᠴᠠᠭ ᠦᠨ ᠵᠡᠬᠦᠦ ᠢᠢᠨ ᠡᠰᠡᠷ ᠬᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠢᠡᠷ ᠬᠦ ᠳᠡᠯᠭᠡ ᠭᠡᠨ ᠬᠢᠵᠦ ᠂ ᠴᠡ ᠭ Q ᠨᠢ ᠰᠧᠺᠦ ᠨᠳ᠋ ᠪᠦᠷᠢ 2 ᠨᠢ ᠭᠡᠴᠢ ᠢᠢᠨ ᠬᠤᠷᠳ ᠤᠴᠠ ᠪᠠᠷ ᠴᠠᠭ ᠦᠨ ᠵᠡᠬᠦᠦ ᠢᠢᠨ ᠲᠠᠭ ᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠢᠡ ᠷ ᠬᠦ ᠳᠡᠯᠭᠡᠭᠡᠨ ᠬᠢᠨ ᠡ ᠂ ᠴᠡ ᠭ P , Q ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ 11 ᠤᠳ ᠠᠭ ᠠ ᠠᠭᠤᠯᠵᠠᠬᠤ ᠳᠤ ᠲᠡᠬᠦ ᠨ ᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ (2) (3) (4) (5) (6) O A B ᠨᠢ_ __ __ _ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ︵ ᠨᠠᠢᠲᠡ 46 ᠬᠤᠪᠢ ︶ 19 .( 8 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ᠲ ᠦ᠋ᠷᠪᠡ ᠯᠵᠢᠯ ᠡᠭᠡᠰᠤ ᠲᠡᠢ ᠴᠠᠭᠠᠰᠤᠨ ᠲᠡᠭᠡᠷ ᠠ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ AB C ᠪᠠᠶᠢᠨ ᠠ ᠃ (1) ∆ AB C ᠢᠢ ᠲᠡᠭᠡᠭ ᠰᠢ 3 ᠨᠢᠭᠡᠴᠢ ᠣ ᠷᠳᠤ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠵ ᠤ ᠂ ᠲᠠᠬᠢᠭᠠ ᠳ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠲᠡᠶᠢᠰᠢ 2 ᠨᠢ ᠭᠡᠴᠢ ᠤᠷᠳ ᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠤᠭᠡᠳ ᠤᠯᠤᠭᠰ ᠠᠠ ᠭᠤᠷ ᠪᠠᠯᠵ ᠢᠨ ∆ A ′B ′C ′ ᠢ ᠵᠢᠷᠤ ᠃ (2) ᠴᠡᠭ A ᠪᠠᠷ ᠡᠬᠢᠨ ᠦ ᠴᠡ ᠭ ᠪ ᠣᠯᠭᠠᠨ ᠲᠡᠭᠰᠢ ᠦᠠᠴᠦᠭᠳ ᠦ ᠪ ᠠᠶᠢᠷᠢᠰᠢᠯ ᠦᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠪᠠᠶᠢᠭ ᠤᠯᠤᠨ ᠠ ᠃ ᠲᠠᠷᠠᠭ ᠠ ᠨᠢ ᠬᠤᠭᠤᠰᠤ ᠨ ᠳ ᠤ ᠲᠠᠭ ᠯᠠ ᠃ B( __ _,_ _) B ′(_ __ _,_ __ _) 20 .( 8 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ᠂ ᠬᠠᠪ ᠳᠠᠭᠠᠢ ᠢᠢᠨ ᠲᠡᠭᠰᠢ ᠦᠠᠴ ᠦᠭᠳᠦ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠦᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠲᠤᠳ ᠤᠷ ᠠ ᠂ ∆ AB C ᠪ ᠠᠶᠢᠨ ᠠ ᠃ (1) ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ᠤ ᠡᠯ ᠡ ᠤᠷ ᠤᠢ ᠢᠢᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ ᠪᠢᠴ ᠢ ᠃ (2) ∆AB C ᠢᠢ ᠲᠡᠭᠡᠭᠰ ᠢ 2 ᠨᠢᠭᠡᠴᠢ ᠣᠷᠳ ᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠵ ᠤ ᠂ ᠲᠠᠬᠢᠭᠠ ᠳ ᠵᠡᠬᠦᠨ ᠲᠠᠯ ᠠ ᠲᠡᠶᠢᠰᠢ 1 ᠨᠢ ᠭᠡᠴᠢ ᠤᠷᠳ ᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠤᠭᠡᠳ ᠤᠯᠤᠭᠰ ᠠᠠ ᠭᠤᠷ ᠪᠠᠯᠵ ᠢᠨ ∆ A ′B ′C ′ ᠢ ᠵᠢᠷᠤ ᠃ ᠪᠠᠰᠠ A ′, B ′ , C ′ ᠦᠨ ᠪᠠᠶᠢᠷ ᠢᠰᠢᠯ ᠢ ᠪᠢᠴ ᠢ ᠃ (3) ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ AB C ᠢᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠢᠢ ᠣ ᠯ ᠃ 21 .(10 ᠬᠤᠪᠢ ) ᠭᠡᠷ ᠪᠡ ᠬᠤᠢ ᠨ ᠠ ᠡᠴᠠ ᠵᠡᠬᠦᠨᠰᠢ 30 ° ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠲᠠᠭ ᠤ 50m ᠶᠠᠪᠤᠬ ᠤ ᠶᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠴᠡ ᠭ A (30 ° ,50) ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢᠶᠠᠷ ᠢ ᠯᠡᠷᠭᠡᠢ ᠯᠡᠪᠠᠯ ᠂ ᠬᠤᠢᠨ ᠠ ᠡᠴᠠ ᠪᠠᠷ ᠠᠭᠤᠠᠰᠢ 45 ° ᠤᠨ ᠴᠢᠭᠯᠡᠯ ᠡᠰᠡᠷ ᠬᠦ 20m ᠶᠠᠪᠤ ᠬᠤ ᠶᠢ ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠴᠡ ᠭ B (− 45 °,− 20 ) ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢᠶᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢᠯ ᠡᠪᠠᠯ ᠂ ᠠ ᠰᠠᠭᠤᠬ ᠤ ᠨᠢ᠄ (1) (− 75 °,− 15 ) , (10 °,− 25 ) ᠲᠤᠰ ᠲᠤᠰ ᠶᠠᠭᠤ ᠶᠢ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠵᠤ ᠪᠠᠢᠬ ᠤ ᠪᠤᠢ︖ (2 ) ᠵᠢᠷ ᠤᠭ ᠲᠡᠭᠡᠷ ᠠ ( 60 °,− 30 ) ( −30 °,40 ) ᠪᠣ ᠯᠬᠤ ᠴᠡᠭ ᠢ ᠣ ᠯᠵᠣ ᠲᠡᠮᠲᠡᠭᠯᠨ᠃ (30°,50) (−45°,−20) ᠬᠣᠢᠨ ᠠ ᠵᠡᠭᠦ ᠨ 22 . ( 10 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ᠲᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠯ ᠡᠭᠡᠰᠤ ᠲᠡ ᠢ ᠴᠠ ᠭᠠᠰᠤᠨ ᠲᠡ ᠭᠡᠷ ᠠ ∆AB C ᠪᠠᠶᠢᠨ ᠠ ᠃ (1) ∆ AB C ᠢᠢ ᠲᠤᠷ ᠤᠭᠰ ᠢ 4 ᠨᠢᠭᠡᠴᠢ ᠣᠷᠳ ᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠵ ᠤ ᠂ ᠲᠠᠬᠢᠭᠠ ᠳ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠲᠡᠶᠢᠰᠢ 3 ᠨᠢ ᠭᠡᠴᠢ ᠤᠷᠳ ᠤ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠬᠦᠯᠤᠭᠡᠳ ᠤᠯᠤᠭᠰ ᠠᠠ ᠭᠤᠷ ᠪᠠᠯᠵ ᠢᠨ ∆ A ′B ′C ′ ᠢ ᠵᠢᠷᠤ ᠃ (2) ᠭᠡᠷ ᠴᠢᠮ AC ᠡᠴᠠ ᠭᠡᠷᠴᠢᠮ 𝐴 ′ 𝐶 ′ ᠣ ᠯᠬᠣ ᠶᠠᠪᠤᠴᠠ ᠳ ᠤ ᠱ ᠦ᠋ᠬᠦ ᠷᠳᠡᠨ ᠭᠠᠷᠤᠭᠰ ᠠᠠ ᠭᠠ ᠵᠠᠷ ᠤ ᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶᠢ ᠣᠯ᠃ 23 . ( 10 ᠬᠤᠪᠢ ) ᠲᠡ ᠭᠰᠢ ᠦᠠᠴᠦᠭ ᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤ ᠴᠡᠭ P ᠨᠢ ᠡᠬ ᠢᠨ ᠦ ᠴᠡᠭ Oᠡᠴᠠ ᠡᠬᠢᠯ ᠡᠵᠤ ᠬᠦᠳᠡ ᠯᠬᠦ ᠪ ᠦᠭᠡᠳ ᠪᠠᠷᠠᠭ ᠤᠨᠰᠢ ᠪᠤᠶᠤ ᠲᠡᠭᠡᠭᠰ ᠢ ᠬᠦ ᠳᠡᠯᠭᠡ ᠭᠡᠨ ᠬᠢᠨ ᠡ ᠃ ᠨᠢ ᠭᠡᠳᠤᠭᠡᠷ ᠤᠳᠠ ᠭ ᠠ ᠨᠢ (0 ,1) ᠂ ( 1,0 ) ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠦ ᠬᠦᠷᠦ ᠨ ᠡ᠃ ᠬᠤᠶ ᠠᠳᠤᠭᠠ ᠷ ᠤᠳᠠ ᠭ ᠠ (2 ,0) ᠂(1 ,1) ᠂ ( 0,2 ) ᠭᠤᠷᠪ ᠠᠨ ᠴᠡᠭ ᠲᠦ ᠬᠦ ᠷᠦᠨ ᠡ᠃ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠤᠳᠠ ᠭ ᠠ (3 ,0) ᠂(2 ,1) ᠂(1 ,2) ᠂(0 ,3) ᠲᠦ᠋ᠷᠪ ᠠᠨ ᠴᠡᠭ ᠲᠦ ᠬᠦ ᠷᠦᠨ ᠡ᠃ ᠲᠡᠭᠡᠪ ᠠᠯ (1) ᠲᠦ᠋ ᠷᠪᠡᠳᠦ ᠭᠡᠷ ᠤᠳᠠ ᠭ ᠠ_ __ __ __ _ ᠴᠡᠭ ᠲᠦ ᠬᠦ ᠷᠦᠨ ᠡ᠃ (2) ᠲᠡ ᠭᠰᠢ ᠦᠠᠴᠦᠭᠳᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠲᠡᠭᠡᠷ ᠠ ᠨᠡ ᠢᠮᠠᠳᠤᠭᠠ ᠷ ᠤᠳᠠ ᠭ ᠠ ᠬᠦᠳᠡ ᠯᠬᠦ ᠴᠡᠭ ᠢ ᠣᠯᠵ ᠤ ᠵᠢᠷ ᠤᠭᠠᠳ ᠪᠠ ᠰᠠ ᠠᠵᠢ ᠭᠯᠠᠵ ᠤ ᠪᠠ ᠢᠷᠢ ᠨᠢ ᠶᠠᠮᠠ ᠷ ᠤᠨᠴᠠ ᠯᠢᠭ ᠲᠡᠢ ᠪᠠᠢᠬ ᠤ ᠶᠢ ᠵᠢᠭᠠᠵᠤ ᠭᠠᠷᠭ ᠠ᠃ (3) ᠴᠡᠭ Pᠨᠢ ᠡᠬᠢᠨ ᠦ ᠴᠡ ᠭ ᠡᠴᠠ __ __ _ᠤᠳᠠ ᠭ ᠠ ᠬᠦᠳᠡᠯᠪᠠᠯ ᠣᠯ ᠬᠤ ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (13 ,5)