ᠬᠤᠷᠢᠨ ᠨᠠᠢᠮᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠬᠤᠷᠴᠠ ᠦᠨᠴᠦᠭ ᠤᠨ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠤ ᠹᠦᠩᠺᠼ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠬᠣᠷ ᠢᠨ ᠨᠠᠢ ᠮᠠᠳᠤ ᠭᠠᠷ ᠪᠥᠯᠥ ᠭ ᠦᠨ ᠨᠢᠭᠡ ᠴᠢ ᠶᠢ ᠨ ᠪᠠᠢᠴᠠ ᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ (90 ᠮᠢᠨᠦ᠋ ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ ᠤᠪᠢ ) ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰ ᠡᠲᠤᠪ᠃ (ᠰ ᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤ ᠪᠢ) 1. ᠦᠭ᠍ ᠭᠦᠭ᠍ᠳ᠋ᠡᠭ᠍ ᠰᠡᠨ ᠨᠢ Rt ∆A BC ᠳ᠋ᠥ ᠂ ∠C = 90° , AC = 6 , BC = 8 ᠲᠡᠭᠡᠪ ᠡᠯ sinA ᠵᠢᠨ ᠬᠡᠮᠵᠢ ᠭ᠍ᠳ᠋ᠡᠯ ᠨᠢ ( ) A. 35 B. 34 C. 45 D. 43 2. ∆AB C ᠵᠢᠨ ᠲᠡᠭ᠍ᠰᠢ ᠲᠦ᠋ᠷᠪ ᠡᠭᠡᠯᠵ ᠢᠭᠡᠰᠤ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠪᠠᠢᠷ ᠢ ᠨᠢ ᠵᠢᠷᠤᠭ 1 ᠮᠡᠲᠤ ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ c os ∠AB C ᠵᠢᠨ ᠬᠡᠮᠵᠢᠭ᠍ ᠳ᠋ᠡᠯ ᠨ ᠢ ( ) A. √55 B. 2 √55 C. √ 5 D. 23 3. ᠦᠭ᠍ ᠭᠦᠭ᠍ᠳ᠋ᠡᠭ᠍ᠰᠡᠨ ᠨᠢ Rt ∆A BC ᠳ᠋ᠥ ᠂ ∠C = 90° , 𝑠𝑖𝑛 A= 513 ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ 𝑡𝑎𝑛 B= ( ) A. 1213 B. 512 C. 1312 D. 125 4. ᠦᠭ᠍ ᠭᠦᠭ᠍ᠳ᠋ᠡᠭ᠍ᠰᠡᠨ ᠨᠢ Rt ∆AB C ᠳ᠋ᠥ᠂ ∠C = 90° , ∠A = 40° , BC = 3 ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ AC = ( ) A. 3sin40 ° B. 3sin50 ° C. 3tan 50 ° D. 3tan 40 ° 5. Rt ∆AB C ᠳ᠋ᠤ ᠂ ∠C = 90° , ∠A , ∠B , ∠C ᠵᠢᠨ ᠡᠰᠡᠷ ᠬᠦ ᠤᠷ ᠤᠰᠢᠬ ᠤ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 𝑎 , 𝑏 , 𝑐 ᠪᠤ ᠯᠤᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠲᠤᠤᠷᠠ ᠬᠢ ᠳ᠋ᠤ᠂ ᠪᠠ ᠢᠭᠤᠯ ᠤᠭᠳ᠋ᠠᠬᠤ ᠦ ᠭᠡᠢ ᠨᠢ ( ) A. b = 𝑎∙ tan B B. 𝑎 = c∙ cos B C. c = 𝑎sinA D. 𝑎 = b∙ cos A 6. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠲᠤ ᠂ ∠ 1 ᠵᠢᠨ ᠲᠠᠨᠵᠠᠢᠨ ᠲ ᠵᠢ ᠨ ᠬᠡ ᠮᠵᠢᠭ᠍ ᠳ᠋ᠡᠯ ᠨᠢ ( ) A. 13 B. 12 C.3 D.2 7. ∆AB C ᠳ᠋ᠤ᠂ 2∠B ＝∠A ＋∠C ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ sinB ＋tan B= ( ) A. 1 B. 32 √ 3 C. 12 √3 D. ᠳᠣᠭᠲᠠᠭ ᠠᠵᠤ ᠴᠢᠳ᠋ ᠠᠬᠤ ᠦᠭᠡᠢ 8. ᠵᠢᠷᠤᠭ 3 ᠮᠡᠲᠤ᠂ Rt ∆AB C ᠳ᠋ᠦ᠂ ∠A CB = 90° , CD ⊥ AB ᠡᠭᠴᠠ ᠵᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ D ᠂ ᠬᠡ ᠷᠪᠡ AC = 2 , BC = 1 ᠪᠤᠯ ᠂ sin∠A CD = ( ) A. √53 B. 2√ 55 C. √52 D. 23 9. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠲᠤ ᠂ ᠦ ᠭ᠍ᠭᠦᠭ᠍ᠳ᠋ᠡ ᠭ᠍ᠰᠡᠨ ᠨᠢ ⊙ O ᠵᠢᠨ ᠷᠠᠳ᠋ᠢᠤᠰ ᠨᠢ 5c m ᠂ ᠬᠦ ᠪᠴᠢ AB ᠵᠢᠨ ᠤᠷᠲᠤ ᠨᠢ 6c m᠂ ᠴᠡᠭ P ᠨᠢ A B ᠵᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠂ BP = 2c m ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ tan ∠O PA =( ) A. 35 B. 45 C. 23 D. 54 (1) (3) (2) 10. ᠵᠢᠷ ᠤᠭ 5 ᠮᠡ ᠲᠤ ᠂ Rt ∆AB C ᠵᠢᠨ ᠵᠢᠰᠢᠭ ᠦᠤ ᠲᠠ ᠯ ᠠ AB ᠵᠢ D ᠬᠦᠷᠳ᠋ᠠᠯ ᠠ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠬᠦᠯ ᠬᠦ ᠳ᠋ᠤ ᠪᠠᠨ BD = AB ᠪᠤᠯ ᠭᠠᠨ ᠠ ᠂CD ᠶᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠂ ᠬᠡᠷ ᠪᠡ tan ∠B CD = 13 ᠪᠣᠯ ᠂ t an A = ( ) A. 32 B.1 C. 13 D. 23 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣ ᠭᠣᠰ ᠤᠨ ᠳ᠋ᠥ ᠲᠠᠭᠯᠠ ᠬᠤ ᠰᠡ ᠳ᠋ᠦ᠋ᠪ ᠃ ( ᠰ ᠡᠳ᠋ᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠡᠢᠲᠠ 24 ᠬᠤᠪᠢ ) 11 . 3t an 30° ᠦ᠋ᠨ ᠬᠡᠮᠵᠢ ᠭ᠍ᠳ᠋ᠡᠯ ᠨᠢ 12 . ᠦᠭ᠍ ᠭᠦᠭ᠍ᠳ᠋ᠡᠭ᠍ ᠰᠡᠨ ᠨᠢ Rt ∆AB C ᠳ᠋ᠥ᠂ ∠C = 90° , sinA = 12 , AC = 2√ 3 ᠲᠡ ᠭᠡᠪᠡᠯ BC ᠵᠢᠨ ᠬᠡᠮᠵᠢ ᠭ᠍ᠳ᠋ᠡᠯ ᠨᠢ 13 . Rt ∆AB C ᠳ᠋ᠥ ᠂ ∠C = 90° , BC = √5 , AC = √15 ᠲᠡᠭᠡᠪ ᠡᠯ ∠A ᠵᠢᠨ ᠭᠷᠠᠳ᠋ᠦ᠋ᠰ ᠦ᠋ ᠨ ᠲᠤᠭ ᠠ ᠨᠢ 14 . ∆ ABC ᠶᠢ ᠵᠢᠷᠤᠭ 6 ᠮᠡᠲᠤ ᠦᠵ ᠡᠭᠦᠯᠪᠡ ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ cos ∠B ᠵᠢᠨ ᠬᠡᠮᠵᠢᠭ᠍ ᠳ᠋ᠡᠯ ᠨᠢ 15 . ᠵᠢᠷ ᠤᠭ 7 ᠮᠡ ᠲᠤ ᠂ ⊙ O ᠨᠢ△ AB C ᠵᠢᠨ ᠪᠠᠭᠲᠠᠭ ᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠤᠷᠢᠭ ᠂ AD ᠨᠢ ⊙ O ᠵᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠸᠲ᠋ᠷ ᠂ ᠬᠡᠷ ᠪᠡ ⊙ O ᠵᠢᠨ ᠷᠠᠳ᠋ᠢᠤᠰ ᠨᠢ 2 , AC = 3 ᠪ ᠣᠯ ᠂ si nB ᠵᠢᠨ ᠬᠡᠮᠵᠢᠭ᠍ᠳ᠋ᠡᠯ ᠨᠢ 16 . ᠵᠢᠷ ᠤᠭ 8 ᠮᠡ ᠲᠤ ᠂ AB ᠨᠢ ⊙ O ᠵᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠸᠲ᠋ᠷ ᠂ ᠬᠦᠪ ᠴᠢ AC ᠪᠠ BD ᠨᠢ ᠴᠡᠭ E ᠳ᠋ᠤ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ ᠂ ᠬᠡᠷ ᠪᠡ ∠B EC = 60° , C ᠨᠢ ᠨᠤᠮᠤ BD ᠵᠢᠨ ᠲᠤᠮᠳ᠋ ᠠ ᠴᠡ ᠭ ᠪᠤ ᠯ ᠂ tan ∠A CD = 17 . ᠵᠢᠷ ᠣᠭ 9 ᠮᠡᠲᠤ ᠂ ᠷᠤᠮ ᠪ A BC D ᠳ᠋ᠤ ᠂ DE ⊥ AB , ᠡᠭᠴᠡ ᠵᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ E ,DE = 6c m , sinA = 35 ᠲᠡᠭᠡᠪ ᠡᠯ ᠷᠤᠮᠪ ABC D ᠵᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ᠃ 18 . ᠵᠢᠷ ᠤᠭ 10 ᠮᠡᠲᠤ᠂ ⊙ O ᠨᠢ ∆ AB C ᠵᠢᠨ ᠪᠠᠭᠲᠠᠭ ᠠᠭᠰᠠᠨ ᠲᠤᠭ ᠤᠷᠢᠭ ᠂ AD ᠨᠢ ⊙ O ᠵᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠸᠲ᠋ᠷ ᠂ CD ᠶᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠂ ᠬᠡ ᠷᠪᠡ AD = 3, AC = 2 ᠪ ᠣᠯ᠂ cos B ᠵᠢᠨ ᠬᠡᠮᠵᠢᠭ᠍ᠳ᠋ᠡᠯ ᠨᠢ ᠃ ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳ᠋ᠤᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡᠳ᠋ᠦ᠋ ᠪ ᠃ (ᠨᠡ ᠢᠲᠡ 46 ᠬᠤᠪᠢ) 19 . ( 8ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ ᠂ ᠬᠦᠳ᠋ᠡᠷ ᠠᠰᠠᠷ ᠲᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ A ᠠ᠋ᠴᠠ ᠠᠰᠠᠷ ᠤᠨ ᠡᠮ ᠤᠨ ᠠ ᠪᠠᠢᠭ ᠠ ᠨᠢ ᠭᠡᠨ ᠮᠤᠳ᠋ᠤ CD ᠵᠢᠨ ᠣᠷ ᠤᠢ C ᠶᠢ ᠬᠠᠷ ᠠᠭᠰᠠᠨ ᠰᠢᠷ ᠭᠤᠭᠰ ᠠᠨ ᠦᠨᠴᠦ᠋ ᠭ ᠨᠢ 60° ᠂ ᠲᠦ᠋ᠪᠰᠢᠨ ᠵᠠᠢ BD ᠨᠢ 10 m ᠪᠤᠯᠬ ᠤ ᠶᠢ ᠮᠡᠳ᠋ᠡᠨ ᠠ ᠂ ᠠᠰᠠᠷ ᠤᠨ ᠦᠨᠳ᠋ᠤᠷ AB = 24m ᠲᠡᠭᠡᠪ ᠡᠯ ᠮᠤᠳ᠋ᠤᠨ ᠦ ᠦᠨᠳ᠋ᠤᠷ ᠨ ᠢ ᠬᠡ ᠳ᠋ᠤᠢ ᠪᠤᠢ ︖ (5) (6) (7) (8) (9) (10) 20 . (8 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ ᠂ ᠳ ᠣᠭ ᠤᠨ ᠡᠰᠢ AB ᠶᠢᠨ ᠦᠨᠳ᠋ᠤᠷ ᠢ ᠬᠡᠮᠵᠢᠬᠦ ᠪᠡᠷ ᠪᠣᠯᠪᠠ᠂ ᠴᠡᠭ D ᠡᠴᠡ ᠴᠡᠭ A ᠵᠢ ᠬᠠᠷ ᠠᠭᠰᠠᠨ ᠥᠯ ᠢᠢᠭ᠍ᠰᠡᠨ ᠥᠨᠴ ᠦ᠋ᠭ ᠨᠢ 30° ᠳᠣᠭ ᠤᠨ ᠲᠡ ᠢᠰᠢ 20 ᠮᠸᠲ᠋ᠷ ᠠᠯᠬ ᠤᠭᠠᠳ ᠴᠡᠭ C ( ᠴᠡ ᠭ D , C , B ᠨᠢ ᠢᠵᠢᠯ ᠨᠢᠭᠡ ᠨ ᠰ ᠢᠯᠤᠭᠤᠨ ᠳᠡ ᠭᠡᠷ ᠡ ᠣᠷᠤᠰ ᠢᠨ ᠠ ) ᠵᠢᠨ ᠪᠠᠢᠷ ᠢᠨ ᠬᠦᠷᠪᠡ᠂ ᠴᠡᠭ C ᠡᠴᠡ ᠴᠡᠭ A ᠵᠢ ᠬᠠᠷ ᠠᠭᠰᠠᠨ ᠥ ᠯᠢᠢᠭ᠍ᠰᠡᠨ ᠥᠨᠴᠦ᠋ ᠭ ᠨᠢ 60° ᠂ ᠲᠡ ᠭᠡᠪᠡᠯ ᠳᠣᠭ ᠤ ᠨ ᠡ ᠰᠢ ᠵᠢᠨ ᠦᠨᠳ᠋ᠤᠷ ᠨ ᠢ ᠬᠡᠳ᠋ᠦ᠋ ᠢ ᠮᠸ ᠲ᠋ᠷ ᠪᠣᠢ ︖ 21 .(10 ᠬᠤᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠂ 6√ 3 ᠮᠸ ᠲ᠋ᠷ ᠤᠷᠲᠤ ᠨᠢᠭᠡᠨ ᠮᠤᠳ᠋ᠤᠨ ᠰᠢᠳ᠋ᠠ ᠮ ( AB ) ᠵᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠭᠠᠳ᠋ᠠ ᠷᠭᠤ (OM ) ᠳ᠋ᠤ ᠡᠭᠴᠠ ᠪᠤᠯᠬ ᠤ ᠬᠡ ᠷᠡᠮ( ON) ᠢ ᠲᠦ᠋ ᠰᠢᠭᠦ ᠯᠦ᠋ᠨ ᠲᠠᠯᠪᠢᠵᠠᠢ ᠂ ᠡᠨ ᠡ ᠦᠶ ᠡᠰ ᠮᠤᠳ᠋ᠤᠨ ᠰᠢᠳ᠋ᠠ ᠮ ᠤᠨ ᠦᠯ ᠢᠭ᠍ᠰᠡᠨ ᠦᠨᠴ ᠦ᠋ᠭ (∠ AB O) ᠨᠢ 60 ° ᠪᠠᠢᠪ ᠠ ᠂ ᠮᠤᠳ᠋ᠤᠨ ᠰᠢᠳ᠋ᠠ ᠮ ᠦ᠋ ᠨ A ᠦᠵ ᠦ᠋ᠭᠦᠷ ᠨᠢ ᠲᠤᠷ ᠤᠭᠰᠢ ᠭᠤᠯ ᠭᠤᠭᠠ ᠳ A ΄ ᠴᠡ ᠭ ᠲᠣ ᠬᠦ ᠷᠬᠦ ᠦᠶ ᠡᠰ B ᠦᠵ ᠦ᠋ᠭᠦᠷ ᠨᠢ ᠭᠠᠵᠠᠷ ᠤᠨ ᠭᠠ ᠳ᠋ᠠᠷᠭ ᠤ ᠲᠡᠭᠡ ᠷ ᠠ ᠪ ᠠᠷᠠᠭ ᠤᠨᠰᠢ ᠭᠤᠯ ᠭᠤᠵ ᠤ B ΄ ᠴᠡ ᠭ ᠲᠣ ᠬᠦᠷ ᠴᠠᠢ ᠃ (1) O B ᠵᠢᠨ ᠤᠷᠲᠤ ᠶᠢ ᠤᠯ ᠃ (2) AA ΄= 1 ᠮᠸ ᠲ᠋ᠷ ᠪᠠᠢᠬ ᠤ ᠦᠶ ᠡᠰ BB ΄ ᠵᠢᠨ ᠤᠷᠲᠤ ᠨᠢ ᠬᠡᠳ᠋ᠤᠢ ᠮᠸ ᠲ᠋ᠷ ᠪᠤᠯᠬ ᠤ ᠪᠤᠢ︖ (ᠪᠤᠳ᠋ᠤᠯ ᠭ ᠠ ᠵᠢᠨ ᠶᠠᠪᠤᠴᠠ ᠳ᠋ᠥ ᠪᠦᠷ ᠷᠠᠲᠢᠺ ᠠᠯ ᠦᠯ ᠡᠳ᠋ᠡᠭᠡ ) B / A / N MO B A A B C D 30° 60° 22. (1 0 ᠬᠤᠪᠢ) ᠨᠢᠭᠡᠨ ᠤᠩᠭᠤ ᠴᠠ ᠦᠳ᠋ᠠ ᠵᠢᠨ ᠡᠮᠤᠨᠡᠬ ᠢ 8 ᠴᠠᠭ ᠲᠣA ᠪᠠᠢᠷ ᠢᠨ ᠠ᠋ᠴᠠ ᠪᠠᠷᠠᠭ ᠤᠨ ᠡᠮᠤᠨ ᠠ ᠴᠢᠭᠯᠡᠯ ᠳ᠋ᠡᠬᠢ ᠬᠡᠷᠡᠯᠳ᠋ᠦ ᠴᠠᠮᠬᠠᠭ B ᠶᠢ ᠤᠯᠵ ᠤ ᠬᠠᠷ ᠠᠪᠡ ᠃ ᠡᠨ ᠡ ᠦᠶ ᠡᠰ ᠤᠩᠭ ᠤᠴᠠ ᠪᠠ ᠬᠡᠷᠡᠯᠳ᠋ᠦ ᠴᠠᠮᠬᠠᠭ ᠤ ᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳ᠋ᠤᠬ ᠢ ᠵᠠ ᠢ ᠨᠢ 36 √2 ᠮᠢᠯ ᠃ ᠤᠩᠭ ᠤᠴᠠ ᠴᠠᠭ ᠪᠦᠷᠢ 20 ᠮᠢ ᠯ ᠤᠨ ᠬᠤᠷᠳ᠋ᠤᠴᠠ ᠪᠠᠷ ᠡᠮᠦ᠋ᠨ ᠠ ᠠ᠋ᠴᠠ ᠪᠠᠷᠠᠭ ᠤᠨᠰᠢ 24° ᠦ᠋ᠨ ᠬᠡᠯᠪᠡ ᠢᠭᠦᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡ ᠷ ᠠ ᠶᠠᠯᠠᠭ ᠠᠳ ᠴᠡᠭ C ᠳ᠋ᠤ ᠬᠦᠷᠬᠦ ᠳ᠋ ᠤ ᠬᠡ ᠷᠡᠯᠳ᠋ᠦ ᠴᠠ ᠮᠬᠠᠭ ᠨᠢ ᠤᠩᠭᠤᠴᠠᠨ ᠦ ᠠᠰ ᠬᠤᠢᠨ ᠠ ᠴᠢᠭ ᠯᠡᠯ ᠳ᠋ᠤ ᠪᠠᠢᠪᠠ ᠃ ︵ ᠯᠠᠪᠯᠬ ᠤ ᠲᠤᠭ ᠠ ᠪᠠᠷᠢᠮᠲᠠ ᠄ sin24 °≈ 0.4 ,cos 24 °≈ 0.9 ︶ (1) ᠤᠩᠭ ᠤᠴᠠ ᠬᠡᠳ᠋ᠦ᠋ ᠢ ᠴᠠ ᠭ ᠦ᠋ᠨ ᠦ ᠶᠡᠰ C ᠴᠡᠭ ᠲᠣ ᠬᠦᠷᠦ᠋ ᠭ᠍ᠰᠡᠨ ᠪᠤᠢ ︖ (2) ᠤᠩᠭ ᠤᠴᠠ ᠴᠡᠭ C ᠳ᠋ᠤ ᠬᠦᠷᠬᠦ ᠦᠶ ᠡᠰ᠂ ᠤᠩᠭ ᠤᠴᠠ ᠪᠠ ᠬᠡᠷᠡᠯᠳ᠋ᠦ ᠴᠠᠮᠬ ᠠᠭ ᠤᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳ᠋ᠤᠬ ᠢ ᠵᠠᠢ ᠨᠢ ᠬᠡᠳ᠋ᠤᠢ ᠪᠤᠢ ︖ 23 . (1 0 ᠬᠤᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠂ ᠬᠡ ᠷᠴᠢᠮ AB ᠪᠠ CD ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠠ ᠂ ᠡ ᠬᠤᠶ ᠠᠷ ᠪᠠᠷᠢᠯ ᠭ ᠠ ᠵᠢᠨ ᠦᠨᠳ᠋ᠤᠷ ᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠨ ᠠ ᠂ AB ⊥ BC , DC ⊥ BC ᠴᠡᠭ B ᠠ᠋ᠴᠠ ᠴᠡᠭ D ᠦᠭᠡᠳ᠋ᠡ ᠦᠯ ᠢᠭ᠍ᠰᠡᠨ ᠦᠨᠴᠦ᠋ ᠭ 𝑎 ᠨᠢ 60° ᠂ ᠴᠡ ᠭ A ᠠ᠋ᠴᠠ ᠴᠡᠭ D ᠦᠭᠡᠳ᠋ᠡ ᠦᠯ ᠢᠭ᠍ᠰᠡᠨ ᠦᠨᠴ ᠦ᠋ᠭ β ᠨᠢ 30° ᠃ ᠮᠡ ᠳ᠋ᠡᠬᠦ ᠨᠢ ᠠ ᠪᠠᠷ ᠢᠯᠭ ᠠ ᠵᠢᠨ ᠦᠨᠳ᠋ᠤᠷ AB = 36 ᠮᠸᠲ᠋ᠷ ᠃ (1) ᠪᠠᠷᠢᠯ ᠭ ᠠ ᠡ ᠵᠢᠨ ᠦᠨᠳ᠋ᠤᠷ DC ᠶᠢ ᠤᠯ ᠃ (2) ᠪᠠᠷᠢᠯ ᠭ ᠠ ᠠ ᠡ ᠵᠢᠨ ᠬᠤᠭᠤᠷ ᠤᠨᠳ᠋ᠤᠬ ᠢ ᠵᠠ ᠢ BC ᠶᠢ ᠤᠯ ᠃ ( ᠳ᠋ᠦ᠋ᠩ ᠢ ᠨᠢ 0.0 1 ᠮᠸᠲ᠋ᠷ ᠬ ᠦᠷᠲᠠᠯ ᠠ ᠨᠠ ᠷᠢᠪᠴᠢ ᠯᠠ ) ( ᠯᠠᠪᠯᠠ ᠬᠤ ᠲᠤᠭ ᠠ ᠪᠠᠷ ᠢᠮᠲᠠ ᠄ √ 2≈ 1.414 ,√ 3≈ 1.732 ) C A B D ᠵᠡᠭᠦᠨ ᠤᠮᠠᠷ᠎ᠠ A B C D α β