ᠬᠤᠷᠢᠨ ᠲᠦᠷᠪᠡᠳᠤᠭᠡᠷ ᠪᠦᠯᠦᠭ ᠲᠤᠭᠤᠷᠢᠭ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠬᠣᠷ ᠢᠨ ᠳᠥ ᠷᠪᠡᠳᠦᠭᠡᠷ ᠪᠥ ᠯᠥᠭ ᠦᠨ ᠪ ᠠᠢᠴ ᠠᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵12 0 ᠮᠢ ᠨᠦ᠋ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬᠤᠪᠢ ︶ ᠨᠢᠭᠡ᠂ ᠰᠤ ᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ( ᠰᠡᠲᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ) 1. ᠲᠣᠭ ᠣᠷᠢᠭ ᠤ ᠨ ᠬᠠᠮᠤᠭ ᠤᠷᠳᠤ ᠬᠥ ᠪᠴᠢ ᠨᠢ1 2cm ᠂ ᠬᠡ ᠷᠪᠠ ᠰᠢᠯᠤᠭᠤᠨ ᠪᠣᠯ ᠤᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠨᠢ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠬᠤ ᠮ ᠥᠷᠲᠡᠭᠡᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲᠥ ᠪ ᠡᠴᠡ ᠰᠢᠯ ᠤᠭᠤᠨ ᠬᠦᠷᠬ ᠦ ᠵᠠᠢ ᠨ ᠢ 𝑑 ᠪᠣᠯ ( ) A. 𝑑 12 cm 2. ᠵᠢᠷᠤᠭ 1 ᠮᠡ ᠳᠤ᠂ Rt △ AB C ᠳ᠋ ᠤ ᠂ ∠ ACB = 90° ᠂ AC = 4᠂ BC = 3᠂ ᠬᠡᠷᠪᠠ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ᠢ ᠲᠠᠯ ᠠ AB ᠶ᠋ᠢᠨ ᠣᠷᠤᠰ ᠢᠬᠤ ᠰᠢᠯᠤᠭᠤᠨ ᠤ ᠳᠠᠭ ᠠᠤ ᠪᠥᠳᠦ ᠨ ᠨᠢ ᠭᠡ ᠡᠷ ᠬᠢᠭᠦᠯ ᠪᠡᠯ᠂ ᠤᠯ ᠬᠤ ᠭ ᠸᠣᠮᠧ ᠲ᠋ᠷᠢ ᠶ᠋ ᠢᠨ ᠪ ᠠᠶ ᠡ ᠶ᠋ ᠢᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠶ᠋ ᠢᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ( ) A. 84 𝜋5 B. 164 𝜋5 C. 12 𝜋 D. 24 𝜋 3. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠳᠤ ᠂ ⊙ C ᠨᠢ ᠡᠬᠢᠨ ᠤ ᠴᠡᠭ ᠢ ᠳᠠᠭ ᠠᠷᠢᠬ ᠤ ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠬᠤᠶ ᠠᠷ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠤ ᠨ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ A ᠂ B ᠳ᠋ᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ A ᠶ᠋ᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( 0,3 )᠂ M ᠨᠢ ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠮᠥᠴᠡ ᠳ᠋ ᠤ ᠣ ᠷᠤᠰᠢᠬ ᠤ ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠨᠤᠮᠤ OB ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ ∠B MO = 120° ᠪᠣᠯ ⊙ C ᠶ᠋ ᠢᠨ ᠷᠠᠳᠢᠦ ᠰ ᠤ ᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) A. 6 B. 5 C. 3 D. 3√ 2 4. ᠳᠣᠣᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠪᠤᠷᠤᠭ ᠤ ᠪᠣ ᠯᠬᠤ ᠨᠢ ( ) A. ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠲᠥᠪ ᠨᠢ ᠯᠠᠪᠲᠠ ᠢ ᠭ ᠤᠷᠪᠠᠯᠵ ᠢᠨ ᠤ ᠭᠠᠳᠠ ᠨ ᠠ ᠣᠷᠤ ᠰᠢᠬ ᠤ ᠪᠢ ᠰᠢ᠃ B. ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠳ᠋ ᠢᠶᠠᠮᠧᠲ᠋ ᠷ ᠪᠢᠰ ᠢ ᠬ ᠤᠶᠠᠷ ᠬᠥᠪ ᠴᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠲᠠᠯᠠ ᠯᠠᠨ ᠬᠤᠪᠢᠶ ᠠᠬᠤ ᠪᠣᠯᠤᠯᠴᠠ ᠭ ᠠ ᠦᠭᠡᠢ᠃ C. ᠬᠤ ᠶᠠᠷ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ᠤ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠲᠥᠪ ᠨᠢ ᠢ ᠵᠢᠯᠢ ᠨᠢ ᠭᠡᠨ ᠶᠡᠷᠦ ᠩᠬᠡᠢ ᠴᠡᠭ ᠪᠠᠢᠬᠤ ᠪᠣᠯᠤᠯ ᠴᠠᠭ ᠠ ᠲᠠᠢ᠃ D. ᠠᠯ ᠢᠪᠠ ᠱᠠᠳᠤ ᠯᠵᠢᠨ ᠳ᠋ᠤ ᠴᠥᠮ ᠪᠠᠭᠲᠠᠭ ᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠪᠠᠢᠬᠤ ᠦᠭᠡᠢ ᠃ 5. 10 ×1 0 ᠶ᠋ ᠢᠨ ᠳᠡᠭᠰᠢ ᠳ ᠦ᠋ᠷᠪᠠ ᠯᠵᠢᠨ ᠳ ᠥᠷᠪᠡ ᠯᠵᠢᠯ ᠡᠭᠡᠰᠦ ᠳ᠋ᠤ ᠵᠢᠵᠢᠭ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷ ᠪᠠᠯᠵ ᠢᠨ ᠪᠦᠷᠢ ᠶ᠋ ᠢᠨ ᠲᠠᠯ ᠠ ᠶ᠋ ᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ 1᠂ ᠬᠡᠷ ᠪᠠ ᠲᠤᠰ ᠳᠥᠷᠪ ᠡᠯᠵᠢ ᠯᠡᠭᠡᠰᠦ ᠶ᠋ ᠢᠨ ᠲᠥ ᠪ ᠢᠶᠠᠷ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲ ᠥᠪ ᠪᠤᠯᠭ ᠠᠵᠦ᠂ 5 ᠪᠡᠷ ᠷᠠ ᠳᠢᠦᠰ ᠪᠣ ᠯᠭᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ᠲᠤᠰ ᠲ ᠣᠭᠤᠷ ᠢᠭ ᠳᠡᠭᠡᠷ ᠠ ᠣ ᠷᠤᠰᠢ ᠬᠤ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠯ ᠡᠭᠡᠰᠦ ᠴᠡᠭ ᠨᠢ ( ) ᠪᠠᠢᠨ ᠠ᠃ A. 4 B .8 C .12 D. 16 6. ᠵᠢᠷᠤᠭ 3 ᠮᠡᠳᠤ ᠂ ⊙ O ᠶ᠋ ᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ ᠨᠢ 10 ᠂ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲᠥᠪ O ᠡᠴᠡ ᠬᠥᠪᠴ ᠢ AB ᠬᠦᠷ ᠬᠦ ᠵᠠ ᠢ OM ᠶ᠋ᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 4 ᠪᠣᠯ ᠬᠥᠪᠴ ᠢAB ᠶ᠋ᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 3 B . 6 C. 4 D. 8 7. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠤ ᠂ ⊙ O ᠶ᠋ ᠢᠨ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ 6 ᠂ ∆ AB C ᠨᠢ (1) (2) (4) M BA (3) ⊙ O ᠳ᠋ ᠤ ᠪ ᠠᠭᠲᠠᠭ ᠰᠠᠨ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ᠂ OB ᠂ OC ᠵᠢ ᠬ ᠣᠯᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ∠ BA C ᠪᠣᠯᠤᠨ ∠BO C ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠ ᠨ ᠨᠦ᠋ ᠭᠦᠪᠤᠷ ᠢ ᠥᠨᠴᠥ ᠭ ᠪᠣᠯ ᠬᠡᠷ ᠴᠢᠮ BC ᠶ᠋ᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 3 √3 B. 3 C. 6 √3 D. 6 8. ᠵᠢᠷᠤᠭ 5 ᠮᠡᠳᠤ ᠂ ∆AB C ᠳ᠋ᠤ ᠂ AB = AC ᠂ ∠ AB C= 45° ᠂ AB ᠪᠡᠷ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ⊙ O ᠨᠢ BC ᠲᠠᠢ D ᠳ᠋ ᠤ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ BC = 4 ᠪ ᠣᠯ ᠵᠢᠷᠤᠭ ᠲᠤ ᠰᠡᠭᠦᠳᠡ ᠷ ᠲ ᠠᠢ ᠬᠡᠰᠡᠭ ᠤ ᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 𝜋 + 1 B. 𝜋 + 2 C. 2 𝜋+ 2 D. 4 𝜋+ 1 9. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ ᠂ ∆ AB C ᠶ᠋ ᠢᠨ ᠭᠤᠷᠪ ᠠᠨ ᠲᠠᠯ ᠠ ᠪᠣᠯ ᠣᠨ⊙ O ᠶ᠋ ᠢᠨ ᠤᠭᠳᠤ ᠯᠤᠯᠴᠠ ᠭᠠᠳ ᠣᠯᠤᠭ ᠰᠠᠨ ᠭᠤᠷᠪ ᠠᠨ ᠬᠥᠪᠴ ᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠪᠣᠯ ᠳᠣᠣᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠡᠴᠡ ᠵᠥᠪ ᠪᠣᠯᠬ ᠤ ᠨᠢ ( ) A. ᠴᠡᠭ O ᠨᠢ ∆AB C ᠶ᠋ ᠢᠨ ᠳᠤᠳᠤᠭ ᠠᠳᠤ ᠲᠥᠪ B. ᠴᠡᠭ O ᠨᠢ ∆AB C ᠶ᠋ ᠢᠨ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠲᠥᠪ C. ∆AB C ᠪ ᠣᠯ ᠵᠥᠪ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ D. ∆ AB C ᠨᠢ ᠠᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ 10 . ᠵᠢᠷ ᠤᠭ 7 ᠮᠡᠳᠤ ᠂ AB ᠨᠢ ☉O ᠶ᠋ᠢᠨ ᠳ᠋ᠢ ᠶᠠᠮᠧᠲ᠋ ᠷ᠂ C ᠨᠢ ☉O ᠶ᠋ᠢᠨ ᠭᠠᠳᠠ ᠨ ᠠ ᠣᠷᠤᠰ ᠢᠬᠤ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ ᠴᠡ ᠭ C ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ ☉O ᠶ᠋ᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤᠨ ᠢ ᠪ ᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠰᠢᠬ ᠠᠯᠲᠠ ᠶ᠋ ᠢᠨ ᠴᠡᠭ ᠨᠢ B ᠂ AC ᠵᠢ ᠬᠣᠯᠪᠤᠪᠠ ᠯ ☉O ᠲᠠᠢ D ᠳ᠋ᠤ ᠣᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ∠C =3 8°᠂ ᠴᠡᠭ E ᠨᠢ AB ᠶ᠋ᠢᠨ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠳ᠋ᠠᠬ ᠢ ᠬᠠᠭ ᠠᠰ ᠲᠣᠭ ᠤᠷᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠥᠳᠡ ᠯᠬᠦ ᠨᠢ ᠭᠡ ᠴᠡᠭ (A , B ᠲᠠᠢ ᠳᠠ ᠪᠬᠤᠴᠠᠬ ᠤ ᠦᠭᠡ ᠢ) ᠪ ᠣᠯ ∠AED ᠶ᠋ᠢᠨ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠤ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 19 ° B. 38 ° C.52° D.76 ° ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳ᠋ᠤ ᠲᠠᠭ ᠯᠠᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ ︵ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 24 ᠬᠤᠪᠢ ︶ 11 . ᠵᠢᠷ ᠤᠭ 8 ᠮᠡᠳᠤ ᠂ OA ᠂ OB ᠨᠢ ⊙ O ᠶ᠋ ᠢᠨ ᠷᠠᠳᠢᠦ ᠰ᠂ ᠴᠡᠭ C ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷ ᠡ ᠣ ᠷᠤᠰᠢ ᠨ ᠠ᠂ AC ᠂ BC ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ∠AOB = 120° ᠪᠣᠯ ∠ ACB = 12 . ᠵᠢᠷ ᠤᠭ 9 ᠮᠡᠳᠤ ᠂ Rt ∆A BC ᠳ᠋ᠤ ∠A CB = 90° , AC = BC = 2√ 2 ᠬᠡ ᠷᠪᠠ Rt ∆A BC ᠵᠢ AB ᠶ᠋ᠢᠨ ᠣᠷᠤᠰ ᠢᠬᠤ ᠰᠢᠯᠤᠭᠤ ᠨ ᠤ ᠳᠠ ᠭᠠᠤ ᠨᠢᠭᠡ ᠪᠥ ᠳᠦᠨ ᠡᠷᠭᠢ ᠭᠦᠯᠵ ᠦ ᠤ ᠯᠬᠤ ᠭᠸᠣᠮᠧ ᠲ᠋ᠷᠢ ᠶ᠋ ᠢᠨ ᠪᠠᠶ ᠡ ᠶ᠋ ᠢᠨ ᠭᠠᠳᠠ ᠷᠭᠤ ᠶ᠋ ᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ __ __ __ __ _ ( ᠳ᠋ ᠦ᠋ᠩ ᠳ᠋ ᠤ π ᠦᠯ ᠡᠳᠡᠭᠡ ) 13. ᠵᠢᠷ ᠤᠭ 10 ᠮᠡ ᠳᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶ᠋ᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ 𝑥Oy ᠳᠤ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ 2 ᠪᠣᠯᠬᠤ ⊙ P ᠤ ᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲᠥᠪ P ᠤ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢ ᠯ ᠨᠢ (− 3,0 )᠂ ⊙ P ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠤ ᠨ ᠡ ᠶᠡᠷᠭᠦ ᠴᠢᠭ ᠯᠡᠯ ᠳᠠᠭ ᠠᠤ ᠰ ᠧᠺᠦ ᠨᠲ ᠪᠦᠷᠢ 0.5 ᠨᠢ ᠭᠡᠴᠢ ᠪᠡᠷ ᠵᠡᠷᠭᠡᠴᠡᠭᠡ ᠰᠢᠯ ᠵᠢᠬᠦ ᠯᠪᠡᠯ᠂ _ __ __ ᠰᠧᠺᠦ ᠨᠲ ᠶ᠋ᠢᠨ ᠳᠠ ᠷᠠᠭ ᠠ ⊙ P (1) (1) (1) (1) (1) (1) ᠪᠣᠯᠤᠨ y ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ᠃ 14 . ᠵᠢᠷ ᠤᠭ 11 ᠮᠡᠳᠤ ᠂ CD ᠨᠢ ⊙ O ᠶ᠋ ᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ᠂ BD ᠨᠢ ᠬᠥᠪᠴ ᠢ᠂ DC ᠵᠢ A ᠬᠦᠷᠲᠡᠯ ᠡ ᠦᠷ ᠭᠦᠯᠵ ᠢᠯᠡᠭᠦ ᠯᠬᠦ ᠳ᠋ ᠡᠭᠡᠨ ∠AB D = 120° ᠪᠣᠯᠭ ᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ᠨᠢᠭᠡ ᠨᠦ᠋ᠬᠦ ᠴᠠᠯ ᠨᠡᠮᠡᠵ ᠦ AB ᠵᠢ ⊙ O ᠶ᠋ ᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤᠨ ᠪᠤᠯᠭ ᠠᠨᠶ ᠠ ᠬᠡᠪᠠᠯ ᠳᠣᠣᠷ ᠠᠬᠢ ᠳᠦ᠋ᠷ ᠪᠠᠨ ᠨᠥᠬᠦᠴᠡ ᠯ ᠡᠴᠡ ①AC =BC ②AC = OC ③OC = BC ④AB = BD ᠨᠡᠮᠡᠵ ᠤ ᠪᠣᠯᠬ ᠤ ᠨᠢ ( ᠵᠥ ᠪᠬᠡᠨ ᠳ᠋ᠤᠭᠠ ᠷ ᠢ ᠪᠢᠴᠢ ) 15 . ᠵᠢᠷ ᠤᠭ 12 ᠮᠡᠳᠤ ᠂ AB ᠨᠢ ⊙ O ᠶ᠋ ᠢᠨ ᠬᠥᠪ ᠴᠢ᠂ ᠷᠠᠳᠢᠦ ᠰ OC ⊥ AB ᠂ ᠡᠭᠴᠡ ᠶ᠋ ᠢᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨᠢ D ᠂ AB =2 √3 ᠂ ∠ A= 30 ° ᠪᠣᠯ ⊙ O ᠶ᠋ ᠢᠨ ᠳ᠋ ᠢᠶᠠᠮᠧᠲ᠋ ᠷ ᠨᠢ 16 . ᠵᠢᠷ ᠤᠭ 13 ᠮᠡ ᠳᠤ᠂ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ 1 ᠪᠣᠯᠬᠤ ᠬᠠᠭ ᠠᠰ ᠲᠣᠭ ᠤᠷᠢᠭ ᠬᠡᠯᠪᠠ ᠷᠢᠲᠠᠢ ᠴᠠᠭᠠᠰᠤᠨ ᠬᠠᠪ ᠲᠠᠰᠤ ᠵᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠨᠤᠭ ᠤᠯᠪᠠ ᠯ ᠨᠤᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠤ ᠳᠠ ᠷᠠᠭᠠ ᠬᠢ ᠬᠠᠭ ᠠᠰ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠳᠤᠮ ᠳᠠ ᠴᠡᠭ M ᠪᠣᠯ ᠤᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲ ᠥᠪ O ᠨᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠨ ᠠ᠃ ᠵᠢᠷᠤᠭ ᠲᠤ ᠰᠡᠭᠦᠳᠡ ᠷ ᠲᠠ ᠢ ᠬᠡ ᠰᠡᠭ ᠤ ᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ᠭᠤᠷᠪ ᠠ᠂ ᠪ ᠣᠳᠣᠵ ᠤ ᠬ ᠠᠷᠢᠭ ᠤᠯᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ ︵ ᠨᠠᠢᠲᠡ 72 ᠬᠤᠪᠢ ︶ 17 . (6 ᠬᠤᠪᠢ ) ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ᠂ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ A , B , C ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠭᠤᠷᠪ ᠠᠨ ᠴᠡ ᠭ᠃ ∠ 𝐴𝑂𝐵 = 50 °᠂ ∠𝑂𝐵 𝐶 = 40 °᠃ ᠣᠯᠬ ᠤ ᠨᠢ᠄ ∠𝑂 𝐴𝐶 ᠶ᠋ᠢᠨ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠤᠨ ᠲᠣᠭ ᠠ 18 . ( 7 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ⊙ O ᠳ᠋ ᠤ CD ᠨᠢ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ᠂ ᠬᠥ ᠪᠴᠢ AB ᠶ᠋ᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠨᠢ 𝐸 ᠂ CD ᠨᠢ AB ᠵᠢ ᠲᠠᠯᠠ ᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ᠃ OE = 3c m᠂ AB = 8c m ᠪᠣᠯ ⊙ O ᠶ᠋ᠢᠨ ᠷᠠᠳᠢ ᠦᠰ ᠢ ᠣᠯ᠃ (1) (1) (1) D O A B C 19. (7 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ △ABC ᠳ᠋ᠦ AB ᠪᠠᠷ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠬᠠᠭ ᠠᠰ ⊙ O ᠨᠢ AC ᠲᠠᠢ ᠴᠡ ᠭ D ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ BC ᠲᠠᠢ ᠴᠡᠭ E ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ BE = CE ᠂ ∠ C= 70° ᠪᠣᠯ ∠D OE ᠤ ᠨ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠤ ᠨ ᠲᠣᠭ ᠠ ᠵᠢ ᠣᠯ᠃ 20. (8 ᠬᠤᠪᠢ ) ᠨᠢᠭᠡ ᠬᠠ ᠪᠴᠢᠭ ᠰᠠᠨ ᠥᠨᠴᠦ ᠭ ᠨᠢ 12 0° ᠪᠣᠯᠬᠤ ᠬᠡᠷ ᠡᠮ ᠤ ᠨ ᠪ ᠤᠯᠤᠩ ᠳᠤ ᠨᠢᠭᠡᠨ ᠪᠥᠭᠡᠷ ᠡᠩᠬᠡᠢ ᠪᠠᠭᠠᠨ ᠠ ᠪ ᠠᠶ ᠡ ᠵᠢ ᠳᠠ ᠯᠪᠢᠪ ᠠ᠂ ᠬᠠᠵᠠᠭ ᠤ ᠡᠴᠡ ᠤᠯᠵ ᠦ ᠬᠠ ᠷᠠᠭᠰᠠᠨ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷᠤᠭ ᠨᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠬᠠᠷ ᠠᠭᠠᠨ ᠵᠢᠷᠤᠭ ᠲᠤ ᠲᠣᠭᠤᠷ ᠢᠭ ᠪᠣᠯᠤᠨ ᠬᠡᠷ ᠡᠮ ᠤ ᠨ ᠬᠤ ᠶᠠᠷ ᠲᠠᠯ ᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ B ᠂ C ᠳ᠋ ᠤ ᠰ ᠢᠬᠠᠯ ᠴᠠᠨ ᠠ᠃ ( ᠪ ᠥᠭᠡᠷ ᠡᠩᠬᠡᠢ ᠪᠠᠭ ᠠᠨ ᠠ ᠪᠠᠶ ᠡ ᠶ᠋ ᠢᠨ ᠳ᠋ ᠢᠶᠠᠮᠧᠲ᠋ ᠷ ᠢ ᠰᠢᠭ ᠤᠳ ᠬᠡᠮᠵᠢᠵ ᠦ ᠳᠡ ᠢᠯᠬᠦ ᠦᠭᠡᠢ ) (1) ᠲᠤᠰ ᠵᠢᠷᠤᠭ ᠲᠤ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠬᠡᠷ ᠴᠢᠮ ᠢ ᠪᠢᠴ ᠢ᠃ (2) ᠴᠢ ᠪᠣᠳᠤ ᠯᠭ ᠠ ᠪᠠᠷ ᠳᠠᠮᠵᠢᠨ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ ᠢ ᠤᠯᠬ ᠤ ᠨᠢ ᠭᠡ ᠳᠦ᠋ᠰᠤ ᠪ ᠳᠤᠭᠲᠠᠭ ᠠᠵᠦ ᠪᠤᠳᠤᠵ ᠦ ᠦᠵ ᠡ᠃ ( ᠪ ᠤᠳᠤᠬ ᠤ ᠭ ᠣᠣᠯ ᠶᠠᠪᠤᠴᠠ ᠵᠢ ᠪᠢᠴᠢ᠃ ) 21. (1 0 ᠬ ᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠪᠠᠯᠪᠠᠷ ᠬᠠᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠬᠠᠪᠲᠠᠰᠤ ᠵᠢ ᠪ ᠥᠳᠦ ᠨ ᠪᠣᠯ ᠭ ᠠ᠃ ᠨᠤᠮᠤ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠭᠤᠷᠪ ᠠᠨ ᠴᠡ ᠭ A ᠂ B ᠂ C ᠃ (1) B AĈ ᠶ᠋ᠢᠨ ᠣᠷᠤᠰ ᠢᠬᠤ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲᠥᠪ O ᠵᠢ ᠳᠤᠭᠲᠠᠭ ᠠ᠃ ( ᠵᠢᠷᠤ ᠭ ᠵᠢᠷ ᠤᠭᠰ ᠠᠨ ᠣᠷᠤᠮ ᠪ ᠠᠨ ᠦ ᠯᠡᠳᠡᠭᠡ ) (2) ∆ AB C ᠨᠢ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠᠭ ᠤᠲᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ᠂ ᠰᠠᠭ ᠤᠷᠢ BC ＝8cm ᠂ ᠬᠠᠵᠠᠭᠤ AB ＝2√ 5cm ᠪᠣᠯ ᠲᠣᠭᠤᠷ ᠢᠭ ᠤ ᠨ ᠷᠠ ᠳᠢᠦᠰ R ᠵᠢ ᠣᠯ᠃ 22 . ( 10 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ AB ᠨᠢ ⊙ O ᠶ᠋ ᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ᠂ ᠬᠡ ᠷᠴᠢᠮ AD ᠂ BC ᠪᠠ CD ᠨᠢ ⊙ O ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ A ᠂ B ᠪᠠ E ᠴᠡᠭ ᠳ᠋ᠤ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ᠂ DA = 3.6 ᠂ CB = 6.4 ᠃ (1) CO ᠪᠣᠯᠤᠨ OD ᠨᠢ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠡᠭᠴᠡ ᠪᠣᠯᠤᠨ ᠠ ᠤ ᠤ ︖ (2) ⊙ O ᠤ ᠨ ᠷᠠ ᠳᠢᠦᠰ ᠪᠣᠯ ᠤᠨ ᠵᠢᠷᠤᠭ ᠳᠡᠬᠢ ᠰᠡᠭᠦᠳᠡ ᠷ ᠲ ᠠᠢ ᠬᠡ ᠰᠡᠭ ᠤ ᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠵᠢ ᠣ ᠯ᠃ ( 0 .01 ᠬᠦᠷᠲᠡᠯ ᠡ ᠨᠠ ᠷᠢᠪᠴᠢ ᠯᠠ ) 23 . ( 12 ᠬᠤᠪ ᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠰ ᠢᠯᠤᠭᠤᠨ 𝑙 ᠪ ᠣᠯᠤᠨ ⊙ O ᠨᠢ ᠰᠠᠯᠤ ᠩᠭᠢ᠂ OA ⊥ 𝑙 ᠡᠭᠴᠡ ᠶ᠋ ᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ A ᠂ ⊙ O ᠲᠠᠢ P ᠳ᠋ ᠤ ᠣ ᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ OA = 5᠂ AB ᠪᠣᠯ ᠤᠨ⊙ O ᠨᠢ ᠴᠡᠭ B ᠳ᠋ᠤ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ᠂ BP ᠶ᠋ᠢᠨ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠯ ᠱ ᠤᠭᠤᠮ ᠨᠢ ᠰᠢᠯ ᠤᠭᠤᠨ 𝑙 ᠲᠠᠢ C ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠃ (1) ᠪᠠᠲᠤ ᠯᠠ᠄ AB = AC (2) ᠬᠡᠷ ᠪᠠ PC = 2 ᠪ ᠣᠯ ⊙ O ᠶ᠋ ᠢᠨ ᠷᠠ ᠳᠢᠦᠰ ᠢ ᠣᠯ᠃ 24. (1 2ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ABCD ᠨᠢ ᠷᠣᠮᠪ ᠂ ᠳᠠ ᠰᠢᠭᠤᠪ ᠴᠢ AC ᠂ BD ᠨᠢ E ᠳ᠋ ᠤ ᠣ ᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ F ᠨᠢ ᠲᠠ ᠯ ᠠ BA ᠶ᠋ᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ EF ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ EF ᠪᠡᠷ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ⊙ O ᠵᠢ ᠪ ᠠᠢᠭ ᠤᠯᠪᠠ ᠯ DC ᠲᠠᠢ D ᠂ G ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ AD ᠨᠢ EF ᠂ GF ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ I᠂ H ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠲᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ ᠃ (1) ∠ FD E ᠤ ᠨ ᠭᠷ ᠠᠳᠦ᠋ ᠰ ᠤ ᠨ ᠲᠣᠭ ᠠ ᠵᠢ ᠣᠯ᠃ (2) ᠳᠥᠷ ᠪᠡᠯᠵ ᠢᠨ FA CD ᠶ᠋ᠢᠨ ᠬᠡᠯ ᠪᠠᠷᠢ ᠳᠦ᠋ᠷᠰᠦ ᠵᠢ ᠮᠠ ᠭᠠᠳᠯ ᠠᠵᠦ ᠳ᠋ᠦ᠋ᠩ ᠨᠡᠯᠲᠡ ᠪᠠᠨ ᠪᠠᠳᠤᠯ ᠠᠵᠦ ᠦᠵ ᠡ᠃ (3) ᠬᠡᠷ ᠪᠠ G ᠨᠢ ᠬᠡᠷᠴᠢᠮ DC ᠶ᠋ᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠦ ᠶᠡᠰ᠂ ① ᠪᠠᠲᠤᠯ ᠠ᠄ FD =FI ②AC =2m ᠂ BD =2n ᠬᠡ ᠵᠦ ᠵᠢᠱᠢᠪᠠ ᠯ ⊙ O ᠤ ᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠪᠣᠯᠤᠨ ᠷᠣᠮᠪ ABCD ᠤ ᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠶ᠋ᠢᠨ ᠬᠠᠷ ᠢᠴᠠᠯ ᠢ ᠣ ᠯ᠃