ᠬᠤᠷᠢᠨ ᠲᠦᠷᠪᠡᠳᠤᠭᠡᠷ ᠪᠦᠯᠦᠭ ᠲᠤᠭᠤᠷᠢᠭ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠬᠣᠷ ᠢᠨ ᠳᠥ ᠷᠪᠡᠳᠦᠭᠡᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠪᠠᠢᠴ ᠠᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵90 ᠮᠢᠨᠦ᠋ ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ ᠤᠪᠢ ︶ ᠨᠢᠭ ᠡ᠂ ᠰᠤᠩᠭ ᠤᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ ᠰᠡ ᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ ︶ 1. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠳᠤ ᠂ ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ AB ⊥CD ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡ ᠭ E ᠂ ᠳᠣᠣᠷ ᠠᠬᠢ ᠳ᠋ᠦ᠋ᠩᠨᠡ ᠯᠲᠡ ᠡᠴᠡ ᠪᠤᠷᠤᠭ ᠤ ᠪ ᠣᠯᠬᠤ ᠨ ᠢ ( ) A. CE =D E B . A E=O E C. 𝐵𝐶̂ = 𝐵𝐷̂ D . ∆ OCE ≌ ∆ODE 2. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠳᠤ ᠂ A , B , C ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠴᠡᠭ ᠂ 𝐴𝐶𝐵̂ ᠨᠢ ᠤᠷ ᠳᠤ ᠨᠤᠮ ᠤ᠂ ᠳᠣᠣᠷ ᠠᠬᠢ ᠡᠴᠡ ∠AOB ᠲᠠᠢ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠪᠣᠯᠬ ᠤ ᠨᠢ ( ) A. 2∠ C B. 4∠ B C. 4∠ A D. ∠ B+ ∠ C 3. ᠵᠢᠷᠤᠭ 3 ᠮᠡ ᠳᠤ᠂ PA ᠂ PB ᠂ DE ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ⊙ O ᠲᠠᠢ ᠴᠡᠭ A ᠂ B ᠂ C ᠳ᠋ᠤ ᠰᠢᠬ ᠠᠯᠴᠠᠨ ᠠ ᠃ DE ᠨᠢ PA ᠂ PB ᠲᠠᠢ ᠴᠡᠭ D ᠂ E ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠃ P ᠡᠴᠡ ᠳᠡ ᠲᠠᠭᠰᠠᠨ ⊙ O ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤᠨ ᠤ ᠤᠷᠳᠤ ᠨᠢ 8c m ᠪᠣᠯ ∆PDE ᠶᠢᠨ ᠬᠦᠷᠢᠶᠡᠨ ᠤ ᠤᠷ ᠳᠤ ᠨᠢ ( ) A. 16 cm B. 14 cm C. 12 cm D . 8 cm 4. ᠬᠡᠷᠪᠠ ᠳᠡᠭ ᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠢ ᠪᠠᠭᠲᠠ ᠭᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠷᠠ ᠳᠢᠦᠰ ᠨᠢ 2 ᠪᠣᠯ ᠳᠤᠳᠤᠭ ᠠᠳᠤ ᠡᠴᠡ ᠰᠢᠬ ᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠷᠠᠳᠢ ᠦᠰ ᠨᠢ ( ) A. √ 2 B. 2√ 2 C. √22 D .1 5. ᠨᠢᠭᠡ ᠨᠥᠭᠡᠨ ᠵᠠᠮ ᠤ ᠨ ᠬ ᠦᠨᠳᠡᠯᠡ ᠨ ᠣ ᠭᠲᠤᠯ ᠤᠭᠠᠰᠤ ᠨᠢ ᠵᠢᠷ ᠤᠭ 4 ᠮᠡ ᠳᠤ᠂ ᠲᠡᠭᠦᠨ ᠤ ᠬᠡ ᠯᠪᠠᠷ ᠢ ᠳᠦ᠋ᠷᠰᠤ ᠨ ᠢ ᠴᠡ ᠭ O ᠪᠠᠷ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠲ ᠥᠪ ᠪᠣᠯᠭ ᠠᠨ 5 ᠪᠠᠷ ᠷᠠ ᠳᠢᠦᠰ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤ ᠨ ᠨᠢ ᠭᠡ ᠬᠡᠰᠡᠭ ᠪᠣᠯ ᠤᠨ ᠠ᠂ ᠴᠡᠭ M ᠨᠢ ⊙ O ᠶᠢᠨ ᠬᠥᠪᠴ ᠢ CD ᠶᠢᠨ ᠳᠤᠮ ᠳᠠ ᠴᠡᠭ ᠂ EM ᠨᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥᠪ O ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠬ ᠤ ᠮᠥᠷᠲᠡ ᠭᠡᠨ ⊙ O ᠲᠠᠢ ᠴᠡᠭ E ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ CD = 6 ᠪ ᠣᠯ ᠨᠥᠭᠡᠨ ᠵᠠᠮ ᠤ ᠨ ᠦᠨᠳᠤᠷ ( ME ᠶᠢᠨ ᠤᠷᠳᠤ ) ᠨᠢ ( ) A. 4 B. 6 C. 8 D. 9 6. ᠵᠢᠷᠤᠭ 5 ᠮᠡᠳᠤ ᠂ ⊙ O ᠶᠢ ᠨ ᠬᠥ ᠪᠴᠢ CD ᠪᠣᠯ ᠤᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ AB ᠨᠢ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ∠A CD = 35° ᠪᠣᠯ ∠B AD = ( ) A. 55 ° B. 40 ° C . 3 5° D .30 ° 7. ᠲᠤᠤᠷᠠ ᠭᠢ ᠥᠭ ᠦᠯᠡᠯ ᠡᠴᠡ ᠬᠠᠭ ᠤᠷᠮᠠᠭ ᠥ ᠭᠦᠯᠡᠯ ᠪᠣᠯ ᠬᠤ ᠨᠢ ( ) A. ᠲᠠᠪᠤᠨ ᠥᠨᠴᠦᠭᠲᠦ ᠶᠢᠨ ᠭᠠᠳᠠ ᠭᠠᠳᠤ ᠥᠨᠴᠦ ᠭ ᠤᠨ ᠨᠡᠢ ᠯᠡᠪᠤᠷᠢ ᠨᠢ 36 0° B. ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯ ᠤᠭᠤᠨ ᠨᠢ ᠰᠢᠬ ᠠᠯᠲᠠ ᠶᠢᠨ ᠴᠡᠭ ᠢ ᠳᠠ ᠭᠠᠷᠢᠭᠰᠠᠨ ᠷᠠᠳᠢᠦ ᠰ ᠲ ᠤ ᠡᠭ ᠴᠡ C. ᠴᠡᠭ (3 ,− 2) ᠤᠨ y ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠤᠨ ᠬᠤᠪᠢ ᠳ᠋ᠤ ᠳᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠴᠡᠭ ᠨᠢ (− 3,2 ) D. ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ y = 𝑥 2 − 4𝑥 + 2017 ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠤ ᠨ (4) (1) (2) (3) (5) ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠨᠢ ᠰᠢ ᠯᠤᠭᠤᠨ 𝑥 = 2 8. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ ᠂ ᠴᠡ ᠭ A ᠂ B ᠂ C ᠂ D ᠨᠢ ᠴᠥᠮ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ 2 ᠪᠣᠯᠬ ᠤ ⊙ O ᠳᠡᠭᠡᠷ ᠠ ᠣᠷᠤᠰ ᠢᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ OA ⊥BC ᠂ ∠ CD A=3 0° ᠪᠣᠯ ᠬᠥᠪᠴ ᠢ BC ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) A. 4 B. 2√ 2 C . √ 3 D . 2 √3 9. ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ CD = 10 cm ᠂ AB ᠨᠢ ⊙ O ᠶᠢᠨ ᠬᠥᠪ ᠴᠢ᠂ AB = 8c m ᠮᠥᠷᠲᠡ ᠭᠡᠨ A B ⊥ CD ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ M ᠪᠣᠯ AC ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) A. 2 √5 cm B. 4 √5 cm C. 2 √5 cm ᠪᠤᠶ ᠤ 4 √5 cm D . 2 √3 cm ᠪᠤᠶ ᠤ 4 √3 cm 10 . ᠵᠢᠷ ᠤᠭ 7 ᠮᠡ ᠳᠤ᠂ AB ᠨᠢ ⊙ O ᠤᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ ᠪᠣᠯ ᠳᠣᠣᠷ ᠠᠬᠢ ᠨᠥᠬᠦᠴᠡ ᠯ ᠤᠳ ᠡᠴ ᠡ ᠰᠢᠯᠤᠭᠤᠨ AT ᠨᠢ ⊙ O ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢ ᠯᠤᠭᠤᠨ ᠪᠣᠯᠬ ᠤ ᠵᠢ ᠮᠠᠭ ᠠᠳᠯᠠ ᠵᠦ ᠳᠡ ᠢᠯᠬᠦ ᠦᠭᠡᠢ ᠨᠢ ( ) A. AB = 4᠂ AT = 3᠂ BT = 5 B. ∠ B= 45 °᠂ AB = AT C. ∠ B= 55 °᠂ ∠ TA C=5 5° D . ∠ AT C= ∠ B ᠬᠤᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠲᠠᠭ ᠯᠠᠬᠤ ᠰᠡᠳ ᠤᠪ᠃ (ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠ ᠢᠲᠡ 18 ᠬᠤᠪᠢ ) 11 . ᠵᠢᠷ ᠤᠭ 8 ᠮᠡ ᠳᠤ᠂ AB ᠨᠢ ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ᠂ C ᠂ D ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠬᠤᠶ ᠠᠷ ᠴᠡᠭ ᠂ CD ⊥AB ᠃ ᠬᠡᠷᠪᠡ ∠ DAB = 65° ᠂ ᠪᠣᠯ ∠A OC =_ __ __ __ __ __ 12 . ⊙ O ᠳᠤᠳᠤᠷ ᠠ ᠤᠷ ᠳᠤ ᠨᠢ 24 ᠪᠣᠯᠬᠤ ᠬᠥᠪ ᠴᠢ ᠶᠢᠨ ᠬᠥᠪ ᠴᠢ ᠲ ᠥᠪ ᠤᠨ ᠵᠠᠢ ᠨᠢ 5 ᠪᠣᠯ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ __ __ __ __ __ _ 13 . ⊙ O ᠤᠨ ᠷᠠᠳᠢᠦ ᠰ r= 5 cm ᠂ ᠴᠡᠭ P ᠨᠢ ᠰᠢᠯᠤᠭᠤᠨ 𝑙 ᠳᠡ ᠭᠡᠷ ᠠ ᠣᠷᠤᠰ ᠢᠨ ᠠ᠂ ᠬᠡᠷᠪᠡ OP = 5 cm ᠪᠣᠯ ᠰᠢᠯᠤᠭᠤᠨ 𝑙 ᠪᠣᠯᠤ ᠨ ⊙ O ᠤ ᠨ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ ᠨᠢ_ __ __ __ __ __ __ 14 . ᠵᠢᠷ ᠤᠭ 9 ᠮᠡᠳᠤ ᠂ ⊙ O ᠳ᠋ᠤ ᠷᠠᠳᠢᠦ ᠰ OA = √2 ᠂ ᠬᠥᠪᠴ ᠢ AB = 2 ᠂ ∠B AD = 18° ᠂ OD ᠪᠣᠯ ᠤᠨ AB ᠨᠢ ᠴᠡᠭ C ᠳ᠋ᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ ᠃ ∠A CO =_ __ __ __ __ _᠃ 15 . ᠵᠢᠷ ᠤᠭ 10 ᠮᠡᠳᠤ ᠂ ᠵᠡ ᠭᠦᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠵᠢᠷ ᠤᠭ ᠪᠣᠯ ᠤᠯᠠᠨ ᠠᠷ ᠪᠢᠨ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ᠡᠴᠡ ᠪᠦᠷᠢᠯ ᠳᠦᠭᠰᠡ ᠨ ᠨᠢ ᠭᠡ ᠰᠠᠢᠬᠠᠨ ᠬᠡ ᠵᠢᠷ ᠤᠭ ᠤᠨ ᠨᠢᠭᠡ ᠬᠡᠰᠡᠭ ᠪᠣᠯ ᠤᠨ ᠠ᠃ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ᠲᠤ ᠷᠠᠳᠢᠦ ᠰ OA = 2𝑐𝑚 ᠂ ∠A OB = 120° ᠪᠣᠯ ᠪᠠᠷᠠᠭ ᠤᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠵᠢᠷᠤᠭ ᠤᠨ ᠬᠦᠷᠢᠶ ᠡᠨ ᠤ ᠤᠷ ᠳᠤ ᠨᠢ __ __ __ __ cm ( ᠳ᠋ ᠦ᠋ᠩ ᠳ᠋ᠤ π ᠦᠯ ᠡᠳᠡᠭᠡ ) 16 . ᠵᠢᠷ ᠤᠭ 11 ᠮᠡ ᠳᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰᠢᠰᠲ᠋ ᠧᠮ ᠳ᠋ᠤ ᠴᠡᠭ A (1 ,0) ᠂ B( 1− 𝑎 , 0) ᠂ 𝐶( 1+ 𝑎 ,0 ) ( 𝑎 0) ᠪᠠᠢᠨ ᠠ᠂ ᠴᠡᠭ P ᠨᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠨᠢ 𝐷( 4,4 ) ᠷᠠ ᠳᠢᠦᠰ ᠨᠢ 1 ᠪᠣᠯᠬ ᠤ ᠲᠣᠭ ᠣᠷᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠬᠥᠳᠡ ᠯᠵᠦ ᠪᠣᠯᠬᠤ ᠨᠢᠭᠡ ᠴᠡᠭ ∠ BP C= 90° ᠪᠣᠯ ᠬᠤ ᠨᠥᠬᠦᠴᠡ ᠯ ᠢ ᠬᠠᠩᠭ ᠠᠬᠤ ᠪᠣᠯ 𝑎 ᠶᠢᠨ ᠬᠠᠮᠤᠭ ᠤᠨ ᠡᠡᠬᠡ C O A B D (6) (7) (8) (10) (11) (9) ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠨᠢ __ __ __ ᠭᠤᠷᠪ ᠠ᠂ ᠱᠠᠭ ᠠᠷᠳᠠᠯ ᠭ ᠠ ᠡᠣᠰᠣ ᠭᠠᠷ ᠪᠠᠶᠠᠯ ᠡᠭᠦᠯᠭ ᠦ ᠰᠡᠳ ᠤᠪ᠃ ( 9 ᠪᠠᠭ ᠠ ᠰᠡᠲᠤᠪ ᠂ ᠰᠡᠳᠤᠪ ᠪᠦᠷᠢ 8 ᠬᠤᠪᠢ᠂ ᠨᠠ ᠢᠲᠡ 72 ᠬᠤᠪᠢ ) 17. (8 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡ ᠳᠤ᠂ ᠳᠡᠭᠰ ᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢ ᠨ AB CD ᠶᠢᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳ ᠤ ᠨᠢ 1᠂ ᠴᠡᠭ A ᠪᠠᠷ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ 1 ᠪᠡᠷ ᠷᠠᠳᠢᠦ ᠰ ᠪ ᠣᠯᠭᠠᠭᠰᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠪᠣᠯᠤᠨ ᠰᠢᠯᠤᠭᠤᠨ BC ᠶᠢᠨ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠬᠣᠯ ᠪᠤᠭᠳᠠᠯ ᠨᠢ ᠡᠠᠮᠠ ᠷ ᠪᠣᠯ ᠬᠤ ᠪᠤᠢ ︖ ᠴᠡᠭ A ᠪᠠᠷ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ ᠬᠡᠳᠤᠢ ᠦ ᠶᠡᠰ ᠲᠣᠭ ᠤᠷᠢᠭ ᠪ ᠣᠯᠤᠨ ᠰᠢᠯ ᠤᠭᠤᠨ BD ᠨᠢ ᠰᠢᠬ ᠠᠯᠴᠠᠬ ᠤ ᠪᠤᠢ ︖ 1 8. (8 ᠬᠤᠪᠢ) ᠥᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ᠄ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ⊙ O ᠳ᠋ᠤ ᠬᠥᠪ ᠴᠢ AB ᠪᠠ CD ᠨᠢ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ AC ᠂ BD ᠵᠢ ᠬ ᠣᠯᠪᠤ ᠨ ᠠ᠂ AC = BD ᠃ ᠪᠠᠲᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ AB = CD 19. (1 0 ᠬᠤᠪᠢ) ᠬᠡᠷ ᠪᠠ ᠨᠢᠭᠡ ᠬᠥᠮᠦᠨ ᠤ ᠲᠣᠯ ᠤᠭᠠᠢ ᠶᠢᠨ ᠣᠷᠤᠢ ᠪᠠ ᠬᠥᠯ ᠤᠨ ᠤᠯᠠ ᠵᠢ ᠲᠤᠰ ᠲᠤᠰ ᠨᠢᠭᠡ ᠴᠡᠭ ᠪᠡᠷ ᠦᠵ ᠡᠵᠦ᠂ ᠳᠡᠯᠡᠬ ᠡᠢ ᠶᠢᠨ ᠪᠥᠮᠪᠦᠷ ᠴᠡᠭ ᠶᠢᠨ ᠡᠷᠭᠢ ᠯᠳᠠ ᠶᠢᠨ ᠵᠠᠮ ᠱᠤᠭᠤᠮ ᠢ ᠨᠢ ᠭᠡ ᠲᠣᠭ ᠤᠷᠢᠭ ᠢᠶᠡ ᠷ ᠦᠵ ᠡᠪᠠᠯ ᠪᠠᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠨᠢ 1.5 𝑚 ᠶᠢᠨ ᠪᠠᠳᠤ ᠳᠡ ᠯᠡᠬᠡᠢ ᠶᠢᠨ ᠪᠥᠮᠪᠦᠷ ᠴᠡᠭ ᠤ ᠨ ᠡᠷᠭᠢᠯ ᠲᠡ ᠶᠢᠨ ᠵᠠᠮ ᠱᠤᠭᠤᠮ ᠢ ᠳᠠ ᠭᠠᠵᠦ ᠨᠢᠭᠡ ᠪᠦᠳᠤᠨ ᠡᠷ ᠭᠢᠭᠦ ᠳ᠋ᠤ ⑴ᠳᠡᠭᠦᠨ ᠤ ᠲᠣᠯ ᠤᠭᠠᠢ ᠶᠢᠨ ᠣᠷᠤᠢ ᠨᠢ ᠬᠥᠯ ᠤᠨ ᠤᠯᠠ ᠡᠴᠡ ᠪᠠᠨ ᠬᠡᠳᠤᠢ ᠢᠯ ᠡᠭᠦᠦ ᠡᠠᠪᠤᠭᠰᠠᠨ ᠢ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠮᠠ ᠭᠠᠳᠯ ᠠᠵᠦ ᠳᠡᠭᠢᠨ ᠪᠤᠳᠤ ᠵᠦ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠣᠷ ᠬᠠᠢ ᠯᠠ᠃ ⑵ ᠬᠡ ᠷᠪᠠ ᠪᠠᠳᠤ ᠨᠢ ᠭᠡᠨ ᠷᠠᠳᠢᠦ ᠰ ᠨᠢ 1 𝑘𝑚 ᠤᠨ ᠣᠳᠤᠨ ᠭᠠᠷ ᠠᠭ ᠲᠤ ᠣ ᠳᠤᠨ ᠭᠠᠷᠠᠭ ᠤᠨ ᠡᠷᠭᠢᠯ ᠲᠡ ᠶᠢᠨ ᠵᠠᠮ ᠱᠤᠭᠤᠮ ᠵᠢ ᠳᠠᠭ ᠠᠵᠦ ᠨᠢᠭᠡ ᠪᠦᠲ ᠦᠨ ᠡᠷᠭᠢᠭ ᠦ ᠳ᠋ᠤ ᠲᠡᠭᠦᠨ ᠤ ᠲᠣᠯ ᠤᠭᠠᠢ ᠶᠢᠨ ᠣᠷᠤ ᠢ ᠨᠢ ᠬᠥᠯ ᠤᠨ ᠤᠯᠠ ᠡᠴᠡ ᠪᠠᠨ ᠪᠠᠰᠠ ᠬᠡᠳᠤᠢ ᠢᠯᠡᠭᠦᠦ ᠡᠠᠪ ᠤᠭᠰ ᠠᠨ ᠪ ᠤᠢ ︖ ⑶ ᠷᠠ ᠳᠢᠦᠰ ᠨᠢ 10 𝑘𝑚 ᠤᠨ ᠣᠳᠤᠨ ᠭᠠᠷ ᠠᠭ ᠲᠤ ᠡᠠᠮᠠ ᠷ ᠪᠣᠯᠬᠤ ᠪᠤᠢ ︖ 20 . (8 ᠬᠤᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ AB ᠨᠢ ᠬᠠᠭ ᠠᠰ ᠲᠣᠭ ᠤᠷᠢᠭ O ᠶᠢᠨ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ᠂ ᠴᠡ ᠭ C , D ᠨᠢ ᠬᠠᠭ ᠠᠰ ᠲᠣᠭ ᠤᠷᠢᠭ O ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠬᠤᠶ ᠠᠷ ᠴᠡ ᠭ ᠮᠥ ᠷᠲᠡᠭᠡᠨ OD ∥ BC ᠂ OD ᠪᠣᠯ ᠤᠨAC ᠨᠢ ᠴᠡᠭ E ᠳ᠋ ᠤ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠃ (1) ᠬᠡᠷ ᠪᠠ ∠B=7 0° ᠪᠣᠯ ∠CA D ᠤ ᠨ ᠭᠷ ᠠᠳᠦ᠋ ᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠬᠡᠳᠤᠢ ᠪᠤᠢ ︖ (2) ᠬᠡᠷ ᠪᠠ AB = 4᠂ AC = 3 ᠪ ᠣᠯ᠂ DE ᠶᠢᠨ ᠤᠷᠳᠤ ᠵᠢ ᠣ ᠯ᠃ 21 . (8 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ∆ AB C ᠳ᠋ᠦ ᠂ AB = AC ᠂ ᠴᠡ ᠭ B ᠂ C ᠪᠠᠷ ᠲᠤᠰ ᠲᠤᠰ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ ᠪᠣ ᠯᠭᠠᠨ BC ᠶᠢᠨ ᠤᠷᠳᠤ ᠪᠠᠷ ᠷᠠᠳᠢᠦ ᠰ ᠪᠣ ᠯᠭᠠᠨ BC ᠶᠢᠨ ᠳᠣᠣᠷ ᠠ ᠲᠠ ᠯ ᠠ ᠳ᠋ ᠤ ᠨᠤᠮᠤ ᠵᠢᠷᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ᠬᠤᠶ ᠠᠷ ᠨᠤᠮ ᠤ ᠨᠢ ᠴᠡᠭ D ᠳ᠋ᠤ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠵ ᠦ᠂ AB ᠂ AC ᠶᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ ᠴᠡᠭ E᠂ F ᠳ᠋ ᠤ ᠣ ᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ AD ᠂ BD ᠂ CD ᠵᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠲᠤ ᠯᠠᠭᠳᠠ ᠬᠤ ᠨᠢ ᠄ AD ᠨᠢ∠ BA C ᠵᠢ ᠲᠠᠯᠠ ᠯᠠᠨ ᠬᠤᠪᠢᠶ ᠠᠨ ᠠ᠃ (2) ᠬᠡᠷ ᠪᠠ BC = 6 ᠂ ∠ BA C ＝50 ° ᠪ ᠣᠯ ᠨᠤᠮᠤ DE ᠂ ᠨᠤᠮ ᠤ DF ᠶᠢᠨ ᠤᠷᠳᠤ ᠶᠢᠨ ᠨᠡᠢᠯ ᠡᠪᠤᠷᠢ ᠵᠢ ᠣᠯ᠃ ( ᠳ᠋ ᠦ᠋ᠩ ᠳ᠋ᠤ π ᠠ ᠭᠤᠯᠤᠭᠰ ᠠᠨ ᠢ ᠯᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢᠶᠡᠷ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡ᠃ ) FE D CB A 22. (1 0 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ᠂ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 1 ᠪᠣ ᠯᠬᠤ ᠵᠢᠵᠢᠭ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠪᠦᠷᠢᠯᠳ ᠦᠭᠰᠡᠨ 8× 4 ᠤᠷ ᠳᠤ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠲᠥ ᠷᠪᠡᠯ ᠵᠢᠭᠡ ᠰᠤ ᠳ᠋ ᠤ ᠵᠢᠵᠢᠭ ᠳᠡᠭ ᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠣ ᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠢ ᠣ ᠭᠲᠤᠯ ᠤᠯᠴᠠ ᠯ ᠤᠨ ᠴᠡᠭ ᠭᠡᠨ ᠡ᠂ ᠴᠡᠭ A ᠂ B ᠂ C ᠂ D ᠨᠢ ᠴᠥᠮ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠯ ᠤᠨ ᠴᠡᠭ ᠂ ᠴᠡᠭ D ᠵᠢ ᠳᠣᠣᠷ ᠠᠬᠢ ᠮᠡᠳᠤ ᠠᠯ ᠬᠤᠮ ᠳᠠ ᠷᠠᠭ ᠠ ᠪᠠᠷ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠤᠨ ᠡ᠃ (1) ᠴᠡᠭ D ᠵᠢ ᠴᠡᠭ A ᠵᠢ ᠲᠣᠭᠤᠷ ᠢᠭᠤᠯ ᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ 18 0° ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ ᠴᠡ ᠭ D 1 ᠵᠢ ᠣᠯ ᠤᠨ ᠠ᠃ (2 ) ᠴᠡᠭ D 1 ᠵᠢ ᠴᠡᠭ B ᠵᠢ ᠲᠣᠭᠤ ᠷᠢᠭᠤᠯ ᠤᠨ ᠴᠠᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ 90 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ᠴᠡᠭ D 2 ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠃ (3 ) ᠴᠡᠭ D 2 ᠵᠢ ᠴᠡᠭ C ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ 90 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ᠳᠡᠭᠢ ᠨ ᠴᠡ ᠭ D ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠃ (1) ᠭᠤᠷᠳᠢᠭ ᠬᠡᠷᠡᠭᠯᠡᠨ D → D 1→ D 2→ D ᠶᠢᠨ ᠡᠠᠪᠤᠵ ᠦ ᠥᠩᠭᠡᠷ ᠡᠭᠰᠡᠨ ᠵᠠᠮ ᠱᠤᠭᠤᠮ ᠢ ᠵᠢᠷᠤ᠃ (2 ) ᠵᠢᠷ ᠤᠭᠠᠳ ᠣᠯᠤᠭᠰ ᠠᠨ ᠵᠢᠷᠤᠭ ᠳᠦ᠋ᠷᠰᠤ ᠨ ᠢ ᠳᠡ ᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠵᠢᠷ ᠤᠭ ᠳᠦ᠋ᠷ ᠰᠤ ᠪᠣᠯᠤᠨ ᠠ᠃ (3) ᠣᠯᠤᠭ ᠰᠠᠨ ᠵᠢᠷ ᠤᠭ ᠳᠦ᠋ᠷᠰᠤ ᠶ ᠢᠨ ᠬᠦ ᠷᠢᠶᠡᠨ ᠤ ᠤᠷ ᠳᠤ ᠵᠢ ᠤᠯ ᠃ ( ᠳ᠋ᠦ᠋ ᠩ ᠳ᠋ᠤ π ᠦᠯ ᠡᠳᠡᠭᠡ ) 23 . (1 0 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ AB = BC ᠂ AB ᠪᠡᠷ ᠳ᠋ᠢᠶ ᠠᠮᠧᠲ᠋ ᠷ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ⊙ O ᠨᠢ AC ᠲᠠᠢ ᠴᠡᠭ D ᠳ᠋ᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ D ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ DE ⊥BC ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡ ᠭ E ᠃ (1) ᠪᠠᠲᠤ ᠯᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ DE ᠨᠢ⊙ O ᠶᠢᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯᠤᠭᠤᠨ (2) DG ⊥AB ᠪᠡᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ⊙ O ᠲᠠᠢ G ᠳ᠋ᠤ ᠤᠭᠲᠤᠯᠤᠯ ᠴᠠᠵᠦ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ F᠂ ᠬᠡᠷ ᠪᠠ ∠A =3 0° ᠂ AB = 8 ᠪ ᠣᠯ DG ᠶᠢᠨ ᠤᠷᠳᠤ ᠵᠢ ᠤᠯ ᠃ 24 . (1 0 ᠬᠤᠪᠢ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ AB ᠨᠢ ⊙ O ᠶᠢᠨ ᠳ᠋ᠢᠶᠠᠮ ᠧᠲ᠋ᠷ᠂ ᠴᠡᠭ C ᠨᠢ ⊙ O ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ ᠨᠤᠮ ᠤ BC ᠵᠢ ᠰᠢᠯᠤᠭᠤᠨ BC ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠨᠤᠭ ᠤᠯᠪᠠ ᠯ ᠨᠤᠮ ᠤ BC ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ D ᠨᠢ ᠡᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠨ ᠲᠥ ᠪ O ᠲᠠᠢ ᠳᠠ ᠪᠬᠤᠴ ᠠᠨ ᠠ᠂ OC , CD , BD ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠴᠡᠭ C ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ ᠰᠢᠬ ᠠᠭᠴᠢ ᠰᠢᠯ ᠤᠭᠤᠨ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ᠬᠡ ᠷᠴᠢᠮ BA ᠶᠢᠨ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠯ ᠱᠤᠭᠤᠮ ᠴᠡ ᠭ P ᠳ᠋ᠤ ᠣ ᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ AD ᠵᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ PB ᠶᠢᠨ ᠨᠥᠭᠦ ᠭᠡ ᠲᠠᠯ ᠠ ᠳ᠋ ᠤ ∠MPB = ∠ADC ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠃ (1) PM ᠪᠣ ᠯᠤᠨ⊙ O ᠶᠢᠨ ᠪ ᠠᠢᠷᠢ ᠶᠢᠨ ᠬᠣᠯ ᠪᠤᠭᠳᠠᠯ ᠢ ᠮᠠ ᠭᠠᠳᠯ ᠠᠵᠦ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠳᠤᠳᠤᠷ ᠬᠠᠢᠯᠠ᠃ (2) ᠬᠡᠷ ᠪᠠ PC = √3 ᠪᠣᠯ ᠳᠥᠷᠪ ᠡᠯᠵᠢ ᠨ O CD B ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠵᠢ ᠣᠯ ᠃