ᠠᠷᠪᠠᠨ ᠭᠤᠷᠪᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠲᠡᠩᠭᠡᠯᠢᠭ ᠲᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠠᠷᠪ ᠠᠨ ᠭᠤ ᠷᠪᠠᠳ ᠤᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦ ᠨ ᠪᠠ ᠢᠴᠠ ᠭᠠᠯᠲ ᠠ ᠠᠩᠭᠢ ᠪᠥᠯᠥᠭ᠄ ᠨᠡᠷ ᠡ ᠄ ︵90 ᠮ ᠢᠨᠦ᠋ᠲ ᠂ ᠨᠠᠢ ᠲᠡ 120 ᠬᠤ ᠪᠢ ︶ ᠨᠢᠭᠡ ᠂ ᠰ ᠤᠩᠭᠤ ᠬᠤ ᠰ ᠡᠲᠤᠪ ᠃ ︵ ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 30 ᠬᠤ ᠪᠢ ︶ 1. ᠳᠠᠷᠠᠭᠠᠬ ᠢ ᠭ ᠠᠷ ᠤ ᠳᠠᠰᠤ ᠨ ᠤ ᠨᠢᠭᠤ ᠴᠠ ᠨᠣᠮᠧᠷ ᠢ ᠲᠠ ᠢᠯᠬᠤ ᠵᠢᠷᠤᠭ ᠤᠳ ᠲᠤ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠦ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠤ ᠵᠢ ᠷᠤᠭ ᠪᠤᠯᠬᠤ ᠨᠢ ( ) 2. ᠬᠠᠪᠳ ᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢ ᠷᠢᠰᠢ ᠯ ᠤᠨ ᠰᠢ ᠰᠲ᠋ᠧᠮ ᠲᠡᠭᠡᠷ ᠡ᠂ ᠴᠡᠭ A( 2,3 ) ᠪᠠ ᠴ ᠡᠭ B ᠨᠢ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠬᠤ ᠪᠢ ᠳ ᠤ ᠲ ᠡᠭᠰᠢ ᠬ ᠡᠮᠳᠡᠢ᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠴᠡᠭ B ᠦᠨ ᠪᠠᠢ ᠷᠢᠰᠢ ᠯ ᠨᠢ ( ) A. (− 2,3 ) B. (− 2, −3 ) C. (2 ,− 3) D. (− 3, −2 ) 3. ᠵᠢᠷᠤ ᠭ 1 ᠮᠡᠳ ᠤ᠂ ∆ AB C ᠳᠦ᠂ AB = AC , EF // BC ᠂ ∠A = 40 0᠂ ᠲᠡᠭᠡᠪᠡᠯ ∠A EF ᠦᠨ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠦᠨ ᠲ ᠤᠭ ᠠ ᠨᠢ ( ) A. 40 0 B. 50 0 C. 70 0 D. 140 0 4. ᠵᠢᠷᠤᠭ 2 ᠮ ᠡᠳᠤ᠂ ∆AB C ᠳᠦ ᠂ ᠰ ᠢᠯᠤᠭᠤ ᠨ D E ᠪ ᠤᠯAC ᠤᠨ ᠡᠭᠴᠡ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴ ᠢ ᠪᠦᠭᠡᠳ BC, AC ᠲᠠᠢ ᠲᠤᠰ ᠲᠤ ᠰ ᠴ ᠡᠭ D , E ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ∠B = 60 0, ∠C = 25 0 ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ∠B AD ᠦᠨ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠤᠨ ᠲᠤᠭ ᠠ ᠨ ᠢ ( ) A. 50 0 B. 70 0 C. 75 0 D. 80 0 5. ᠬᠠᠪᠳ ᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠦᠨ ᠴᠦᠭᠳ ᠦ ᠪᠠ ᠢᠷᠢ ᠰᠢᠯ ᠤᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠲᠡᠭᠡᠷ ᠡ᠂ A , B ᠬ ᠤᠶᠠᠷ ᠴᠡᠭ ᠪᠤᠢ᠂ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠡᠭᠡᠷ ᠡ ᠨ ᠢᠭᠡ ᠴᠡᠭ C ᠶᠢ ᠡ ᠷᠢᠬᠦ ᠳᠦ ᠪᠡᠨ A , B ᠬᠤᠶ ᠠᠷ ᠴ ᠡᠭ ᠬᠦᠷᠬ ᠦ ᠵᠠᠢ ᠶᠢ ᠬᠠᠮᠤᠭ ᠤᠨ ᠤᠬᠤ ᠷ ᠪᠠ ᠢᠯᠭᠠ ᠨ ᠠ᠂ ᠵ ᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠲᠦ᠋ ᠷᠪᠡᠨ ᠵᠦ᠋ ᠢᠯ ᠦᠨ ᠲᠦ᠋ ᠰᠦᠯ ᠪᠤᠢ᠂ ᠵ ᠦ᠋ᠪ ᠪ ᠤᠯᠬᠤ ᠨᠢ ( ) 6. ᠵᠢᠷᠤ ᠭ 3 ᠮᠡᠳ ᠤ᠂ ᠨᠢᠭᠡᠨ ᠲᠠᠯ ᠠᠢ ᠶᠢ ᠨ ᠬ ᠠᠢᠭᠤ ᠯ ᠤ ᠩᠭᠤ ᠴᠠ A ᠭᠠᠵᠠᠷ ᠠᠴᠠ ᠮᠤ ᠷᠳᠠᠵᠤ᠂ ᠡᠮᠦᠨ ᠡ ᠡ ᠴᠡ ᠪᠠᠷᠠᠭᠤᠨᠰᠢ 40 0 ᠤ ᠨ ᠴ ᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 40 ᠲ ᠠᠯᠠᠢ ᠶᠢᠨ ᠮᠢᠯ ᠶᠠᠪᠤᠭᠠᠳ B ᠭᠠᠵᠠᠷ ᠬᠦᠷᠪᠡ᠂ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠨ B ᠭ ᠠᠵᠠᠷ ᠠᠴᠠ ᠮᠤᠷᠳ ᠠᠵᠤ᠂ ᠬ ᠤᠢᠨ ᠠ ᠡ ᠴᠡ ᠪᠠᠷᠠᠭᠤᠨᠰᠢ 20 0 ᠤᠨ ᠴᠢᠭᠯ ᠡᠯ ᠢᠶ ᠡᠷ4 0 ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮᠢᠯ ᠶᠠᠪᠤᠭᠠᠳ C ᠭᠠᠵᠠ ᠷ ᠬᠦᠷ ᠪᠡ᠂ ᠲᠡᠭᠡᠪᠡᠯ A , C ᠬᠤᠶ ᠠᠷ ᠭᠠᠵᠠ ᠷ ᠤᠨ ᠬᠤ ᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠨ ᠢ ( ) A. 30 ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮ ᠢᠯ B. 40 ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮ ᠢᠯ C. 50 ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮ ᠢᠯ D. 60 ᠲᠠᠯᠠᠢ ᠶᠢᠨ ᠮ ᠢᠯ 7. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ∠A OB = 30 0 , ∠A OB ᠶᠢᠨ ᠲᠤ ᠳᠤᠨ ᠠ ᠴ ᠡᠭ P ᠪᠤ ᠢ᠂ ᠴ ᠡᠭ 𝑃 ′ ᠪᠠ ᠴᠡᠭ P ᠨᠢ OB ᠶᠢᠨ ᠬᠤ ᠪᠢ ᠳ ᠤ ᠲ ᠡᠭᠰᠢ ᠬᠡᠮᠳ ᠡᠢ᠂ ᠴᠡᠭ 𝑃 ᠌ ᠌ ′′ ᠪᠠ ᠴᠡᠭ P ᠨᠢ O A ᠶᠢᠨ ᠬᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮ ᠳᠡᠢ᠂ ᠲᠡᠭᠡᠪᠡᠯ O , 𝑃 ′ , 𝑃 ᠌ ᠌ ′′ ᠭᠤᠷᠪᠠ ᠨ ᠴ ᠡᠭ ᠢ ᠶᠡᠷ ᠤ ᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠪᠤᠯᠭᠠᠭᠰᠠᠨ (1) (2) (3) ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠨᠢ ( ) A. ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ B. ᠠᠳ ᠠᠯᠢ ᠬᠠᠵᠠ ᠭᠤᠳ ᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ C. ᠲᠡᠭ ᠰᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ D. ᠲᠤ ᠭᠳᠠᠭᠠᠵ ᠤ ᠴ ᠢᠳᠠᠬᠤ ᠦᠭᠡᠢ 8. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠤ᠂ ∆AB C ᠳᠦ᠂ AB = AC ,∠C = 70 0, ∆AB ′C ′ ᠪᠠ ∆ AB C ᠨ ᠢ ᠰ ᠢᠯᠤᠭᠤ ᠨ EF ᠤ ᠨ ᠬ ᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠬ ᠡᠮᠳᠡᠢ᠂ ∠C AF = 10 0, BB ′ ᠶᠢ ᠬᠤᠯ ᠪᠤᠨ ᠠ ᠂ ᠲᠡᠭᠡᠪᠡᠯ ∠AB B ′ ᠦᠨ ᠭᠷ ᠠᠳᠦ᠋ ᠰ ᠦᠨ ᠲᠤᠭ ᠠ ᠨ ᠢ ( ) A. 30 0 B. 35 0 C. 40 0 D. 45 0 9. ᠵᠢᠷᠤᠭ 5 ᠮᠡᠳᠤ᠂ ∆AB C ᠳᠦ ᠂ ∠A CB = 90 0, ∠B = 30 0 ᠂ AD ᠨᠢ ∠C AB ᠶᠢ ᠲᠠᠯᠠᠯ ᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ ᠂ DE ⊥ AB ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨ ᠢ E , C E ᠶ ᠢ ᠬ ᠤᠯᠪᠤ ᠪᠠᠯ᠂ AD ᠲᠠᠢ ᠴᠡᠭ H ᠳᠦ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠵᠢᠷᠤᠭ ᠳᠠᠬ ᠢ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠ ᠭᠤᠳ ᠤ ᠭ ᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠤ ᠪ ᠤᠳᠠᠭᠠᠯ ᠢ ᠲᠤᠭ ᠠ ᠨ ᠢ ( ) A. 5 B. 4 C. 3 D. 2 10. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ᠂ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ∆AB C ᠳᠤ᠂ BD ᠨᠢ ∠A BC ᠶᠢ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠬᠤ ᠪᠦᠭᠡᠳ AC ᠲ ᠠᠢ ᠴᠡᠭ D ᠳᠤ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ D ᠶᠢ ᠲᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨ DE ⊥ BC ᠪᠡᠷ ᠪᠠ ᠢᠭᠤ ᠯᠪᠠᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭᠤ ᠷᠢ ᠨᠢ E ᠪᠦᠭᠡᠳ CE = 2᠂ ᠲᠡᠭᠡᠪᠡᠯ B C ᠶ ᠢᠨ ᠤᠷ ᠳᠤ ᠨ ᠢ ( ) A. 4 B. 6 C. 8 D. 10 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠳ ᠤ ᠲ ᠠᠭᠯᠠᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ︵ᠰᠡᠲ ᠤᠪ ᠪ ᠦᠷᠢ 3 ᠬᠤᠪᠢ ᠂ ᠨ ᠠᠢᠲ ᠡ 24 ᠬᠤ ᠪᠢ ︶ 11. ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ ᠲ ᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠨᠢ ᠬ ᠠᠮᠤᠭ ᠤᠨ ᠴᠦ᠋ᠬ ᠡᠨ ᠪ ᠤᠯᠬᠤ ᠵᠢᠷᠤᠭ ᠨᠢ __ ___ ( ᠨᠣ ᠮᠧᠷ ᠢ ᠨᠢ ᠪᠢᠴ ᠢᠪᠡᠯ ᠪᠤᠯ ᠴᠢᠬᠠᠨ ᠠ ) ᠃ 12. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ P( 𝑎+ 1,2 𝑎− 4) ᠶᠢ ᠨ 𝒳 ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠬᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠤ ᠴ ᠡᠭ ᠨ ᠢ ᠨᠢ ᠭᠡᠳᠤ ᠭᠡᠷ ᠮᠦ᠋ᠴ ᠡ ᠳ ᠦ ᠤ ᠷᠤᠰᠢ ᠨ ᠠ᠂ ᠲᠡᠭᠡᠪᠡᠯ 𝑎 ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪᠴ ᠢᠶ ᠡ ᠨᠢ _____᠃ 13. ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤ ᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠤ ᠨ ᠢᠭᠡ ᠭᠠ ᠳᠠᠭᠠ ᠳᠤ ᠦᠨᠴ ᠦᠭ ᠨ ᠢ 110 0᠂ ᠲᠡᠭᠡᠪᠡᠯ ᠲᠡᠭᠦ ᠨ ᠦ ᠨᠢᠭᠡ ᠰᠠᠭᠤ ᠷᠢ ᠦ ᠨᠴᠦᠭ ᠨᠢ_______ _ 14. Rt ∆AB C ᠳᠦ᠂ ∠A = 30 0 , ∠ 𝐶 = 90 0, AB + BC = 12 cm ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ AB =____c m 15. ᠵᠢᠷᠤᠭ 8 ᠮ ᠡᠳᠤ᠂ ᠲᠡ ᠩᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ∆AB C ᠳᠤ᠂ ᠴᠡᠭ D , E ᠨᠢ ᠲᠤ ᠰ ᠲ ᠤᠰ BC, AC ᠲ ᠡᠭᠡᠷ ᠡ ᠪᠤᠢ᠂ DE // BA , ᠴ ᠡᠭ E ᠶ ᠢ ᠲ ᠠᠭᠠᠷᠢ ᠭᠤᠯᠤ ᠨ EF ⊥ DE ᠪᠠ ᠷ ᠪᠠᠢ ᠭᠤᠯᠪ ᠠᠯ᠂ BC ᠶ ᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲᠠᠢ ᠴᠡᠭ F ᠳᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ ᠲᠡᠭᠡᠪᠡ ᠯ ∠D FE = ___ ___ 16. ᠵᠢᠷᠤᠭ 9 ᠮ ᠡᠳᠤ᠂ ᠭᠡᠷᠡᠯ ᠳᠤ A ᠭᠠᠵᠠ ᠷ ᠠᠴ ᠠ ᠭ ᠠᠷᠴᠤ ᠂ ᠬ ᠤᠢᠨ ᠠ ᠠᠴ ᠠ (8) (9) (4) (5) (6) ᠵᠡᠭᠦᠨᠰ ᠢ 60 0 ᠤ ᠨ ᠴ ᠢᠭᠯᠡᠯ ᠳᠡᠬ ᠢ C ᠭᠠᠵ ᠠᠷ ᠬ ᠦᠷᠬᠦ ᠴᠢ ᠬᠤᠯᠠᠳ ᠠᠢ᠂ ᠲᠡᠷᠡ ᠤᠷᠢ ᠳᠠᠪᠠ ᠷ ᠵᠡᠭᠦᠨᠰ ᠢ 200 m ᠶᠠᠪᠤᠭᠠᠳ B ᠭᠠᠵᠠᠷ ᠬᠦᠷᠴᠤ ᠂ ᠦᠷᠭ ᠦᠯᠵᠢ ᠯᠡᠨ ᠬᠤᠢ ᠨ ᠠ ᠠᠴᠠ ᠵᠡᠭᠦ ᠨᠰᠢ 30 0 ᠤ ᠨ ᠴ ᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠶᠠᠪᠤ ᠭᠠᠳ ᠶᠠᠭ ᠰᠠᠢ ᠬᠠᠨ C ᠭᠠᠵᠠ ᠷ ᠬᠦᠷ ᠪᠡ᠂ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ ᠲᠡᠭᠡᠪᠡᠯ B , C ᠬ ᠤᠶᠠᠷ ᠭᠠᠵᠠ ᠷ ᠤᠨ ᠬᠤ ᠭᠤᠷᠤᠨᠳ ᠤᠬᠢ ᠵᠠᠢ ᠨᠢ ______ m 17. ᠵᠢ ᠷᠤᠭ 10 ᠮᠡᠳ ᠤ᠂ AD ᠪᠤᠯ ᠠᠳᠠᠯ ᠢ ᠬ ᠠᠵᠠᠭᠤ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ AB C ᠶ ᠢᠨ ᠵᠢᠰᠢ ᠬᠦᠦ ᠲᠠᠯ ᠠ ᠲ ᠡᠭᠡᠷᠡᠬ ᠢ ᠲᠤᠮ ᠳᠠ ᠱᠤᠭᠣ ᠮ᠂ P ᠪ ᠤᠯ DA ᠶᠢᠨ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠣᠭᠤ ᠮ ᠲ ᠡᠭᠡᠷᠡᠬ ᠢ ᠨ ᠢᠭᠡ ᠴᠡᠭ ᠂ ∠PB A = ___ ᠦᠶ ᠡᠰ᠂ ∆PB C ᠨ ᠢ ᠲ ᠡᠩᠴ ᠡᠬᠦᠦ ᠲᠠᠯ ᠠᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠃ 18. ᠵᠢᠷᠤᠭ 11 ᠮᠡᠳ ᠤ᠂ ∠B OC = 10 0᠂ ᠴᠡᠭ A ᠨᠢ OB ᠲᠡᠭᠡᠷ ᠡ ᠪᠤ ᠢ᠂ O A= 1᠂ ᠲᠠᠷᠠᠭᠠ ᠬᠢ ᠱ ᠠᠭᠠᠷᠳ ᠠᠯᠭ ᠠ ᠪᠡᠷ ᠵᠢᠷᠤᠨ ᠠ᠄ ᠴᠡᠭ A ᠪᠠᠷ ᠲᠤᠭᠤ ᠷᠢᠭ ᠤᠨ ᠷᠠᠳᠢ ᠤᠰ ᠪᠤᠯᠭᠠᠨ ᠨᠤ ᠮᠤ ᠵᠢᠷᠤᠪᠠ ᠯ᠂ OC ᠲ ᠠᠢ ᠴᠡᠭ A1 ᠳᠤ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠵᠤ᠂ ᠨᠢᠭᠡᠳ ᠤᠭᠡᠷ ᠵᠤᠷᠪ ᠤᠰ ᠬᠡᠷᠴᠢ ᠮ A A1 ᠶᠢ ᠣᠯᠣ ᠨ ᠠ᠃ ᠲᠠᠬᠢ ᠨ ᠴ ᠡᠭ A1 ᠪᠠᠷ ᠲᠤ ᠭᠤᠷᠢ ᠭ ᠤᠨ ᠲ ᠥᠪ ᠪᠤᠯ ᠭᠠᠨ ᠂ 1 ᠪᠠᠷ ᠷᠠᠳᠢ ᠦᠰ ᠪ ᠤᠯᠭᠠᠨ ᠨᠤᠮᠤ ᠵ ᠢᠷᠤᠪᠠ ᠯ᠂ OB ᠲᠠᠢ ᠴᠡ ᠭ A 2 ᠳ ᠤ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠵᠤ᠂ ᠬᠤ ᠶᠠᠳᠤ ᠭᠠᠷ ᠵᠤ ᠷᠪᠤᠰ ᠬ ᠡᠷᠴᠢ ᠮ A1 A2 ᠶᠢ ᠣᠯᠣ ᠨ ᠠ ᠁ ᠡᠨᠡ ᠮᠡᠳᠤ ᠪᠠᠷ ᠦᠷᠬ ᠦᠯᠵᠢ ᠯᠡᠨ ᠵᠢᠷᠤ ᠵᠤ 𝑛 ᠳ ᠦᠭᠡᠷ ᠵᠤᠷᠪᠤ ᠰ ᠬᠡᠷᠴ ᠢᠮ ᠢ ᠤᠯ ᠪᠠᠯ᠂ ᠲᠠᠬ ᠢᠭᠠᠳ ᠥᠭᠡᠷ ᠡ ᠭᠡᠷᠴ ᠢᠮ ᠵᠢᠷᠤᠵᠤ ᠤ ᠯᠵᠤ ᠲᠡᠢ ᠯᠬᠦ ᠦᠭᠡᠢ ᠪᠤᠯ ᠤᠨ ᠠ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ n =_____ ᠭᠤᠷᠪᠠ᠂ ᠪᠣᠳ ᠣᠵᠤ ᠬᠠᠷᠢ ᠭᠤᠯᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ (ᠨᠠᠢ ᠲᠡ 66 ᠬᠤ ᠪᠢ) 19. (8 ᠬᠤ ᠪᠢ) ᠨᠢᠭᠡᠨ ᠰᠢᠨᠵᠢ ᠯᠡᠭᠡ ᠮᠡᠷᠭᠡᠵᠢ ᠯ ᠦᠨ ᠺᠤ ᠮᠫᠠᠨᠢ ᠨ ᠢ ᠮᠠᠰ ᠢᠨ ᠲ ᠡᠷᠭᠡᠨ ᠦ ᠲ ᠦ᠋ᠷᠢ ᠮ ᠵ ᠦ᠋ᠷᠢ ᠴᠡᠨ ᠶᠠᠪᠤᠬᠤ ᠶᠢ ᠰ ᠡᠭᠦᠳ ᠡᠷᠯᠡᠬ ᠦ ᠲ ᠦ᠋ᠬᠦ ᠭᠡᠷᠦᠮᠵ ᠢ ᠪᠦ ᠳᠦᠭᠡᠵᠦ ᠭᠠᠷᠭᠠ ᠪᠠ᠃ ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ᠂ ᠭᠤᠷᠪᠠ ᠨ ᠵ ᠤᠷᠪᠤ ᠰ ᠵ ᠠᠰᠠᠮᠠᠯ ᠵᠠᠮ ᠨᠢ ᠬ ᠤᠶᠠᠷ ᠬᠤᠶ ᠠᠷ ᠢᠶ ᠠᠷ ᠢᠶ ᠠᠨ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠨ ᠠ ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ ᠡᠨᠡ ᠲᠦ᠋ ᠬᠦᠭᠡᠷᠦ ᠮᠵᠢ ᠶᠢ ᠶ ᠠᠮᠠᠷ ᠪᠠᠢ ᠷᠢᠨ ᠳᠤ ᠤ ᠭᠰᠠᠷᠠᠪᠠ ᠯ ᠭᠤᠷᠪᠠ ᠨ ᠵᠠᠮ ᠬᠦᠷ ᠬᠦ ᠵᠠᠢ ᠶᠢ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠪᠠ ᠢᠯᠭᠠᠵᠤ ᠴᠢᠳ ᠠᠬᠤ ᠪᠤᠢ ︖ 20. (8 ᠬᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠳᠤ᠂ ᠵᠢᠵᠢ ᠭ ᠲ ᠡᠭᠰᠢ ᠲᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠲ ᠠᠯᠠᠪᠠ ᠢ ᠨᠢ ᠨᠢᠭᠡ ᠪᠤᠯ ᠬᠤ ᠬᠠᠨᠠᠯ ᠵᠢ ᠵᠢᠷᠤᠭ ᠪᠤᠢ᠂ ∆AB C ᠦᠨ ᠭᠤᠷᠪᠠ ᠨ ᠤ ᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠨᠢ ᠲᠦ᠋ ᠷᠪᠡᠯ ᠵᠢᠨ ᠦ ᠤ ᠷᠤᠢ ᠶᠢᠨ ᠴᠡᠭ ᠲᠡᠭᠡᠷ ᠡ ᠪᠤ ᠢ᠂ ᠨᠢᠭᠡᠨ ᠬᠠᠪᠳ ᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢ ᠷᠢᠰᠢ ᠯ ᠤᠨ ᠰᠢ ᠰᠲ᠋ᠧᠮ 𝒳𝒪𝒴 ᠶᠢ ᠪᠠᠢ ᠭᠤᠯᠬᠤ ᠳᠤ ᠪᠠ ᠨ᠂ ᠴᠡᠭ A ᠦᠨ ᠪᠠ ᠢᠷᠢ ᠰᠢᠯ ᠢ ( 1,4 ) ᠪᠠ ᠢᠯᠭᠠᠨ ᠠ ᠂ ∆AB C ᠪᠠ ∆ A ′B ′C ′ ᠨ ᠢ y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠦᠨ ᠬᠤᠪᠢ ᠳᠤ ᠲᠡᠭᠰ ᠢ ᠬᠡᠮᠳ ᠡᠢ᠃ (1) ᠬᠠᠪᠳ ᠠᠭᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠳᠦ ᠪᠠᠢ ᠷᠢᠰᠢ ᠯ ᠤᠨ ᠰᠢ ᠰᠲ᠋ᠧᠮ 𝒳𝒪𝒴 ᠪᠠ ∆ A ′B ′C ′ ᠶᠢ (10) ( 11) ᠵᠢᠷᠤ ᠃ (2) y ᠲᠡᠩ ᠬᠡᠯᠢᠭ ᠲᠡᠭᠡ ᠷ ᠡ ᠨᠢᠭᠡ ᠴᠡᠭ P ᠶ ᠢ ᠡ ᠷᠢᠵᠦ᠂ PC + PB ᠦᠨ ᠬᠡᠮᠵ ᠢᠭᠳᠡᠯ ᠢ ᠬ ᠠᠮᠤᠭ ᠪᠠᠭ ᠠ ᠪᠠᠢ ᠯᠭ ᠠ ᠂ ᠪᠠ ᠰᠠ ᠴᠡᠭ P ᠶ ᠢᠨ ᠪᠠ ᠢᠷᠢ ᠰᠢᠯ ᠢ ᠪᠢ ᠴᠢ᠃ 21. (10 ᠬᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠲ ᠦ᠋ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠳᠦ᠂ AB // CD , BC // AD , ∆AB C ᠪᠠ ∆ AE C ᠨ ᠢ ᠲ ᠠᠰᠢᠭᠤ ᠪᠴᠢ AC ᠶᠢᠨ ᠬᠤ ᠪᠢ ᠳᠡᠬᠢ ᠲᠡᠭ ᠰᠢ ᠬᠡᠮᠳ ᠤ ᠵ ᠢᠷᠤᠭ ᠲᠦ᠋ ᠷᠰᠦ᠂ EC ᠪᠠ AD ᠨᠢ ᠴᠡᠭ F ᠳᠤ ᠤᠭᠳ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ ∆A CF ᠨᠢ ᠶᠠᠮᠠᠷ ᠬᠡᠯᠪᠡᠷᠢ ᠲᠦ᠋ ᠷᠰᠦ ᠲᠠᠢ ᠪᠤᠯᠬᠤ ᠶᠢ ᠮ ᠠᠭᠠᠳ ᠯᠠᠵᠤ᠂ ᠰᠢᠯᠳ ᠠᠭᠠᠨ ᠢ ᠨ ᠢ ᠲ ᠤᠳᠤᠷ ᠬᠠᠢ ᠯᠠ᠃ 22. (10 ᠬᠤᠪ ᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠴ ᠡᠭ D , E ᠨᠢ ∆ AB C ᠶ ᠢᠨ ᠲ ᠠᠯ ᠠ BC ᠲᠡᠭᠡᠷ ᠡ ᠪᠤ ᠢ᠂ AD = AE , BD = CE ᠪᠠ ᠳᠤᠯᠠ᠄ ∠B = ∠C 23. (10 ᠬᠤᠪᠢ ) ᠵᠢ ᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ ∆ AB C ᠪᠤᠯ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠂ D , E , F ᠨᠢ ᠲ ᠤᠰ ᠲᠤᠰ BC , AC , AB ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠴ ᠡᠭ᠂ DE ⊥ AC , EF ⊥ AB , FD ⊥ BC ᠃ ᠪᠠᠳᠤ ᠯᠠ᠄ ∆D EF ᠪᠤᠯ ᠲ ᠡᠩᠴ ᠡᠭᠦᠦ ᠲᠠᠯᠠᠳ ᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠃ 24. (10 ᠬᠤ ᠪᠢ) ᠵᠢᠷᠤᠭ ᠮᠡᠳ ᠤ᠂ ᠲ ᠦ᠋ᠷᠪᠡᠯ ᠵᠢᠨ AB CD ᠳᠦ᠂ ∠B = 90 0 , DE // AB ᠪᠦᠭᠡᠳ B C ᠲᠠᠢ ᠴᠡᠭ E ᠳᠦ ᠤ ᠭᠳᠤ ᠯᠤᠯᠴ ᠠᠵᠤ᠂ AC ᠲᠠᠢ ᠴᠡᠭ F ᠳᠦ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠂ ∠C DE = ∠A CB = 30 0᠃ (1) ᠪᠠ ᠳᠤᠯᠠ᠄ ∆FC D ᠪᠤᠯ ᠠᠳᠠᠯ ᠢ ᠬᠠᠵᠠᠭᠤ ᠳᠤ ᠭᠤᠷᠪᠠ ᠯᠵᠢᠨ ᠃ (2) ᠬᠡᠷᠪᠡ BC = DE ᠪᠤ ᠯ᠂∠C AD ᠶᠢᠨ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠤᠨ ᠲᠤᠭ ᠠ ᠶ ᠢ ᠤᠯ᠃ 25. (10 ᠬᠤᠪᠢ ) ᠦᠭ ᠭᠦᠭᠳ ᠡᠭᠰᠡᠨ ᠨᠢ ᠄ ∆ AB C ᠪᠤᠯ ᠲᠡᠩ ᠴᠡᠭᠦ ᠦ ᠲ ᠠᠯᠠᠳᠤ ᠭᠤ ᠷᠪᠠᠯ ᠵᠢᠨ ᠂ ᠵᠢ ᠷᠤᠭ ① ᠮᠡᠳ ᠤ᠂ ᠴ ᠡᠭ M ᠪᠤᠯ ᠲ ᠠᠯ ᠠ BC ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠨ ᠢᠭᠡ ᠴ ᠡᠭ᠂ ᠴᠡᠭ N ᠪᠤᠯ ᠲᠠᠯ ᠠ C A ᠲᠡᠭᠡᠷᠡᠬ ᠢ ᠨ ᠢᠭᠡ ᠴᠡᠭ᠂ BM = CN , B N ᠪᠠ AM ᠨᠢ ᠴ ᠡᠭ Q ᠳᠤ ᠤᠭ ᠳᠤᠯᠤ ᠯᠴᠠᠨ ᠠ᠃ (1) ᠵᠢ ᠷᠤᠭ ① ᠳ ᠡᠬᠢ ∠A QN ᠤ ᠭᠷᠠᠳ ᠦ᠋ᠰ ᠤᠨ ᠲᠤᠭ ᠠ ᠶᠢ ᠲᠠᠭᠠᠮᠠᠭᠯ ᠠᠬᠤ ᠶᠢᠨ ᠬ ᠠᠮᠳᠤ ᠰᠢᠯᠳ ᠠᠭᠠᠨ ᠢ ᠲ ᠤᠳᠤᠷᠬ ᠠᠢᠯᠠ᠃ (2) ᠬᠡᠷᠪᠡ M , N ᠬᠤᠶ ᠠᠷ ᠴ ᠡᠭ ᠨ ᠢ ᠲ ᠤᠰ ᠲᠤᠰ ᠬᠡᠷᠴ ᠢᠮ BC , C A ᠶᠢᠨ ᠦ ᠷᠬᠦᠯᠵᠢ ᠯᠡᠯ ᠱᠤᠭᠤ ᠮ ᠲ ᠡᠭᠡᠷᠡᠬ ᠢ ᠴ ᠡᠭ᠂ ᠪᠤᠰᠤ ᠳ ᠨᠦ᠋ᠬ ᠦᠴᠡᠯ ᠬᠤᠪᠢ ᠷᠠᠬᠤ ᠦᠭᠡᠢ ᠂ ᠵᠢ ᠷᠤᠭ ② ᠮᠡᠳ ᠤ᠂ ᠲ ᠡᠭᠡᠪᠡᠯ (1 ) ᠳ ᠡᠬᠢ ᠳᠦ᠋ᠩ ᠨᠡᠯᠳᠡ ᠮ ᠦ᠋ᠨ ᠪᠠᠢ ᠭᠤᠯᠤ ᠭᠳᠠᠨ ᠠ ᠤᠤ ︖ ᠬᠡᠷᠪᠡ ᠪᠠᠢ ᠭᠤᠯᠤ ᠭᠳᠠᠪᠠ ᠯ ᠪᠠ ᠳᠤᠯᠠ ᠂ ᠪᠠ ᠢᠭᠤ ᠯᠤᠭᠳ ᠠᠬᠤ ᠦᠭᠡᠢ ᠪᠤᠯ ᠰᠢᠯᠳ ᠠᠭᠠᠨ ᠢ ᠨ ᠢ ᠲ ᠤᠳᠤᠷᠬ ᠠᠢᠯᠠ᠃