ᠬᠤᠷᠢᠨ ᠭᠤᠷᠪᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠡᠷᠬᠢᠯᠳᠡ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 2） .pdf

ᠬᠣᠷ ᠢᠨ ᠭᠤᠷᠪ ᠠᠳᠤᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠪᠠᠢ ᠴᠠᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵12 0 ᠮᠢ ᠨᠦ᠋ᠲ᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬᠤᠪᠢ ︶ ᠨᠢᠭ ᠡ᠂ ᠰᠤᠩᠭ ᠤᠬᠤ ᠰᠡᠲᠤ ᠪ᠃ ( ᠰ ᠡᠳᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠡᠢᠲᠡ 30 ᠬᠤᠪ ᠢ) 1. ᠳᠣᠣᠷ ᠠᠬᠢ ᠲᠡ ᠷᠭᠡᠨ ᠲᠡᠮᠳᠡ ᠭ ᠤᠨ ᠵᠢᠷᠤᠭ ᠳᠤᠳᠤᠷ ᠠ ᠲᠥᠪ ᠲᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠵᠢᠷ ᠤᠭ ᠳᠦ᠋ᠷᠰᠤ ᠪᠣᠯᠬ ᠤ ᠨᠢ ( ) A. B. C. D. 2. ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠦᠭᠲ ᠦ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠴᠡ ᠭ P( －3, m 2+1 ) ᠤᠨ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ ᠤᠨ ᠬᠤᠪ ᠢ ᠳᠤ ᠳᠡᠭᠰ ᠢ ᠬᠡᠮᠳᠤ ᠴᠡᠭ ᠨᠢ ( ) ᠣᠷᠤᠰ ᠢᠨ ᠠ᠃ A. ᠨᠢ ᠭᠡᠳᠤᠭᠡᠷ ᠮᠥᠴᠡ B. ᠬᠣᠶ ᠠᠳᠤᠭ ᠠᠷ ᠮ ᠥᠴᠡ C. ᠭᠤᠷᠪ ᠠᠳᠤᠭ ᠠᠷ ᠮᠥᠴᠡ D. ᠳᠥᠷᠪ ᠡᠳᠦᠭᠡᠷ ᠮᠥ ᠴᠡ 3. ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠰᠢᠰ ᠲ᠋ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠴᠡᠭ A (1 ,√ 3) ᠵᠢ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ ᠢᠶᠡᠷ ᠲᠥᠪ ᠪᠣᠯᠭ ᠠᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢ ᠭᠯᠡᠯ ᠢᠶᠡᠷ 30° ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ ᠴᠡᠭ A ′ ᠢ ᠣᠯᠤ ᠨ ᠠ᠂ ᠴᠡᠭ A ′ ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (√ 3, 1) B. (√ 3, −1 ) C. (2 ,1) D. (0 ,2) 4. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠳᠤ ᠂ ᠴᠡ ᠭ E ᠪᠣᠯ ᠳᠡᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ AB CD ᠶᠢᠨ ᠲᠠᠯ ᠠ DC ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ Δ ADE ᠵᠢ ᠴᠡᠭ A ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡ ᠷᠭᠢᠭᠦᠯ ᠪᠠᠯ ΔAB F ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠂ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ A ECF ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 20 ᠂ DE = 2 ᠪ ᠣᠯ AE ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) A. 4 B. 2√ 5 C. 6 D. 2√ 6 5.ᠳᠣᠣᠷ ᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠳᠦ᠋ᠷᠰᠤ ᠳᠤᠳᠤᠷ ᠠ ᠶᠠᠮᠠ ᠷ ᠨᠢ ᠭᠡᠨ ᠴᠡᠭ ᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ 18 0° ᠡᠷᠭᠢ ᠭᠦᠯᠪᠠ ᠯ ᠥᠪᠡᠷ ᠤᠨ ᠲᠤᠰ ᠪᠠᠶ ᠡ ᠲᠠᠢ ᠪᠠᠨ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠵᠢᠷᠤᠭ ᠳᠦ᠋ᠷᠰᠤ ᠶ ᠢᠨ ᠪᠣᠳᠠᠭ ᠠᠯᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) ᠪᠠᠢᠨ ᠠ᠃ (1 ) ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢ ᠨ ( 2) ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ (3 ) ᠳᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠦ (4) ᠳᠡᠭ ᠰᠢ ᠥᠨᠴ ᠦᠭ (5) ᠵᠢᠰᠢᠭ ᠦᠯᠵᠢᠨ A. 5 B. 4 C. 3 D. 2 6. ᠵᠢᠷᠤᠭ 2 ᠮᠡᠳᠤ ᠂ ᠵᠢᠰᠢ ᠭᠦᠯᠵ ᠢᠨA BCD ᠳᠤ AE ⊥BC ᠂ ᠡᠭ ᠴᠡ ᠶᠢ ᠨ ᠰ ᠠᠭᠤᠷᠢ ᠨ ᠢ E ᠂ ᠴᠡᠭ B ᠪᠠᠷ ᠡᠷ ᠭᠢᠯᠲᠡ ᠶᠢᠨ ᠲᠥᠪ ᠪᠣ ᠯᠭᠠᠨ ᠡᠷᠭᠢᠯ ᠲᠡ ᠶᠢ ᠨ ᠥᠨᠴᠦ ᠭ ᠢ ∠A BC ᠲᠠᠢ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠪᠠᠷ ∆BA E ᠵᠢ ᠴᠠᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ ∆ BA ′ E ′ᠵᠢ ᠣᠯᠤᠨ ᠠ᠂ DA ′ ᠵᠢ ᠬᠣᠯᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ∠ADC = 60 °᠂ ∠AD A ′ = 50° ᠪᠣᠯ ∠D A ′E ′= ( ) A. 13 0° B. 15 0° C. 16 0° D. 17 0° 7.ᠵᠢᠷ ᠤᠭ 3 ᠮᠡ ᠳᠤ᠂ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 1 ᠪ ᠣᠯᠬᠤ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ A BCD ᠨᠢ ᠴᠡ ᠭ A ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠨ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 45° ᠡᠷ ᠭᠢᠪᠠᠯ ᠳᠡ ᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ (1) (2) (3) AB 1C 1D 1 ᠵᠢ ᠣᠯᠤᠨ ᠠ᠂ ᠲᠠᠯ ᠠ B 1C 1 ᠪ ᠣᠯᠤᠨ CD ᠨᠢ ᠴᠡ ᠭ O ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ AB 1OD ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ( ) A. 34 B. √ 2− 1 2 C. √ 2 − 1 D. 1 + √2 8. ᠵᠢᠷᠤᠭ 4 ᠮᠡᠳᠤ ᠂ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ AB CD ᠳᠤ AB = 2√ 5 ᠂ ᠴᠡ ᠭ O ᠨᠢ BC ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠂ ᠴ ᠡᠭ E ᠨᠢ ᠳᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠳᠤᠳᠤᠷ ᠠᠭᠢ ᠨᠢᠭᠡ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡᠭ ᠂ OE = 2 ᠂ DE ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠴᠢᠮ DE ᠨᠢ ᠴᠡᠭ D ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90° ᠡᠷᠭᠢᠪᠠᠯ ᠬᠡ ᠷᠴᠢᠮ DF ᠵᠢ ᠣᠯᠤᠨ ᠠ᠂ AE ᠂ CF ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠬᠡ ᠷᠴᠢᠮ OF ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠶᠢ ᠨ ᠬᠠᠮ ᠤᠭ ᠤ ᠨ ᠪ ᠠᠭ ᠠ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠨᠢ ( ) A. 2√ 5 B √5 + 2 C. 2√ 10 - 2 D. 5√ 2- 2 9. ᠵᠢᠷᠤᠭ 5 ᠮᠡ ᠳᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ 𝑥Oy ᠳᠣᠲ ᠤᠷ ᠠ ᠰᠢᠯᠤᠭᠤ ᠨ y = √3 𝑥 ᠨᠢ ᠨᠢᠭᠡᠳᠦᠭᠡᠷ ᠮᠥᠴᠡ ᠳᠤᠳᠤᠷ ᠠᠭᠢ ᠴᠡᠭ A ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠬ ᠤ ᠮᠥᠷᠲᠡ ᠭᠡᠨ OA ＝4᠂ ᠴᠡᠭ A ᠵᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ AB ⊥ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠢᠶᠡᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ᠡᠭ ᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡ ᠭ B ᠂ ∆ AB O ᠵᠢ ᠴᠡᠭ B ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭᠦᠦ ᠶᠢ ᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡ ᠷ 60 ° ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ ∆ CB D ᠵᠢ ᠣ ᠯᠤᠨ ᠠ᠂ ᠴᠡᠭ C ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (− √ 3 ,2) B. (− √3 ,1 ) C. (− 2, √3 ) D. (− 1, √3 ) 10. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠳᠤ ᠂ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠳᠡᠭᠰᠢ ᠳᠥᠷ ᠪᠡᠯᠵ ᠢᠨO ABC ᠵᠢ ᠴᠡᠭ O ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠤᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 45 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ᠳᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢ ᠨ OA 1B 1C 1 ᠵᠢ ᠣᠯᠤᠨ ᠠ᠂ ᠡᠨᠡ ᠮᠡᠳᠤ ᠪᠠᠷ ᠴᠡᠭ O ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠵᠠᠯ ᠭᠠᠭ ᠠ ᠪ ᠠᠷ 2 01 8 ᠤᠳᠠ ᠭ ᠠ ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷ ᠪᠠᠯᠵ ᠢᠨ OA 2018 B 2018 C 2018 ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ ᠴᠡᠭ A ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (1, 0) ᠪᠣᠯ ᠴᠡᠭ B 2018 ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( ) A. (1 ,1) B. (0 ,√ 2) C. (− √2 ,0 ) D. (− 1,1) ᠬᠣᠶ ᠠᠷ᠂ ᠬᠠᠭ ᠠᠯᠲᠠ ᠳᠤ ᠲᠠᠭ ᠯᠠᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ (ᠰ ᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠ ᠢᠲᠡ 18 ᠬᠤᠪᠢ ) 11 . ᠵᠢᠷ ᠤᠭ 7 ᠮᠡᠳᠤ ᠂ ᠳᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷ ᠪᠠᠯᠵ ᠢᠨ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠭᠡᠰᠤ ᠳᠤᠳᠤᠷ ᠠ Δ AB C ᠨᠢ ᠶᠠᠮᠠ ᠷ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ ᠥᠨᠴᠦ ᠭ 𝑎 (0 𝑎 18 0° ) ᠡᠷ ᠭᠢᠪᠠᠯ Δ A1 B1 C1 ᠵᠢ ᠣᠯᠤᠨ ᠠ᠂ ᠴᠡᠭ A ᠪᠣᠯᠤ ᠨ ᠴᠡ ᠭ A 1 ᠂ ᠴᠡᠭ B ᠪᠣᠯᠤᠨ B1 ᠂ ᠴᠡᠭ C ᠪᠣᠯ ᠤᠨ ᠴᠡᠭ C 1 ᠨᠢ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡᠭ ᠪᠣᠯ 𝑎= ___ ___ _ 12. ᠵᠢᠷ ᠤᠭ 8 ᠮᠡᠳᠤ ᠂ ᠲᠡ ᠩᠴᠡᠭᠦᠦ ᠲᠠ ᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ABC ᠶᠢᠨ ᠳᠤᠳᠤᠷ ᠠ ᠨᠢ ᠭᠡ ᠴᠡᠭ P ᠪᠠᠢᠨ ᠠ᠂ AP ᠂ BP ᠂ CP ᠵᠢ ᠲᠤᠰ ᠲᠤᠰ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ AP = 6 ᠂ BP = 8 ᠂ CP = 10 ᠪᠣᠯ (4) (5) (6) (7) (8) S△ AB P + S△ BP C ＝___ ___ 13. ᠵᠢᠷ ᠤᠭ 9 ᠮᠡ ᠳᠤ᠂ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ Rt ΔA BC ᠶᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭ ᠤ ᠨ ᠣ ᠷᠤᠢ ᠶᠢ ᠨ ᠴᠡ ᠭ C ᠶᠢ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠨᠢ (1 ,0) ᠂ ᠴᠡᠭ A ᠨᠢ x ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠤ ᠨ ᠡᠶᠡᠷᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡ ᠤᠷ ᠤᠰᠢᠬ ᠤ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ AC = 2 . Δ AB C ᠵᠢ ᠤᠷᠢᠳᠠ ᠪᠠᠷ ᠴᠡᠭ C ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶ ᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90 ∘ ᠡᠷ ᠭᠢᠭᠦᠯ ᠵᠦ ᠳᠡᠭᠢᠨ ᠵᠡᠭᠦᠨᠰᠢ 3 ᠨᠢ ᠭᠡᠴᠢ ᠵᠡᠷ ᠭᠡᠴᠡᠭᠡ ᠰᠢᠯᠵ ᠢᠭᠦᠯᠪᠠ ᠯ ᠴᠡ ᠭ A ᠶᠢᠨ ᠬᠤᠪᠢᠷ ᠠᠭᠠᠯᠲ ᠠ ᠬᠢᠭᠰ ᠡᠨ ᠤ ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡᠭ ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ_ ___ ___ 14. ᠵᠢᠷ ᠤᠭ 10 ᠮᠡ ᠳᠤ᠂ Rt △ABC ᠳᠤ ∠A CB =9 0°᠂ AC =5 cm ᠂ BC =1 2cm ᠂ △ABC ᠵᠢ ᠴᠡᠭ B ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠳᠠ ᠭᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 6 0° ᠡᠷᠭᠢ ᠭᠦᠯᠪᠠ ᠯ ∆ BD E ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠂ DC ᠵᠢ ᠬᠣᠯ ᠪᠤᠪᠠᠯ AB ᠲᠠ ᠢ ᠴᠡᠭ F ᠲᠠ ᠢ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠᠨ ᠠ᠂ ∆A CF ᠪᠣᠯᠤᠨ ∆B DF ᠤ ᠬᠦ ᠷᠢᠶᠠᠨ ᠤ ᠤᠷ ᠳᠤ ᠶᠢᠨ ᠨᠡᠢ ᠯᠡᠪᠤᠷᠢ ᠨᠢ_ ___ __cm 15. ᠨᠢ ᠭᠡ ᠬᠣᠣ ᠰ ᠭᠤᠷᠪ ᠠᠯᠵᠢ ᠨ ᠬᠠ ᠪᠲᠠᠰᠤ ᠵᠢ ᠵᠢᠷ ᠤᠭ 11 ᠮᠡᠳᠤ ᠨᠡᠢᠯ ᠡᠭᠦᠯᠵ ᠦ ᠲᠠ ᠯᠪᠢᠪ ᠠᠯ ᠬᠤ ᠶᠠᠷ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠬᠠᠪᠲᠠᠰᠤ ᠶᠢᠨ ᠬᠤᠶ ᠠᠷ ᠡᠭ ᠴᠡ ᠲᠠᠯ ᠠ AC ᠪᠠ MD ᠨᠢ ᠳᠠ ᠪᠬᠤᠴ ᠠᠨ ᠠ᠂ AB ＝AC ＝8c m᠂ ∆MED ᠵᠢ ᠴᠡᠭ A (M ) ᠵᠢ ᠲᠣᠭᠤᠷ ᠢᠭᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠡᠰ ᠡᠷᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 60 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ( ᠵᠢᠷ ᠤᠭ 12 ᠮᠡᠳᠤ ) ᠬ ᠤᠶᠠᠷ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠳᠠᠪᠬᠤᠴ ᠠᠭᠰᠠᠨ ( ᠰ ᠡᠭᠦᠳᠡ ᠷ ᠲᠠᠢ ᠬᠡᠰᠡᠭ ) ᠬᠡ ᠰᠡᠭ ᠤ ᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ___ ___ _cm 2 16. ᠵᠢᠷ ᠤᠭ 13 ᠮᠡ ᠳᠤ᠂ ᠷᠣᠮᠪ A BCD ᠳᠤ᠂ ∠B =60° ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ AC ᠨᠢ ∠ BA D ᠵᠢ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ᠂ ᠴᠡᠭ P ᠨᠢ ∆AB C ᠶᠢᠨ ᠳᠤᠳᠤᠷ ᠠ ᠣ ᠷᠤᠰᠢᠬ ᠤ ᠨᠢ ᠭᠡ ᠴᠡᠭ ᠂ PA ᠂ PB ᠂ PC ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠂ ᠬᠡᠷᠪᠠ PA =6 ᠂ PB =8 ᠂ PC =10 ᠪᠣᠯ ᠷᠣᠮᠪ ABCD ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ___ ___ _ ᠭᠤᠷᠪ ᠠ᠂ ᠪ ᠣᠳᠣᠵ ᠤ ᠬ ᠠᠷᠢᠭ ᠤᠯᠬᠤ ᠰᠡᠲ ᠤᠪ᠃ ( ᠨᠠ ᠢᠲᠡ 72 ᠬᠤᠪ ᠢ) 17. (5 ᠬᠤᠪᠢ ) ∆AMB ᠳᠤ ∠AMB ＝90 °᠂ ∆AMB ᠵᠢ ᠴᠡᠭ B ᠪᠡᠷ ᠲᠥ ᠪ ᠪᠣᠯᠭ ᠠᠨ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠳᠠᠭ ᠠᠤ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 90 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ∆CN B ᠵᠢ ᠣ ᠯᠤᠨ ᠠ᠃ ᠪᠠᠲᠤᠯ ᠠ᠄ AM ∥ NB (9) (10) (11) (12) (13) 18. ( 6 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠠ ᠳᠠᠯᠢ ᠬᠠᠵᠠᠭᠤᠳᠤ ∆AB C ᠳᠤ ∠A CB = 90° ᠂ ᠴᠡᠭ D ᠨᠢ CB ᠶᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡ ᠴᠡᠭ ᠂ ᠴᠡ ᠭ A ᠵᠢ ᠳᠠ ᠭᠠᠷᠢᠭ ᠤᠯᠤᠨ AE ⊥AD ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠬᠤ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ AE = AD ᠪᠣᠯ ᠭᠠᠨ ᠠ᠂ BE ᠪᠣᠯ ᠤᠨAC ᠶᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠨᠢ ᠴᠡᠭ P ᠳᠤ ᠣ ᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠃ ᠪᠠᠲᠤ ᠯᠠ᠄ PB = PE 19. ( 5 ᠬᠤᠪᠢ ) ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠴᠡ ᠭ A( 3,0 )᠂ B (2 ,3) ᠂ ᠴᠡᠭ B ᠶᠢᠨ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ ᠤᠨ ᠬᠤᠪᠢ ᠳᠤ ᠳᠡᠭᠰᠢ ᠬᠡᠮᠳᠤ ᠴᠡᠭ ᠨᠢ ᠴᠡᠭ C ᠃ (1) ᠴᠡᠭ C ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠪᠢᠴ ᠢ᠃ (2) ∆AB C ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠵᠢ ᠣᠯ᠃ 20. ( 7 ᠬ ᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ (1) ᠪᠣᠯ ᠮᠠᠨ ᠤ ᠤᠯᠤᠰ ᠤᠨ ᠡᠷᠳᠡᠨ ᠤ ᠨᠢᠭᠡ ︽ ᠵᠤᠤ ᠱᠤᠸᠠᠩ ᠵᠢᠷᠤᠬ ᠠᠢ ︾ ᠶᠢᠨ ᠲᠥ ᠯᠦᠪ ᠵᠢᠷᠤᠭ ᠪᠣᠯᠤᠨ ᠠ᠃ ᠲᠡ ᠷᠡ ᠨᠢ ᠳᠦ᠋ᠷᠪ ᠠᠡ ᠪᠦᠷᠢᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠬᠦ ᠷᠢᠶᠠᠯ ᠡᠭᠳᠡᠨ ᠪᠦᠲ ᠦᠭᠰᠡᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ 74 ᠪᠣᠯ ᠬᠤ ᠳᠡᠭᠰᠢ ᠳᠦ᠋ᠷ ᠪᠠᠯᠵ ᠢᠨ ᠪᠣᠯᠤᠨ ᠠ᠃ Rt △ABC ᠳᠤ ᠬᠡᠷᠪᠠ ᠡᠭ ᠴᠡ ᠲᠠᠯ ᠠ BC = 5᠂ ᠳᠦ᠋ᠷ ᠪᠠᠨ ᠳᠡ ᠭᠰᠢ ᠥᠨ ᠴᠦᠭᠲᠦ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠤ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 5 ᠪ ᠣᠯᠬᠤ ᠡᠭᠴᠡ ᠲᠠᠯ ᠠ ᠵᠢ ᠲᠤᠰ ᠲᠤᠰ ᠭᠠᠳᠠ ᠭᠰᠢ ᠨᠢᠭᠡ ᠳᠠᠬᠢᠨ ᠦᠷᠭᠦ ᠯᠵᠢᠯ ᠡᠭᠦᠯᠪᠠ ᠯ ᠵᠢᠷᠤᠭ (2) ᠮᠡᠳᠤ ︽ ᠲᠣᠭ ᠠᠨ ᠤ ᠰᠠ ᠯᠬᠢᠨ ᠬᠦᠷᠳᠤ ︾ ᠵᠢ ᠤᠯᠵ ᠦ ᠪᠣᠯ ᠤᠨ ᠠ᠃ (1) ᠲᠤᠰ ᠰᠠᠯᠬ ᠢᠨ ᠬᠦᠷᠳᠤ ᠨᠢ ᠠᠳᠠ ᠭ ᠲᠠ ᠭᠠᠨ ᠬᠡᠳᠤᠢ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠡᠷ ᠭᠢᠪᠠᠯ ᠦᠪᠠᠷ ᠤᠨ ᠲᠤᠰ ᠪᠡᠶ ᠠ ᠲᠠᠢ ᠪᠠᠨ ᠳᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠪᠤᠢ ︖ (2) ᠲᠤᠰ ᠰᠠᠯᠬ ᠢᠨ ᠬᠦᠷᠳᠤ ᠶᠢᠨ ᠭᠠᠳᠠᠭ ᠠᠳᠤ ᠬᠦᠷᠢᠶ ᠠᠨ ᠤᠷᠳᠤ ᠵᠢ ᠣᠯ᠃ (ᠵᠢᠷ ᠤᠭ( 2) ᠳᠡᠬᠢ ᠪᠤᠳᠠᠳᠤ ᠱᠤᠭᠤᠮ ) 21. ( 8 ᠬ ᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ∆ BAD ᠪᠣᠯ ∆B EC ᠵᠢ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠳᠡ ᠭᠡᠷ ᠠ ᠴᠡᠭ B ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠ ᠭ ᠤᠨ ᠵᠡᠭᠦᠦ ᠤ ᠨ ᠡ ᠰᠡᠷᠭᠦ ᠴᠢᠭᠯᠡᠯ ᠢᠶᠡᠷ 60 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠵᠦ ᠣᠯᠤᠭ ᠰᠠᠨ ᠮᠦ᠋ᠷᠳ ᠡᠭᠡᠨ AB ⊥BC ᠂ BE ＝CE ᠂ DE ᠵᠢ ᠬᠣ ᠯᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠳᠤ ᠯᠠ᠄ △BD E≌ △BCE (2) ᠳᠦ᠋ᠷ ᠪᠠᠯᠵ ᠢᠨA BE D ᠤ ᠬᠡᠯᠪᠠ ᠷᠢ ᠳᠦ᠋ᠷᠰᠤ ᠵᠢ ᠮᠠᠭᠠᠳᠯ ᠠᠵᠦ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠲᠣᠳᠤᠷ ᠬᠠᠢ ᠯᠠ᠃ 22. ( 9 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ᠴ ᠡᠭ O ᠨᠢ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠲᠠᠯ ᠠᠳᠤ ∆AB C ᠳᠤᠳᠤᠷ ᠠᠭᠢ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ ∠ AOB =1 10 °᠂ ∠BO C= 𝑎᠂ ∆B OC ᠵᠢ ᠴᠡᠭ C ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ ᠢᠶᠡᠷ 60 ° ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ∆ADC ᠢ ᠣ ᠯᠤᠨ ᠠ ᠂ OD ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠃ (1) ᠪᠠᠳᠤ ᠯᠠ᠄ △CO D ᠪᠣᠯ ᠲᠡᠩ ᠴᠡᠭᠦᠦ ᠲᠠᠯᠠᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠃ (2) 𝑎= 15 0° ᠦᠶ ᠡᠰ∆ AO D ᠶᠢᠨ ᠬᠡᠯᠪᠠ ᠷᠢ ᠳᠦ᠋ᠷᠰᠤ ᠵᠢ ᠮᠠᠭᠠᠳᠯ ᠠᠵᠦ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠳᠤᠳᠤᠷ ᠬᠠᠢᠯᠠ᠃ (3) ᠰᠦ᠋ᠪᠡ ᠭᠴᠢᠯ ᠡᠬᠦ᠄ 𝑎 ᠨᠢ ᠬᠡᠳᠤᠢ ᠭᠷᠠᠳᠦ᠋ ᠰ ᠦᠶ ᠡᠰ ∆A OD ᠨᠢ ᠠᠳᠠ ᠯᠢ ᠬᠠᠵᠠᠭ ᠤᠳᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ ᠪᠣᠯᠬᠤ ᠪᠤᠢ ︖ 23. ( 9 ᠬᠤᠪᠢ ) ᠵᠢᠷᠤᠭ ᠮᠡᠳᠤ ᠂ ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨ ᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠴᠡ ᠭ O ᠨᠢ ᠡᠬᠢᠨ ᠤ ᠴᠡ ᠭ᠂ ᠴᠡᠭ A ᠨᠢ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠤ ᠨ ᠰ ᠥᠭᠡᠷ ᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠠ ᠤᠷᠤᠰ ᠢᠵᠦ᠂ ᠴᠡ ᠭ B ᠨᠢ y ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠤ ᠨ ᠡ ᠶᠡᠷᠭᠦ ᠬᠠᠭ ᠠᠰ ᠲᠡᠩᠭᠡ ᠯᠢᠭ ᠳᠡᠭᠡᠷ ᠠ ᠣ ᠷᠤᠰᠢᠨ ᠠ᠂ AO = 2OB ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠬᠡᠷᠴᠢᠮ OB ᠤᠨ ᠤᠷᠳᠤ ᠨᠢ ᠳᠡᠭᠰ ᠢᠳᠬᠡᠯ 𝑥 2 − 2𝑥 − 8= 0 ᠶᠢᠨ ᠨᠢᠭᠡ ᠢᠵᠠᠭ ᠤᠷ᠃ (1) ᠰᠢᠯᠤᠭᠤᠨ AB ᠤᠨ ᠹᠦᠩᠺᠼ ᠤ ᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ∆AB D ᠵᠢ ᠴᠡ ᠭ O ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠡᠰᠡᠷ ᠭᠦ ᠴᠢ ᠭᠯᠡᠯ ᠢᠶ ᠡᠷ 9 0° ᠡᠷᠭᠢᠭ ᠦᠯᠪᠠ ᠯ∆ ED O ᠵᠢ ᠣᠯᠤᠨ ᠠ᠃ ᠰᠢᠯᠤᠭ ᠤᠨ ED ᠨᠢ ᠬᠡ ᠷᠴᠢᠮ AB ᠲᠠᠢ ᠴᠡ ᠭ C ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ ᠴᠡᠭ F ᠨᠢ ᠰᠢᠯ ᠤᠭᠤᠨ CE ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡ ᠴᠡᠭ ᠂ ᠴᠡ ᠭ E ᠂ F ᠵᠢ ᠲᠤᠰ ᠲᠤᠰ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠪᠠ 𝑦 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠠᠢ ᠵᠡ ᠷᠭᠡᠴᠡᠭᠡ ᠰ ᠢᠯᠤᠭᠤᠨ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ ᠴᠡ ᠭ G ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ ∆EF G ᠵᠢ EF ᠶᠢᠨ ᠳᠠ ᠭᠠᠤ ᠨᠤᠭ ᠤᠯᠪᠠ ᠯ ᠴᠡ ᠭ G ᠶᠢᠨ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡ ᠭ ᠨᠢ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷ ᠠ ᠤᠨᠠᠨ ᠠ᠂ ᠴᠡᠭ F ᠤ ᠨ ᠪ ᠠᠢᠷᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (3) ᠰᠡᠳᠤᠪ (2 ) ᠤᠨ ᠨᠥᠬᠦᠴᠡ ᠯ ᠳ ᠣᠣᠷ ᠠ ᠴᠡᠭ M ᠨᠢ DO ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠂ ᠴᠡᠭ N ᠂ P ᠂ Q ᠨᠢ ᠰ ᠢᠯᠤᠭᠤᠨ BD ᠪᠤᠶ ᠤ y ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠠ ᠣ ᠷᠤᠰᠢᠨ ᠠ᠂ ᠳᠦ᠋ᠷᠪ ᠠᠯᠵᠢᠨ MN PQ ᠢ ᠳᠡᠭᠰᠢ ᠥᠨᠴᠦ ᠭᠲᠦ ᠪᠣᠯᠭ ᠠᠬᠤ ᠴᠡᠭ P ᠣ ᠷᠤᠰᠢᠨ ᠠ ᠤᠤ ︖ ᠬᠡ ᠷᠪᠡ ᠣᠷᠤᠰ ᠢᠪᠠᠯ ᠵᠢᠷᠤ ᠭ (3 ) ᠳᠡ ᠭᠡᠷ ᠠ ᠵᠢᠷᠤᠭ ᠢ ᠵᠢᠷᠤᠵ ᠦ ᠴᠡ ᠭ P ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴ ᠢ᠂ ᠬᠡᠷᠪᠠ ᠣᠷᠤ ᠰᠢᠬ ᠤ ᠦᠭᠡᠢ ᠪᠣᠯ ᠰᠢᠯᠳᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳᠤᠷ ᠬᠠᠢᠯᠠ᠃ 24. ( 1 2 ᠬᠤᠪᠢ ) ᠵᠢᠷ ᠤᠭ ᠮᠡᠳᠤ ᠂ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ AB CD ᠳᠤ AB ⊥AC ᠂ ᠳᠠ ᠰᠢᠭᠤᠪ ᠴᠢ AC ᠂ BD ᠨᠢ ᠴᠡ ᠭ O ᠳᠤ ᠣᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ ᠰᠢᠯᠤᠭᠤ ᠨAC ᠵᠢ ᠴᠡ ᠭ O ᠵᠢ ᠲᠣᠭ ᠤᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ ᠨᠢᠭᠡ ᠥᠨᠴ ᠦᠭ 𝑎 (0° 𝑎≤ 90° ) ᠡᠷ ᠭᠢᠭᠦᠯ ᠪᠠᠯ ᠬᠡᠷᠴᠢᠮ BC ᠂ AD ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ E᠂ F ᠳᠤ ᠣᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠨ ᠠ᠂ BF ᠵᠢ ᠬᠣᠯ ᠪᠤᠨ ᠠ᠃ (1) ᠵᠢᠷᠤᠭ 1 ᠮᠡ ᠳᠤ᠂ ᠪᠠᠲᠤᠯ ᠠ᠄ OE ＝OF (2) ᠵᠢᠷ ᠤᠭ 2 ᠮᠡᠳᠤ ᠂ ᠰ ᠢᠯᠤᠭᠤ ᠨ A C ᠡᠷ ᠭᠢᠭᠡᠳ 9 0° ᠪᠣᠯᠬ ᠤ ᠦᠶ ᠡᠰ ᠳᠥᠷᠪ ᠡᠯᠵᠢ ᠨ ABE F ᠶᠢᠨ ᠬᠡᠯᠪᠠ ᠷᠢ ᠳᠦ᠋ᠷᠰᠤ ᠵᠢ ᠮᠠᠭ ᠠᠳᠯᠠᠵᠦ ᠪᠠᠰᠠ ᠣᠯ ᠤᠭᠰᠠᠨ ᠳᠦ᠋ᠩᠨᠡ ᠯᠲᠡ ᠪᠠᠨ ᠪᠠᠲᠤᠯ ᠠ᠃ (3) ᠬᠡᠷᠪᠠ AB ＝1᠂ BC ＝√5 ᠮᠥᠷᠲᠡ ᠭᠡᠨ BF ＝DF ᠪᠣᠯ ᠡᠷ ᠭᠢᠯᠲᠡ ᠶᠢᠨ ᠥᠨᠴᠦ ᠭ 𝑎 ᠤᠨ ᠶᠠᠬᠡ ᠪᠠ ᠭ ᠠ ᠵᠢ ᠣ ᠯ᠃ 25. ( 11 ᠬᠤᠪᠢ )ᠬᠠᠪ ᠲᠠᠭᠠᠢ ᠶ ᠢᠨ ᠳᠡᠭᠰᠢ ᠥᠨᠴᠦᠭᠲᠦ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰ ᠢᠰᠲ᠋ ᠧᠮ ᠳᠤᠳᠤᠷ ᠠ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ A OBC ᠶᠢᠨ ᠴᠡᠭ O (0 ,0) ᠂ ᠴᠡᠭ A (5, 0) ᠂ ᠴᠡ ᠭ B (0 , 3 ) ᠪᠣᠯᠤᠨ ᠠ᠂ ᠴᠡᠭ A ᠪᠠᠷ ᠲᠥᠪ ᠪᠣᠯ ᠭᠠᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡ ᠭᠦᠦ ᠶᠢᠨ ᠴᠢᠭᠯᠡᠯ ᠳᠠᠭᠠᠤ ᠳᠡᠭ ᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ A OBC ᠵᠢ ᠡᠷ ᠭᠢᠭᠥᠯᠪᠠᠯ ᠳᠡᠭ ᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ ADEF ᠵᠢ ᠣᠯᠤ ᠨ ᠠ᠂ ᠴᠡᠭ O ᠂ B ᠂ C ᠶᠢᠨ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡᠭ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ D ᠂ E ᠂ F ᠃ (1) ᠵᠢᠷᠤᠭ ① ᠮᠡ ᠳᠤ᠂ ᠴᠡᠭ D ᠨᠢ BC ᠲᠠ ᠯ ᠠ ᠳᠡᠭᠡᠷ ᠠ ᠤᠨᠠ ᠬᠤ ᠦᠶ ᠡᠰ ᠴᠡᠭ D ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (2) ᠵᠢᠷ ᠤᠭ ② ᠮᠡ ᠳᠤ᠂ ᠴᠡᠭ D ᠬᠡᠷ ᠴᠢᠮ BE ᠳᠡᠭᠡᠷ ᠠ ᠪ ᠠᠢᠬᠤ ᠦ ᠶᠡᠰ AD ᠪᠣᠯ ᠤᠨ BC ᠨᠢ ᠴᠡ ᠭ H ᠳᠤ ᠣᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠃ ① ᠪᠠᠲᠤᠯ ᠠ᠄ ∆ADB ≌ ∆AOB ② ᠴᠡᠭ H ᠤᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (3) K ᠪᠠᠷ ᠳᠡᠭᠰᠢ ᠥᠨᠴ ᠦᠭᠲᠦ A OBC ᠶᠢᠨ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠶᠢᠨ ᠣ ᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠯ ᠤᠨ ᠴᠡ ᠭ ᠶᠢ ᠲᠡᠮᠳᠡ ᠭᠯᠡᠵᠦ᠂ S ᠪᠠᠷ ∆K DE ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠵᠢ ᠲᠡᠮᠳᠡ ᠭᠯᠡᠪᠠ ᠯ S ᠤᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬ ᠤ ᠬᠡᠪᠴᠢᠶ ᠠ ᠵᠢ ᠣᠯ᠃ (ᠰ ᠢᠭᠤᠳ ᠳ ᠦ᠋ᠩ ᠢ ᠪᠢᠴ ᠢᠪᠠᠯ ᠪᠣᠯ ᠴᠢᠬ ᠠᠨ ᠠ )