ᠠᠷᠪᠠᠨ ᠨᠠᠢᠮᠠᠳᠤᠭᠠᠷ ᠪᠦᠯᠦᠭ ᠵᠢᠰᠢᠬᠦᠯᠵᠢᠨ （ᠨᠢᠭᠡᠴᠢ ᠶᠢᠨ ᠪᠠᠢᠴᠠᠭᠠᠯᠲᠠ 1） .pdf

ᠠᠷ ᠪᠠ ᠨᠠᠢ ᠮᠠᠳᠤ ᠭᠠᠷ ᠪᠥᠯ ᠥᠭ ᠦᠨ ᠨᠢᠭᠡ ᠴᠢ ᠶᠢ ᠨ᠊ ᠪᠠᠢᠴᠠ ᠭᠠᠯᠲᠠ ᠠᠩᠭᠢ ᠪᠥᠯ ᠥᠭ᠄ ᠨᠡᠷ ᠡ᠄ ︵90 ᠮᠢᠨᠦ᠋ ᠲ᠊᠂ ᠨᠠᠢᠲᠡ 12 0 ᠬ᠊ ᠤᠪᠢ︶ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤᠬ ᠤ ᠰᠡ ᠲᠤᠪ᠃ ︵ᠰᠡᠲᠤᠪ ᠪᠦ ᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 30 ᠬᠤᠪᠢ︶ 1. ᠵᠢᠷᠤᠭ 1 ᠮᠡᠲᠦ᠂ ᠵᠢᠰᠢᠭ ᠦᠯᠵᠢ ᠨ᠊ AB CD ᠲᠤ᠂ ᠳᠣᠣᠷ ᠠᠬᠢ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠠᠴᠠ ᠵᠥᠪ ᠪᠢᠰᠢ ᠮᠠᠭᠠᠳ ᠨᠢ︶ ︵A. ∠1 + ∠2 ＝18 0° B. ∠2 + ∠3 ＝18 0° C. ∠3 + ∠4 ＝180 ° D. ∠2 + ∠4 ＝18 0° 2. ᠵᠢᠷᠤᠭ 2 ᠮᠡ ᠲᠤ᠂ ᠵᠢᠰᠢᠭᠦᠯ ᠶᠢᠨ AB CD ᠲᠤ᠂ EF ∥AB , GH ∥AD , EF ᠪᠠ GH ᠨᠢ ᠴᠡ ᠬ᠋᠊ O ᠲᠤ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠵᠢᠷᠤᠭ ᠲᠠᠬᠢ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ ᠦ ᠤ ᠤᠳᠠᠭ ᠠᠯᠢ ᠲᠤᠭ ᠠ ᠨᠢ ︵︶ A.7 B.8 C.9 D. 11 3. ᠵᠢᠷᠤᠭ 3 ᠮᠡᠲᠦ᠂ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ AB CD ᠲᠤ᠂ ∠ B= 110°᠂ AD ᠢ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠭᠦ ᠯᠦᠨ F ᠲᠤ ᠬᠦᠷᠭᠡᠨ ᠡ᠂ CD ᠢ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡ ᠭᠦᠯᠦ ᠨ᠊ E ᠲᠤ ᠬᠦᠷᠭᠡᠨ ᠡ᠂ EF ᠢ ᠬᠣ ᠯᠪᠤᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ∠ E+ ∠F =︶ ︵A. 110° B .30° C. 50° D. 70° 4. ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠨᠢ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠡᠭᠴᠡ ᠲᠠ ᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠬᠤ ᠮᠦ᠋ᠷᠲᠡ ᠭᠡᠨ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠲᠡᠩ ᠴᠡᠬᠦ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠨᠢ ᠯᠠᠪᠲᠠ ᠢ︶ ︵A. ᠲᠡᠭᠰ ᠢ ᠳ᠋᠊ ᠦᠷᠪᠡ ᠯᠵᠢᠨ B. ᠷᠤᠮᠪ C. ᠲᠡᠭᠰ ᠢ ᠦᠨᠴ ᠦᠭᠲᠦ D . ᠠ ᠳᠠᠯᠢ ᠬᠠᠵ ᠠᠭᠤᠲᠤ ᠱᠠᠲᠤ ᠯᠵᠢᠨ 5. ᠳᠤᠤᠷ ᠠᠬᠢ ᠬᠡᠯᠡᠯ ᠭᠡ ᠡᠴ ᠡ ᠵᠥ ᠤ ᠪᠤ ᠯᠬᠤ ᠨᠢ︶ ︵A. ᠲᠡ ᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢ ᠨ᠊ ᠨᠢ ᠲᠡ ᠩᠬᠡᠯ ᠢᠭ ᠲᠤ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠲᠦ ᠵᠢᠷ ᠤᠭ ᠳᠦ᠋ᠷᠰᠦ ᠮᠦ᠋ᠷᠲᠡ ᠭᠡᠨ ᠳᠦ᠋ᠷᠪ ᠠ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠬ ᠡᠯᠢᠭ ᠲᠠᠢ B. ᠲᠡᠭ ᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢᠨ ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠤ ᠣᠯ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦ ᠨ᠊ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ C. ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ ᠪᠣᠯ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠲᠤ ᠲᠡᠭᠰᠢ ᠭᠡᠮᠲᠦ ᠵᠢᠷ ᠤᠭ ᠲᠤᠷ ᠰᠦ ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠳᠥᠷᠪ ᠠ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠭᠡ ᠯᠢᠭ ᠪᠠᠢᠨ ᠠ D. ᠷᠤᠮᠪ ᠶᠢᠨ ᠲᠠᠰᠢᠭᠤ ᠪᠴᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠤ 6. ᠷᠣᠮᠪ᠂ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥᠭᠲᠦ᠂ ᠲᠡ ᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠲᠤ ᠪᠦᠷ ᠪᠠᠢᠬᠤ ᠰᠢᠨᠵᠢ ᠴᠢᠨ ᠠᠷ ᠪᠣᠯ︶ ︵A. ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠲᠡᠩ ᠴᠡᠭᠦᠤ B. ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠡᠭᠴᠡ C. ᠲᠠᠰᠢᠭᠤᠪ ᠴᠢ ᠨᠢ ᠬᠠᠷ ᠢᠯᠴᠠᠨ ᠲᠠᠯ ᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ D. ᠲᠠᠰᠢᠭᠤ ᠪᠴᠢ ᠨᠢ ᠨᠢ ᠭᠡ ᠬᠤᠣ ᠰ᠊ ᠡᠰᠡ ᠷᠭᠦ ᠦᠨᠴ ᠦᠭ ᠢ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠨ ᠠ 7. ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ᠄ ᠵᠢᠷᠤᠭ 4 ᠮᠡᠲᠦ᠂ ᠷᠤᠮᠪ AB CD ᠲᠤ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ AC ᠪᠠ BD ᠨᠢ ᠴᠡᠭ 𝑂 ᠲᠤ ᠤᠭᠲᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ 𝑂𝐸 ∥ 𝐷𝐶᠂ BC ᠲᠠᠢ E ᠴᠡ ᠬ᠋᠊ ᠲᠤ ᠤᠭᠲ ᠤᠯᠤᠯ ᠴᠠᠨ ᠠ᠂ AD = 6c m᠂ ᠲᠡᠭᠡᠪ ᠡᠯ OE ᠦᠨ ᠤᠷᠲᠤ ᠨᠢ︶ ︵图 2 H G D O FE C BA43 2 1 DA CB O G F CD BA H E CD A E F O E B A D C ︵1︶ ︵4︶ ︵2︶ ︵3︶ A. 6 cm B. 4 cm C. 3 cm D. 2 cm 8. ︽ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ︾ ᠭᠡᠬᠦ ᠪᠦᠯᠦ ᠬ᠋᠊ ᠢ ᠰᠤᠷᠤᠯ ᠴᠠᠬᠤ ᠦᠶ ᠡ ᠲᠤ᠂ ᠰᠢᠶᠣᠤ ᠮᠢᠩ ᠦᠨ ᠨᠣᠮ ᠳᠡ ᠭᠡᠷ ᠡ ᠨᠢ ᠨᠢ ᠭᠡ ᠵᠢᠷᠤᠭ ᠬᠢᠴᠢᠶ ᠡᠮᠵᠢ ᠦᠭᠡᠢ ᠠᠴᠠ ᠪᠣᠯᠵᠤ ᠪᠠᠬᠢ ᠲᠤ ᠰᠤᠭᠰᠠ ᠨ᠊ ︵ᠵᠢᠷᠤᠭ 5 ᠮᠡ ᠲᠤ︶ ᠢ ᠬ᠋᠊ ᠦᠰᠦ ᠨ᠊ ᠠᠰᠭᠠᠯᠯ ᠠᠮᠤᠷᠠ ᠳ᠋᠊᠂ ᠲᠣᠳᠣᠷ ᠬᠠᠶ ᠬᠯᠵ ᠤ ᠦᠵ ᠡᠬᠦ ᠦᠭᠡᠢ᠂ ᠪᠠᠬᠢ ᠶᠢᠨ ᠬᠷᠣᠮ ᠲᠤ ᠬᠠᠯᠬ ᠠᠯᠠᠭᠳᠠᠭ ᠰᠠᠨ ᠦᠰ ᠦᠭ ᠨᠢ︶ ︵A. ᠲᠡᠩ ᠴᠡᠭᠦᠤ ᠲᠠ ᠯᠠᠲᠤ ᠭᠤᠷᠪ ᠠᠯᠵᠢᠨ B. ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢ ᠨ᠊ C. ᠠᠳᠠᠯᠢ ᠬᠠᠵ ᠠᠭᠤᠲᠤ ᠱᠠᠲᠤ ᠯᠵᠢᠨ D. ᠷᠤᠮᠪ 9. ᠵᠢᠷᠤᠭ 6 ᠮᠡᠲᠦ᠂ ᠥᠷᠭᠡᠨ ᠨᠢ 20 m᠂ ᠬ᠊ ᠷᠳᠣ ᠨᠢ 30 m ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠲᠦ ᠭᠠᠵᠠᠷ ᠳᠡᠭᠡᠷ ᠡ ᠠᠳᠠᠯᠢ ᠬᠠᠨ ᠬᠤᠶ ᠠᠷ ᠥᠷᠭᠡᠨ ᠵᠠᠮ ᠳᠡᠪᠢᠰᠴᠦ᠂ ᠦᠯ ᠡᠳᠡᠭᠰᠡᠨ ᠬᠡᠰᠡᠭ ᠢ ᠨᠢ ᠲᠠᠷᠢᠶ ᠠᠨ ᠭᠠᠵᠠᠷ ᠪᠤ ᠯᠭᠠᠪ ᠠ᠃ ᠵᠢᠷᠤᠭ ᠲᠠᠬᠢ ᠲᠤᠭ ᠠ ᠤ ᠠᠷᠢᠮᠲᠠ ᠶᠢ ᠦᠨᠳᠦ ᠰᠦᠯᠡᠪ ᠡᠯ᠂ ᠲᠠ ᠷᠢᠶᠠᠨ ᠭᠠᠵᠠᠷ ᠤ ᠨ᠊ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ︶ ︵A. 600 m 2 B. 551 m 2 C. 550 m 2 D. 500 m 2 10. ᠵᠢᠷ ᠤᠭ 7 ᠮᠡᠲᠦ᠂ 4 × 4 ᠵᠢᠵᠢᠭ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠤ ᠦᠷᠢᠯᠳ ᠦᠭᠰᠡᠨ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ ᠦ ᠲᠣᠤᠷ ᠯᠢᠭ ᠳᠣᠲᠣᠷ ᠠ᠂ ᠰ᠊ ᠡᠭᠦᠳᠡ ᠷ᠊ ᠬᠡ ᠰᠡᠭ ᠦᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠪᠣᠯᠤᠨ ᠲᠡ ᠭᠰᠢ ᠳᠥᠷ ᠪᠡᠯᠵ ᠶᠢᠨ AB CD ᠲᠠ ᠯᠠᠪᠠᠢ ᠶᠢᠨ ᠬᠠᠷ ᠢᠴᠠᠯ ᠨᠢ︶ ︵A. 3: 4 B. 5: 8 C. 9: 16 D. 1: 2 ᠬᠤᠶ ᠠᠷ᠂ ᠬᠠᠭᠠ ᠯᠲᠠ ᠲᠠᠭᠯ ᠠᠬᠤ ᠰᠡᠲᠥ ᠪ᠊᠃︶ ᠰ᠊ ᠡᠲᠥᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪ ᠢ᠂ ᠨᠡᠢᠲᠡ 24 ᠬᠤᠪᠢ ︵11. ᠵᠢᠷ ᠤᠭ 8 ᠮᠡᠲᠦ᠂ AB ∥DC , AD ∥BC᠂ ᠬᠡ ᠷᠪᠡ ∠B = 150°᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ∠D ＝ ＿ ＿ ＿°᠃ 12. ᠮᠡᠳᠠᠬ ᠦ ᠨᠢ ᠱᠠᠲᠤ ᠯᠵᠢᠨ AB CD ᠲᠤ AD ∥BC , ∠AB C ＝60 °, BD ＝2√ 3 ,AE ᠨᠢ ᠱᠠᠲᠣᠯᠵᠢᠨ ᠤ ᠦᠨᠳ ᠦᠷ᠂ ᠮᠦ᠋ᠷᠲᠡ ᠭᠡᠨ BE ＝1᠂ ᠲᠡ ᠭᠡᠪᠡᠯ AD ＝ ＿ ＿ ＿ 13. ᠨᠢᠭᠡ ᠵᠢᠰᠢᠭᠦᠯᠵᠢᠨ ᠤ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ 𝑆1 ,𝑆 2 ,𝑆 3 ,𝑆 4 ᠶᠢ ᠨ᠊ ᠳᠥᠷᠪ ᠠ ᠵᠢᠵᠢᠭ ᠵᠢᠰᠢᠬᠦ ᠯᠵᠢᠨ ᠢᠶᠡᠷ ᠪᠦ ᠷᠢᠯᠳᠥ ᠵᠠᠢ︶ ᠵᠢᠷ ᠤᠭ 9 ᠮᠡ ᠲᠤ ︵᠃ CD ᠨᠢ AB ᠶᠢ ᠳᠠᠭ ᠠᠵᠤ ᠵᠡᠭᠦᠨ ᠠᠴᠠ ᠪᠠᠷᠠᠭ ᠤᠨᠰᠢ ᠵᠢᠰᠢᠭᠦ ᠯᠵᠢᠨ ᠳᠣᠲ ᠣᠷ ᠠ ᠵᠡᠷᠭᠡᠴᠡᠭ ᠡ ᠰᠢᠯᠵ ᠢᠯᠲᠡ ᠬᠢᠬᠦ ᠦᠶ ᠡᠰ᠂ 𝑆1 ∙𝑆 4 ᠤ ᠣᠯᠤᠨ 𝑆2 ∙𝑆 3 ᠶ᠊ ᠶᠢᠨ ᠶᠡᠭᠡ ᠪᠠᠭ ᠠ ᠶᠢᠨ ᠬᠤᠯ ᠪᠣᠭᠳᠠᠯ ᠨᠢ ＿ ＿ ＿ 14. ᠵᠢᠷ ᠤᠭ 10 ᠮᠡᠲᠦ᠂ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡ ᠨ᠊ ᠨᠢ AB ∥DC , AE ⊥ DC , AE = 12 ,BD = 15 ,AC = 20᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠱᠠᠲᠤᠯᠵᠢᠨ AB CD ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ＿ ＿ ＿ 15. ᠲᠡᠭ ᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠴᠠᠭ ᠠᠰᠤ AB CD ᠲᠤ᠂ AD ＝4c m , AB ＝10c m᠂ ᠵᠢᠷᠤᠭ 1 1 ᠮᠡᠲᠦ ᠨᠤᠭ ᠤᠯᠵᠤ᠂ ᠴᠡᠭ B ᠪᠠ ᠴᠡᠭ D ᠢ ᠳᠠᠪᠬᠤᠴ ᠠᠭᠤᠯ ᠤᠨ ᠠ᠂ ᠨᠤᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠤ ᠷᠤᠮ ᠨᠢ EF᠂ ᠲᠡ ᠭᠡᠪᠡᠯ DE ＝ ＿ ＿ ＿𝑐𝑚 ᠵᠢᠰ ᠢᠭᠦᠯᠵᠢ ᠨ᠊ ᠲᠡᠭ ᠰᠢ ᠦ ᠨᠴᠦᠭ ᠲᠤ ᠲᠡᠭᠰ ᠢ ᠳ᠋᠊ ᠦᠷᠪ ᠡᠯᠵᠢᠨ BA CD B D C A 𝑆1 𝑆2 𝑆3 𝑆4 E A CD B ︵8︶ ︵9︶ ︵10︶ ︵11︶ ︵5︶ ︵6︶ ︵7︶ 16. ᠲᠡᠭ ᠰᠢ ᠦᠨᠴ ᠦᠭᠲᠦ AB CD ᠲᠤ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ AC ,BD ᠶᠢᠨ ᠤᠭᠲᠤ ᠯᠤᠯᠴᠠ ᠯ᠊ ᠤᠨ ᠴᠡᠭ ᠨᠢ O᠂ ∠ AO B ＝2∠ BO C᠃ ᠬᠡᠷᠪᠡ AC ＝18 𝑐𝑚᠂ ᠲᠡᠭᠡᠪ ᠡᠯ AD ＝ ＿ ＿ ＿𝑐𝑚 17. ᠵᠢᠷ ᠤᠭ 12 ᠮᠡ ᠲᠤ᠂ ᠳᠡᠭᠡᠭᠰ ᠢ ᠥᠨᠴᠥ ᠭᠲᠦ AB CD ᠶᠢᠨ ᠵᠡᠷᠭᠡᠯ ᠡᠳᠡᠭᠡ ᠬᠤᠶ ᠠᠷ ᠲᠠ ᠯ᠊ ᠠ ᠶᠢᠨ ᠤᠷ ᠲᠤ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ 3 𝑐𝑚 ᠪᠤᠯ ᠤᠨ 4 𝑐𝑚 ᠪᠤᠯᠤᠨ ᠠ᠂ ᠳᠡᠰ ᠳᠠᠷᠠᠭ ᠠᠯᠠᠨ ᠳᠡ ᠭᠡᠭᠰ ᠢ ᠥᠨᠴᠥ ᠭᠲᠦ AB CD ᠶᠢᠨ ᠡᠯ ᠡ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠢ ᠬᠤᠯ ᠪᠤᠪᠠᠯ ᠳᠥᠷ ᠪᠡᠯᠵ ᠶᠢᠨ EF GH ᠢ ᠤᠯᠤᠨ ᠠ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ EF GH ᠶᠢᠨ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠲᠤ ᠨᠢ ＿ ＿ ＿𝑐𝑚᠂ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢ ᠨ᠊ EF GH ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠢ ᠨᠢ ＿ ＿ ＿𝑐𝑚 2 18. ᠰᠢᠯ ᠤᠭᠤᠨ 𝑙 ᠳᠡᠭᠡᠷ ᠡ ᠲᠡ ᠰ᠊ ᠳᠠ ᠷᠠᠭᠠᠯᠠᠨ ᠳᠣᠯ ᠣᠭᠠᠨ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠢ ᠲᠠᠯᠪᠢᠵᠤ ᠪᠠᠢᠨ ᠠ︶ ᠵᠢᠷᠤᠭ 13 ᠮᠡᠲ ᠤ ︵᠃ ᠵᠢᠰ ᠢᠭᠦᠦ ᠲᠠᠯᠪᠢᠭᠰᠠ ᠨ᠊ ᠭᠤᠷᠪ ᠠᠨ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠦ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ᠲᠣᠰ ᠪᠦᠷᠢ 1, 2,3᠂ ᠲᠡ ᠭᠰᠢ ᠲᠠᠯᠪᠢᠭᠰᠠᠨ ᠳᠦ᠋ᠷᠪ ᠠ ᠲᠡᠭᠰᠢ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ ᠦ ᠲᠠᠯᠠᠪ ᠠᠢ ᠨᠢ ᠳᠠᠷᠠᠭ ᠠᠯᠠᠨ 𝑆1 ,𝑆 2 ,𝑆 3 ,𝑆 4᠂ ᠲᠡᠭᠡᠪ ᠡᠯ 𝑆1 + 𝑆2 + 𝑆3 + 𝑆4 = ＿ ＿ ＿ ᠭᠤᠷᠪ ᠠ᠂ ᠤ ᠤᠳᠤᠵ ᠤ ᠬᠠᠷ ᠢᠭᠤᠯ ᠬᠤ ᠰᠡ ᠳᠤᠪ᠃︶ ᠨᠡᠢᠲᠡ 58 ᠬᠤᠪᠢ ︵19. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠠᠳᠠᠯᠢ ᠬᠠ ᠵᠠᠭᠤᠲᠤ ᠱᠠ ᠲᠤᠯᠵ ᠶᠢᠨ AB CD ᠲᠤ᠂ AD ∥BC , AD = 3, AB ＝4, BC ＝7 ᠪᠤᠯᠤᠨ ᠠ᠃ ∠ B ᠶᠢᠨ ᠭᠷᠡᠳᠦ᠋ᠰ ᠦ ᠨ᠊ ᠲᠤᠭ ᠠ ᠶᠢ ᠬᠯ᠃ 20. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠳᠥᠷᠪ ᠡᠯᠵᠢ ᠨ᠊ AB CD ᠨᠢ ᠵᠢᠰᠢ ᠭᠦᠯᠵ ᠶᠢᠨ ᠪᠤᠯᠤᠨ ᠠ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ AC ᠪᠤᠯ ᠤᠨ BD ᠨᠢ ᠬᠠᠷᠢᠯ ᠴᠠᠨ ᠴᠡᠭ O ᠲᠤ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠂ ᠴᠡᠭ O ᠶᠢ ᠳᠠ ᠭᠠᠷᠢᠭ ᠤᠯᠤᠨ ᠰᠢᠯ ᠤᠭᠤᠨ EF ᠢ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ AD ,BC ᠲᠣᠰ ᠪᠦᠷᠢ ᠴᠡ ᠬ᠋᠊ E ,F ᠲᠤ ᠤᠭᠲᠤᠯ ᠤᠯᠴᠠᠨ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ OE ＝OF 图 16 E D C O B F A DA B C F O DA CB E ︵13︶ ︵12︶ 21. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ □AB CD ᠲᠤ᠂ ∠AB C ＝5∠ A᠂ ᠴᠡᠭ B ᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ BE ⊥ DC ᠪᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ AD ᠶᠢᠨ ᠦᠷᠭᠦᠯ ᠵᠢᠯᠡᠯ ᠱᠤᠭᠤᠮ ᠲᠠᠢ E ᠴᠡᠭ ᠲᠤ ᠤᠭᠲᠣᠯ ᠣᠯᠴᠠ ᠨ᠊ ᠠ᠃ O ᠨᠢ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠮᠥᠷᠲᠡᠭᠡ ᠨ᠊ DE ＝DA ＝4𝑐𝑚᠃ ᠤᠯᠬ ᠣ ᠨᠢ᠄︶ 1 ︵□AB CD ᠶᠢᠨ ᠬᠦᠷᠢ ᠶᠠᠨ ᠤ ᠤᠷᠲᠤ᠃︶ 2 ︵ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ BD EC ᠶᠢᠨ ᠬᠦᠷᠢᠶ ᠠ ᠦ ᠤᠷᠲᠤ ᠪᠤᠯᠤ ᠨ᠊ ᠲᠠ ᠯᠠᠪᠡᠢ᠃ ︵ᠳᠦ᠋ ᠩ᠊ ᠲᠤ √ ᠠ ᠭᠤᠯᠵᠤ ᠪᠤᠯᠤᠨ ᠠ︶ 22. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ □AB CD ᠶᠢᠨ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ AC ᠶᠢᠨ ᠡᠭᠴᠡ ᠲᠠᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴᠢ ᠨᠢ AD ᠪᠤᠯ ᠤᠨ BC ᠲᠠᠢ ᠲᠤᠰ ᠲᠤᠰ E ,F ᠴᠡᠭ ᠲᠤ ᠬᠭ᠍ᠲᠣ ᠯᠣᠯᠴᠠ ᠨ᠊ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ ᠳ᠋᠊ ᠥᠷᠪᠡ ᠯᠵᠢᠨ AF CE ᠨᠢ ᠷᠤᠮᠪ ᠪᠤᠯᠤᠨ ᠠ᠃ O E CD BA F E O DA CB 23. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠳᠡ ᠭᠰᠢ ᠲᠤᠷ ᠪᠡᠯᠵ ᠶᠢᠨ AB CD ᠲᠤ᠂ ᠴᠡᠭ P ᠨᠢ CD ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠮᠥᠷᠲᠡ ᠭᠡᠨ DF ⊥ AP᠂ BE ⊥ AP ᠪᠤᠯ ᠤᠨ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ AE ＝DF 24. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠳᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠲᠦ AB CD ᠲᠤ ᠴᠡᠭ P ᠨᠢ ᠲᠡᠭᠰᠢ ᠦᠨᠴ ᠦᠭᠲᠦ ᠳᠤᠲᠤᠷᠠ ᠬᠢ ᠨᠢ ᠭᠡᠨ ᠴᠡᠭ ᠮᠥᠷᠲᠡ ᠭᠡᠨ PA ＝PD ᠪᠤᠯ ᠤᠨ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ᠄ PB ＝PC 25. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠦ᠂ ᠱᠠᠳᠤᠯ ᠶᠢᠨ AB CD ᠲᠤ AB ∥ BC , AB = DC = AD , ∠ C= 60° , AE ⏊BD ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ ᠴᠡᠭ E ᠪᠤᠯ ᠤᠨ ᠠ᠂ ᠴᠡᠭ F ᠨᠢ CD ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ᠂ DG ᠨᠢ ᠱᠠᠳᠤᠯ ᠶᠢᠨ AB CD ᠶᠢᠨ ᠦᠨᠳᠦ ᠷ᠊ ᠪᠤ ᠯᠤᠨ ᠠ᠃︶ 1 ︵ᠪᠠᠳᠤ ᠯᠠᠭᠳᠠ ᠬᠤ ᠨᠢ : ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ A EFD ᠨᠢ ᠵᠢᠰᠢᠭ ᠦᠯᠵᠢᠨ ᠪᠤᠯ ᠤᠨ ᠠ᠃︶ 2 ︵ᠬᠡᠷᠪᠡ AE = 𝑥᠂ ᠳᠦ᠋ᠷᠪ ᠡᠯᠵᠢᠨ DEG F ᠶᠢᠨ ᠲᠠᠯᠠᠪ ᠢ ᠶᠢ 𝑦 ᠭᠡᠪᠡᠯ᠂ 𝑦 ᠶᠢᠨ ᠲᠤᠬᠠᠢ 𝑥 ᠦ ᠨ᠊ ᠹᠦ ᠩᠺᠽ ᠦᠨ ᠢᠯ ᠡᠷᠬᠡᠢᠯ ᠡᠯ ᠢ ᠬᠯ᠃