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课例 17 勾股定理

课例 17 勾股定理. A. 如果直角三角形的两直角边长分别为 a ， b ，斜边长为 c ， 那么 a 2 + b 2 ＝ c 2. c. b. C. B. a. “ 弦图” ∵ c 2 － 4× ab ＝ ( a － b ) 2 ， ∴ c 2 ＝ a 2 + b 2. a － b. b. a. c. _. 1. 2. c. c. a. b. b. a. b. a. c. c. a. b. c. a. b. c. a. b. c. a. b. c.

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课例 17 勾股定理

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  1. 课例17 勾股定理

  2. A 如果直角三角形的两直角边长分别为a，b，斜边长为c， 那么 a2 + b2＝ c2 . c b C B a

  3. “弦图” ∵ c2－4× ab＝(a－b)2 ， ∴ c2＝a2 + b2 . a－b b a c _ 1 2

  4. c c a b b a

  5. b a c c a b

  6. c a b c a b

  7. c a b c a b

  8. c a c b b a

  9. 2002 年世界数学家大会会标

  10. a2 b2

  11.  a2 + b2 = c2 c2

  12. 1955年希腊发行的一枚纪念邮票

  13. a－b b a c

  14. b a b a c a a a a c a b b c c b b b c b c a a a b b

  15. _ _ 1 1 2 2 1876 年美国总统加菲尔德的证法 c b c a a b （a+b) (a+b）= c2+2× ab

  16. Ｍ Ｌ G F H A S△ABD = S矩形BDLM， S△FBC = S正方形FBAG， ∵S△ABD = S△FBC， ∴S矩形BDLM = S正方形FBAG． 同理， S矩形CELM = S正方形 ACKH． ∴ S正方形 ABFG + S正方形 ACKH = S正方形 BDEC ． K B C D E

  17. D D x y C B a b C A C a b B A y x c D △ ACD∽△ABC △ CBD∽△ABC => => => b2 = cx ①a2 = cy② ①+②得 a2+b2=c(x+y)=c2 ， =>  a2 + b2 = c2.

  18. B E D A C 在Rt△ABC 中 ，∠C = 90°. 以 B 为圆心 BC 为半径作圆 AB交⊙B于点D，延长AB交⊙B于点E . 根据切割线定理，AC2 = AD · AE . ∴ AC2＝AD(AB+BC)=(AB– BC)(AB+BC) = AB2– BC2. ∴AC2 + BC2 = AB2.

  19. 如图，一根旗杆在离地面 9 m 处断裂，旗杆顶部落在离旗杆底部 12 m 处，旗杆折断之前有多高？

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