课例 17 勾股定理
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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. - PowerPoint PPT PresentationTRANSCRIPT
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课例 17 勾股定理
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A
B Ca
bc
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 .
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“ 弦图”
∵ c2 - 4× ab = (a - b)2 , ∴ c2 = a2 + b2 .
c
a - b
a b
1_2
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a
ab
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c
c
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a
ba
bc
c
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a
b
c
c a
b
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a
b
c
a
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c
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a
b
c
c
a
b
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2002 年世界数学家大会会标
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a2
b2
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c2
a2 + b2 = c2
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1955 年希腊发行的一枚纪念邮票
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c
a b
a - b
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aa a
a
a
c
a
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bb c
cc
c
c
b
a
a
a
a
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b
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1876 年美国总统加菲尔德的证法
cc
a
a b
b
( a+b) (a+b ) = c2+2× ab 1_2
1_2
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A
B C
D E
F
G
K
H
L
M
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 .
2
1
2
1
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A
C
BD
b2 = cx ① a2 = cy ②
①+② 得 a2+b2=c(x+y)=c2 , a2 + b2 = c2 .
c
ab
x y
b
xy
aA C
D
cb
bx
ca
ay
BC
D
△ ACD∽△ABC △ CBD∽△ABC
=>=>
=>=>
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根据切割线定理, AC2 = AD · AE .
∴ AC2 = AD(AB+BC)=(AB – BC)(AB+BC)
= AB2 – BC2.
∴AC2 + BC2 = AB2.
以 B 为圆心 BC 为半径作圆
D
E
AB 交⊙ B 于点 D ,延长 AB 交⊙ B 于点E .
B
AC在 Rt△ABC 中 ,∠ C = 90°.
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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?