Maths (Hard)

👨🏻‍🏫 學術台數學

死亡天才 Death Genius

2009-03-06 2:12 am

Please solve the following question

圖片參考：http://photos-a.ak.fbcdn.net/hphotos-ak-snc1/hs034.snc1/2595_1115424127617_1287361866_341288_6336532_n.jpg

更新1:

I need a detail proof.

更新2:

I need a faster and simplier proof. (NOT COS FORMULA)

回答 (4)

用戶2053

2009-03-11 11:01 am

✔ 最佳答案

圖片參考：http://euler.tn.edu.tw/images/think045.gif

In the figure, AB = BC,ㄥABC = 90, ㄥ2 = b, ㄥ3 = c
soㄥ2+ㄥ3 = b + c = 45, it is very easy to know that a = 45
So a + b + c = 45 + 45 = 90

參考: http://euler.tn.edu.tw/think7.htm

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用戶7639

2009-03-07 7:48 am

the following question
proof:

圖片參考：http://f.imagehost.org/0687/ScreenHunter_01_Mar_06_23_44.gif

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Gabriella Montez

2009-03-07 2:09 am

It is clear that a = 45* but the proof can still be completed without finding b and c. By considering tan b = 1/2 and tan c = 1/3,
tan ( b + c ) = ( tan b + tan c ) / ( 1 - tan b tan c )
= ( 1/2 + 1/3 ) / ( 1 - 1/6 )
= 1
As b + c < 90*, b + c = 45*
Lastly, a + b + c = 45* + 45* = 90*

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用戶106181

2009-03-06 2:34 am

let the side length of each square is l

a=45

l^2=(2l)^2+[l^2+(2l)^2]-2(2l){[l^2+(2l)^2]^(1/2)}cos b
1=(2)^2+[1+(2)^2]-2(2)[5^(1/2)]cos b
1=4+5-4[5^(1/2)]cos b
b=26.56505118

l^2=(3l)^2+[l^2+(3l)^2]-2(3l){[l^2+(3l)^2]^(1/2)}cos c
1=(3)^2+[1+(3)^2]-2(3)[10^(1/2)]cos c
1=9+10-2(3)[10^(1/2)]cos c
c=18.43494882

So, a+b+c=90

參考: 自己

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