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 Post subject: Lil geometry fun...
PostPosted: Sun, 24 Jul 2011 12:04:42 UTC 
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Code:
 
                               D


                      ?
     C
              E                        60
                                       
  28

B                      100                          A

Right triangles ABC and ABC shaue hypotenuse AB.
AC and BD cross at E.
AB = 100, BC = 28 and AD = 60.
Your mission: calculate DE

...hey, it posted FINE! :shock:

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 Post subject: Re: Lil geometry fun...
PostPosted: Sun, 24 Jul 2011 15:53:55 UTC 
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Posts: 6844
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
Denis wrote:
Code:
 
                               D


                      ?
     C
              E                        60
                                       
  28

B                      100                          A

Right triangles ABC and ABC shaue hypotenuse AB.
AC and BD cross at E.
AB = 100, BC = 28 and AD = 60.
Your mission: calculate DE

...hey, it posted FINE! :shock:


Huh? Triangles ABC and ABC? They are the same triangle!

(and what is the meaning of "shaue"?)

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: Lil geometry fun...
PostPosted: Sun, 24 Jul 2011 16:09:42 UTC 
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Geezzzzz....can't find the "EDIT" button....Helmut???

Anyhow, sorry for the 2 typos; corrected:
Right triangles ABC and ABD
"shaue" should be SHARE

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 Post subject: Re: Lil geometry fun...
PostPosted: Sun, 24 Jul 2011 16:31:22 UTC 
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
Denis wrote:
Geezzzzz....can't find the "EDIT" button....Helmut???

Anyhow, sorry for the 2 typos; corrected:
Right triangles ABC and ABD
"shaue" should be SHARE


Edit button is at the bottom of post, to the right (in the row with profile, PM, etc.).

So I assume right-angles are at C and D respectively?

Spoiler:
Pythagoras gives AC=96, BD=80. So we know \tan\angle DAB=\dfrac{4}{3},\quad \tan\angle EAB=\tan\angle CAB=\dfrac{7}{24}, hence \tan\angle DAE=\tan(\angle DAB-\angle EAB)=\dfrac{\frac{4}{3}-\frac{7}{24}}{1+\frac{4}{3}\cdot\frac{7}{24}}=\dfrac{3}{4}, so ED=45.

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: Lil geometry fun...
PostPosted: Sun, 24 Jul 2011 23:58:12 UTC 
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Correct!

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 Post subject: Re: Lil geometry fun...
PostPosted: Mon, 25 Jul 2011 19:44:24 UTC 
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Right triangles BCE and ADE are similar, so
DE/CE = AE/BE = AD/BC = 15/7

BE + DE = 80
CE + AE = 96

Using the simlar triangles OR Pythagoras (with some algebra):
Spoiler:
DE = 45


Turns out BCE & ADE are 3-4-5 ratio right triangles


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 Post subject: Re: Lil geometry fun...
PostPosted: Tue, 26 Jul 2011 00:53:10 UTC 
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Agree Alstat.

If we let a = BC, b = AC, d = AD, e = BD,
then we have as General Case:

DE = d(de - ab) / (d^2 - a^2)

And if we want CE:
CE = a*DE / d

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