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Let a be the length of a side and b be
the length of a diagonal in the regular pentagon
PQRST. Prove that
b/a - a/b=1
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What do you mean by "diagonal in the regular pentagon"?
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1.618034 - 0.618034 = 1 may help.
[sqrt(5) +/- 1] / 2 is golden ratio.
igloo myrtilles fourmis
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A diagonal of a polygon is a line drawn between any two non-adjacent points on it.
Pentagons actually have five diagonals, but they're all rotationally symmetric and so they are all the same length.
The only thing I can think of is to use the cosine rule to say that b² = 2a²[1 - cos(108)] and work from there. I'm not sure how easy that'd be though.
Why did the vector cross the road?
It wanted to be normal.
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The relation between the two lengths in a regular pentagon is
(which is easily established by sketching the pentagon and using elementary trigonometry). Hence
and if you want this to be 1,
will have to satisfy the equation , i.e.So John E. Franklin was spot on. This is what we need to prove.
Last edited by JaneFairfax (2008-03-17 19:37:19)
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I got it. Use the trig identity
Hence
is a root of the equationObviously
. Thus it must be a root of . Problem solved!Last edited by JaneFairfax (2008-03-17 20:18:59)
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Wow! So many trig formulas to know!!
Yes!! 5 times 36 is 180 and cosine of 180 is -1.
That's where the +1 term came from in the 5th
power equation. Then Jane factored it somehow,
and that checks out right by remultiplying I noted.
And then when you get the cosine of 36 solved,
just multiply by 2 to get the diagonal 1.618034,
providing a side is 1.
igloo myrtilles fourmis
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Then Jane factored it somehow
And the key lies in fact that I already knew (from the post I had made before) that one of the factors was going to be
.If I hadnt known that before, all I would have been to manage would be
and there would have been no way I would have known what to do with that quartic expression next. BUT knowing that
was a factor enabled me to divide it out by long division and factorize the quartic successfully.Offline
Very nice. Thanks, I see now.
In post #5, 2nd equation, toggle the plus sign, just a typo
igloo myrtilles fourmis
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