Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

regular pentagon

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

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: regular pentagon

What do you mean by "diagonal in the regular pentagon"?

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: regular pentagon

1.618034 - 0.618034 = 1 may help.
[sqrt(5) +/- 1] / 2 is golden ratio.

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: regular pentagon

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.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: regular pentagon

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)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: regular pentagon

I got it. Use the trig identity

Hence

is a root of the equation

Obviously

. Thus it must be a root of

. Problem solved!

Last edited by JaneFairfax (2008-03-17 20:18:59)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: regular pentagon

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.

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igloo myrtilles fourmis

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: regular pentagon

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 hadn’t 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.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: regular pentagon

Very nice. Thanks, I see now.
In post #5, 2nd equation, toggle the plus sign, just a typo

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igloo myrtilles fourmis