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        riddles >> medium >> Minimum Triangle Area
        
(Message started by: THUDandBLUNDER on Jan 12^th, 2004, 3:24pm)

Title: Minimum Triangle Area
Post by THUDandBLUNDER on Jan 12^th, 2004, 3:24pm

I came across the following puzzle recently.
It looks difficult to solve without some programming.

A triangle with sides a, b, and c has an area A equal to n times its perimeter,
where a, b, c, n, and A are all positive integers.

Also, a = n² + n^1/5

What is the minimum possible value for A?

Title: Re: Minimum Triangle Area
Post by Icarus on Jan 12^th, 2004, 4:01pm

Is x also restricted to integers?

Title: Re: Minimum Triangle Area
Post by THUDandBLUNDER on Jan 12^th, 2004, 4:24pm

Quote:

Is x also restricted to integers?

Iff it is equal to n. ::)

(Thanks, I will correct that now.)

Title: Re: Minimum Triangle Area
Post by Sameer on Jan 30^th, 2004, 4:00pm

on 01/12/04 at 16:01:24, Icarus wrote:

Is x also restricted to integers?

I don't see any x in the questions? :-/ ???

Title: Re: Minimum Triangle Area
Post by THUDandBLUNDER on Jan 30^th, 2004, 4:53pm

Quote:

I don't see any x in the questions?

So my correction was successful then?
Thank you for your positive feedback, Sameer.
;D
(Before Icarus queried it, I think I originally had Also, a = x² + x^1/5)

Title: Re: Minimum Triangle Area
Post by Sameer on Feb 2^nd, 2004, 12:07pm

;D Hee... i spent hours thinking there is something wrong with the problem and then I realised I was considering 'n' as the perimeter all the time keke.. Ok back to board for solving this again!!!

Title: Re: Minimum Triangle Area
Post by Icarus on Feb 2^nd, 2004, 3:50pm

Perhaps I should remove the posts that deal only with correcting the original problem statement, but then again, T&B has lost too many posts lately without me purposely deleting them! :D

Title: Re: Minimum Triangle Area
Post by Sameer on Feb 25^th, 2004, 9:02am

Ok so I just approached it with simple trigonometric formulas and came up with this equation and am at stalemate

2nA + (A/2n)*( (A/2n) - (n²+n^1/5)² = bc( (A/2n) - (n²+n^1/5))

I don't know where to go from here. Anyone?

Also there can be other method ? ???

Title: Re: Minimum Triangle Area
Post by Barukh on Feb 27^th, 2004, 7:22am

on 02/25/04 at 09:02:05, Sameer wrote:

Also there can be other method ? ???

You may try to use the parametric equations (3)-(7) at Mathworld's page on Heronian's triangles (http://mathworld.wolfram.com/HeronianTriangle.html).

Title: Re: Minimum Triangle Area
Post by aero_guy on Mar 8^th, 2004, 3:39pm

Sameer, also do not forget the limitations on the relative size of a, b, and c. Neither of them can be larger than the sum of the other two.

Title: Re: Minimum Triangle Area
Post by Sameer on Mar 9^th, 2004, 7:52am

Hmm I haven't looked at this problem in a while. I shall take a look at it this weekend with Barukh's and aero_guy's hints!! :D