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Topic: Triangular Lawns (Read 1078 times) |
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ThudnBlunder
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Triangular Lawns
« on: Sep 18th, 2008, 2:33pm » |
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1) Dick's lawn is triangular, with each side being an exact number of metres long. Also, the number of square metres in its area is exactly twice the number of metres in its perimeter. The same is true for Harry's lawn. However, the longest side of Dick's lawn is longer than the longest side of Harry's lawn.
What are the dimensions of Harry's lawn?
2) On the other hand, while Tom's lawn is also triangular with each side being an integer number of metres long, the number of square metres in its area is a rational fraction, less than unity, of the number of metres in its perimeter. Even if Tom told you the fraction you still couldn't work out the area of his lawn, as there are two possible answers, one twice the other.
So what is the fraction?
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| « Last Edit: Sep 19th, 2008, 1:18pm by ThudnBlunder » |
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Grimbal
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Re: Triangular Lawns
« Reply #1 on: Sep 19th, 2008, 12:59pm » |
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1)
For Dick's lawn, I can only think of sides (3,4,5). Area is 6, perimeter is 12.
That leaves only a lawn with sides (0,0,0) to Harry.
Not very satisfied with the answer, though.
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SMQ
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Re: Triangular Lawns
« Reply #2 on: Sep 19th, 2008, 1:14pm » |
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on Sep 19th, 2008, 12:59pm, Grimbal wrote:| Not very satisfied with the answer, though. |
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The which is twice the what, now? 
--SMQ
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ThudnBlunder
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Re: Triangular Lawns
« Reply #3 on: Sep 19th, 2008, 2:00pm » |
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Sorry, I think I might have deleted a post of yours by mistake, Grimbal, instead of deleting the screwed-up one of mine below. 
(For example, it had my name in place of yours.) I must re-install SMQ's scripts pronto.
Anyway, I have reposted it below.
on Sep 18th, 2008, 2:33pm, Grimbal wrote:| « Last Edit: Today at 22:18 by ThudanBlunder » |
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on Sep 18th, 2008, 2:33pm, Grimbal wrote:| Are you trying to make a fool of myself? |
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Quote:| Nah - being the pedant that I am, I merely inserted a missing preposition. |
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| « Last Edit: Sep 20th, 2008, 2:39am by ThudnBlunder » |
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Grimbal
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Re: Triangular Lawns
« Reply #4 on: Sep 21st, 2008, 9:56am » |
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on Sep 18th, 2008, 2:33pm, ThudanBlunder wrote:1) Dick's lawn is triangular, with each side being an exact number of metres long. Also, the number of square metres in its area is exactly twice the number of metres in its perimeter. The same is true for Harry's lawn. However, the longest side of Dick's lawn is longer than the longest side of Harry's lawn.
What are the dimensions of Harry's lawn? |
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A small program I wrote shows 18 triangles with this property. So, there are many solutions.
But assuming the area (and therefore the perimeter) is the same for both lawns, the unique solution is 10, 35 and 39 meters (and 14, 30, 40 meters for Dick).
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ThudnBlunder
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Re: Triangular Lawns
« Reply #5 on: Sep 22nd, 2008, 4:50am » |
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on Sep 21st, 2008, 9:56am, Grimbal wrote:
But assuming the area (and therefore the perimeter) is the same for both lawns... |
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Oops... yes, that is the case. 
Those 18 cases can also be obtained by deriving a Diophantine equation of the form pqr = 16(p + q + r)
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Eigenray
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Re: Triangular Lawns
« Reply #6 on: Sep 23rd, 2008, 12:57am » |
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on Sep 22nd, 2008, 4:50am, ThudanBlunder wrote:| Those 18 cases can also be obtained by deriving a Diophantine equation of the form pqr = 16(p + q + r) |
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Nice... {p,q,r} = {s-a,s-b,s-c} where s=(a+b+c)/2 is the semiperimeter. If we had 6 < p <= q <= r, then
(p2-16)(p2-16) <= (pr-16)(pq-16) = p2qr - 16pq - 16pr + 162 = 16(p2+16)
is a contradiction. So p <= 6. But for any fixed p, we can easily find all solutions to (pr-16)(pq-16) = 16(p2+16).
In fact the same argument applies to Area = n*Perimeter, for any fixed n. Much easier than trying to work with the parameterization of Heronian triangles.
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| « Last Edit: Sep 23rd, 2008, 1:04am by Eigenray » |
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