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million dollar dream
New Member

Australia
1 Posts

Posted - 09/14/2007 :  04:47:37  Show Profile
Hi, my child gave me this problem I found it tricky to complete, have a go, post your answers below.

A triangle has the following properties:
It is scalene;
It does not contain a right angle;
It has integer length sides;
Its area is an integer.
Find the triangle with these properties which has the least perimeter.

Happy solving
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Subhotosh Khan
Advanced Member

USA
9117 Posts

Posted - 09/14/2007 :  09:10:31  Show Profile
quote:
Originally posted by million dollar dream

Hi, my child gave me this problem I found it tricky to complete, have a go, post your answers below.

A triangle has the following properties:
It is scalene;
It does not contain a right angle;
It has integer length sides;
Its area is an integer.
Find the triangle with these properties which has the least perimeter.

Happy solving



By brute force - 6,5,5
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sahsjing
Advanced Member

USA
2399 Posts

Posted - 09/14/2007 :  15:52:15  Show Profile
Combine two 3-4-5 right triangles to get the least perimeter.
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sahsjing
Advanced Member

USA
2399 Posts

Posted - 09/14/2007 :  19:59:29  Show Profile
Now I agree with you.

Edited by - sahsjing on 09/14/2007 20:00:00
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someguy
Advanced Member

Canada
143 Posts

Posted - 09/16/2007 :  14:28:44  Show Profile
The initial poster requested a SCALENE triangle.

I found Heron's formula for the area of a triangle
to be quite useful.
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someguy
Advanced Member

Canada
143 Posts

Posted - 09/16/2007 :  23:45:19  Show Profile
The side lengths of the required triangle are 4, 13, and 15.

If you are trying to piece together two right angle triangles,
are you assuming the common side must have integral length?
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sahsjing
Advanced Member

USA
2399 Posts

Posted - 09/17/2007 :  19:14:02  Show Profile
quote:
Originally posted by someguy

The side lengths of the required triangle are 4, 13, and 15.

If you are trying to piece together two right angle triangles,
are you assuming the common side must have integral length? Yes


Yes. Since the base is an integer, and the area is also an integer, then the height must be an integer too.

Edited by - sahsjing on 09/17/2007 19:16:01
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Subhotosh Khan
Advanced Member

USA
9117 Posts

Posted - 09/18/2007 :  21:30:25  Show Profile
quote:
Originally posted by sahsjing

quote:
Originally posted by someguy

The side lengths of the required triangle are 4, 13, and 15.

If you are trying to piece together two right angle triangles,
are you assuming the common side must have integral length? Yes


Yes. Since the base is an integer, and the area is also an integer, then the height must be an integer too.



If base is 4, height need not be an integer.
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Subhotosh Khan
Advanced Member

USA
9117 Posts

Posted - 09/19/2007 :  12:36:11  Show Profile
quote:
Originally posted by David

"If base is 4, height need not be an integer."

It does if the area is to be an integer, as sahsjing pointed out. In fact, in the 4, 13, 15 SCALENE triangle offered by Someguy, the area is 24, and the height from that base would be 12.



A = 1/2 * b * h = 1/2 * 4 * h = 2 * h

'h' could be 3.5 (non-integer) and area could be 7 (integer) - among other possibilities.
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Subhotosh Khan
Advanced Member

USA
9117 Posts

Posted - 09/20/2007 :  07:55:26  Show Profile
quote:
Originally posted by someguy

The side lengths of the required triangle are 4, 13, and 15.

If you are trying to piece together two right angle triangles,
are you assuming the common side must have integral length?



My problem was that I was adding two right-triangles together - did not even think about subtracting (which gives the minimum solution in this case -- 15-9-12 - 13-5-12 --> 15-4-13).
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sahsjing
Advanced Member

USA
2399 Posts

Posted - 09/20/2007 :  17:31:06  Show Profile
quote:
Originally posted by Subhotosh Khan

quote:
Originally posted by David

"If base is 4, height need not be an integer."

It does if the area is to be an integer, as sahsjing pointed out. In fact, in the 4, 13, 15 SCALENE triangle offered by Someguy, the area is 24, and the height from that base would be 12.



A = 1/2 * b * h = 1/2 * 4 * h = 2 * h

'h' could be 3.5 (non-integer) and area could be 7 (integer) - among other possibilities.


An counter example must satisfy all the required conditions. Can you find one?
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someguy
Advanced Member

Canada
143 Posts

Posted - 09/20/2007 :  18:16:45  Show Profile
quote:
Originally posted by sahsjing


Since the base is an integer, and the area is also an integer, then the height must be an integer too.



quote:
Originally posted by sahsjing


An counter example must satisfy all the required conditions. Can you find one?



I think you are asking for an example of a triangle with the following properties.

1) It is a scalene triangle
2) It does not contain a right angle
3) The length of each side is an integer
4) The area of the triangle is an integer
5) The height of the triangle is not an integer (regardless of which side is used as the base)

Why do you think such an object can not exist?
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someguy
Advanced Member

Canada
143 Posts

Posted - 10/02/2007 :  16:49:15  Show Profile
I didn't do anything fancy to find the required triangle.
I just used Heron's formula and a computer to find a triangle with the required properties(minus the minimum one). This gave a bound on the perimeter, and so reduces the search space to a finite set
which can easily be searched to find the minimum.

The problem with trying to find a minimum by a construction method involving right triangles with sides of integer length is that you are not considering the whole space of allowable triangles.
There ARE scalene triangles for which the area and sides lengths are all integers, but the height is not an integer (no matter which side is used as a base). These triangles can not be formed by the construction methods used above. Unless you know that the required triangle with minimum perimeter is not in the set of such triangles, you can't just ignore them.



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