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million dollar dream
New Member

Australia
1 Posts

 Posted - 09/14/2007 :  04:47:37 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

Subhotosh Khan

USA
9116 Posts

 Posted - 09/14/2007 :  09:10:31 quote:Originally posted by million dollar dreamHi, 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 solvingBy brute force - 6,5,5

sahsjing

USA
2399 Posts

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

sahsjing

USA
2399 Posts

 Posted - 09/14/2007 :  19:59:29 Now I agree with you. Edited by - sahsjing on 09/14/2007 20:00:00

someguy

143 Posts

 Posted - 09/16/2007 :  14:28:44 The initial poster requested a SCALENE triangle.I found Heron's formula for the area of a triangleto be quite useful.

someguy

143 Posts

 Posted - 09/16/2007 :  23:45:19 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?

sahsjing

USA
2399 Posts

 Posted - 09/17/2007 :  19:14:02 quote:Originally posted by someguyThe 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? YesYes. 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

Subhotosh Khan

USA
9116 Posts

 Posted - 09/18/2007 :  21:30:25 quote:Originally posted by sahsjingquote:Originally posted by someguyThe 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? YesYes. 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.

Subhotosh Khan

USA
9116 Posts

 Posted - 09/19/2007 :  12:36:11 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.

Subhotosh Khan

USA
9116 Posts

 Posted - 09/20/2007 :  07:55:26 quote:Originally posted by someguyThe 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).

sahsjing

USA
2399 Posts

 Posted - 09/20/2007 :  17:31:06 quote:Originally posted by Subhotosh Khanquote: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?

someguy

143 Posts

 Posted - 09/20/2007 :  18:16:45 quote:Originally posted by sahsjingSince the base is an integer, and the area is also an integer, then the height must be an integer too.quote:Originally posted by sahsjingAn 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 triangle2) It does not contain a right angle3) The length of each side is an integer4) The area of the triangle is an integer5) 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?

someguy

143 Posts

 Posted - 10/02/2007 :  16:49:15 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 setwhich 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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