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A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?

 Feb 12, 2022
 #1
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+1

Pythagorean triples are the right angled triangles with integer leg lengths:

 

 Feb 12, 2022
 #2
avatar+124594 
+1

I think there might be a slight mistake above

 

I think we want  A  = 3L

 

So

 

mn ( m^2 - n^2 )   =  [ 2m (m + n) ]

 

mn ( m + n) ( m - n)  =  6m ( m + n)              since ( m + n)  > 0    we have that

 

mn ( m - n)  =  6m

 

n ( m - n)  =  6

 

(m - n)  = 6 / n

 

Possibilities  (with a little help from WolframAlpha )

m    n                     2mn         m^2 - n^2        m^2 + n^2          Perimeter   Area

5     2                       20              21                   29                     70            210     Yes     

5    3                        30              16                   34                     80            240     Yes

7    1                        14              48                   50                    112           336     Yes   

7    6                        84              13                   85                    182           546     Yes

 

Added areas    =  1332

 

cool cool cool  

CPhill  Feb 12, 2022
edited by CPhill  Feb 12, 2022
edited by CPhill  Feb 12, 2022
 #3
avatar+251 
+5

Thanks for your help!

dolphinia  Feb 12, 2022
 #4
avatar+124594 
0

No prob  !!!

 

 

cool cool cool

CPhill  Feb 12, 2022

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