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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?

 Dec 27, 2021
 #1
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+1

The only triangles that work have legs 10 and 15, and 12 and 12, so the answer is 10*15/2 + 12*12/2 = 147.

 Dec 27, 2021
 #2
avatar+23186 
+2

If the legs have values  x  and  y     its area is  ½·x·y     and the sum of its legs is  x + y

 

The area is equal to three times the sum of its legs:     ½·x·y  =  3(x + y)

Solving:                                                                                xy  =  6(x + y)

                                                                                            xy  =  6x + 6y

                                                                                      xy - 6y  =  6x

                                                                                     y(x - 6)  =  6x

                                                                                              y  =  6x / (x - 6)

 

Substituting values for x to get integral values for y gives:  

     (7, 42)     (8, 24)     (9, 18)     (10, 15)     (12, 12)     (15, 10)     (18, 9)     (24, 8)     (42, 7)

 

Removing the duplicates:     (7, 42)     (8, 24)     (9, 18)     (10, 15)     (12, 12) 

 

Finding the areas of these triangles and summing their answers will give the final answer.

 Dec 27, 2021
 #3
avatar+11 
0

Thanks a lot to you both !!!

oTechz  Dec 27, 2021

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