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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 twice 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?

 
 Oct 12, 2021
 #1
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The only triangle that works is the isosceles triangle with legs 8 and 8, so the sum is 1/2*8*8 = 32.

 
 Oct 12, 2021
 #2
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\(2(x+y)=\frac{1}{2}xy\\ 4x+4y-xy=0\\ 4x+y(4-x)=0\\ y=\frac{4x}{x-4} \)

 

\(\therefore x>4, \;\;y>4 \qquad \text{since x and y must both be positive}\)

 

the only positive integer x values that give positive integer y values are    5,6,8,12,and 20

 

But i just realized that is not what you asked. 

(5,20)    area = 0.5*5*20 = 50

(6,12)    area = 0.5*6*12 = 36

(8,8)      area = 0.5*8*8  = 32

(12,6)    area  is 36

(20,5)    area is 50

 

so sum of all possible areas is   50+36+32 = 118 units squared.

 

 

 

 

 
 Oct 12, 2021

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