+216 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?
The only triangle that works is the isosceles triangle with legs 8 and 8, so the sum is 1/2*8*8 = 32.
+118677 \(2(x+y)=\frac{1}{2}xy\\ 4x+4y-xy=0\\ 4x+y(4-x)=0\\ y=\frac{4x}{x-4} \)
\(\therefore x>4, \;\;y>4 \qquad \text{since x and y must both be positive}\)
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.
